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MATH 0180 Intermediate Calculus 1 MATH 0520 Linear Algebra 2 1 APMA 0350 Applied Ordinary Differential Equations 2 & APMA 0360 and Applied Partial Differential Equations Chair I 3 4 Bjorn Sandstede Select one course on programming from the following: 1 APMA 0090 Introduction to Mathematical Modeling Associate Chair APMA 0160 Introduction to Computing Sciences Kavita Ramanan CSCI 0040 Introduction to Scientific Computing and The Division of Applied Mathematics at Brown University is one of Problem Solving the most prominent departments at Brown, and is also one of the CSCI 0111 Computing Foundations: Data oldest and strongest of its type in the country. The Division of Applied CSCI 0150 Introduction to Object-Oriented Mathematics is a world renowned center of research activity in a Programming and wide spectrum of traditional and modern mathematics. It explores the connections between mathematics and its applications at both CSCI 0170 Computer Science: An Integrated the research and educational levels. The principal areas of research Introduction activities are ordinary, functional, and partial differential equations: Five additional courses, of which four should be chosen from 5 stochastic control theory; applied , and stochastic the 1000-level courses taught by the Division of Applied systems theory; neuroscience and computational molecular ; Mathematics. APMA 1910 cannot be used as an elective. numerical analysis and scientific computation; and the mechanics of Total Credits 10 solids, materials science and fluids. The effort in virtually all research 1 ranges from applied and algorithmic problems to the study of fundamental Substitution of alternate courses for the specific requirements is mathematical questions. The Division emphasizes applied mathematics subject to approval by the division. 2 as a unifying theme. To facilitate cooperation among faculty and Concentrators are urged to consider MATH 0540 as an alternative students, some research programs are partly organized around to MATH 0520. 3 interdepartmental research centers. These centers facilitate funding and APMA 0330, APMA 0340 will sometimes be accepted as substitutes cooperative research in order to maintain the highest level of research for APMA 0350, APMA 0360. APMA 1910 cannot be used as an and education in the Division. It is this breadth and the discovery from elective. mutual collaboration which marks the great strength and uniqueness of the 4 Concentrators are urged to complete their introductory programming Division of Applied Mathematics at Brown. course before the end of their sophomore year. For additional information, please visit the department's website: https:// www.brown.edu/academics/applied-mathematics/ Standard program for the Sc.B. degree. Program Applied Mathematics Concentration Eighteen approved semester courses in mathematics, applied Requirements mathematics, , the natural or social sciences. These classes must include: 1 The concentration in Applied Mathematics allows students to investigate the mathematics of problems arising in the physical, life and social MATH 0090 Introductory Calculus, Part I 2 sciences as well as in engineering. The basic mathematical skills of & MATH 0100 and Introductory Calculus, Part II Applied Mathematics come from a variety of sources, which depend on MATH 0180 Intermediate Calculus 1 the problems of interest: the theory of ordinary and partial differential MATH 0520 Linear Algebra 2 1 equations, matrix theory, statistical sciences, probability and decision APMA 0350 Applied Ordinary Differential Equations 2 theory, risk and insurance analysis, among others. Applied Mathematics & APMA 0360 and Applied Partial Differential Equations appeals to people with a variety of different interests, ranging from those I 3 with a desire to obtain a good quantitative background for use in some future career, to those who are interested in the basic techniques and Select one senior seminar from the APMA 1930 or APMA 1940 1 series, or an approved equivalent. approaches in themselves. The standard concentration leads to either 4 the A.B. or Sc.B. degree. Students may also choose to pursue a joint Select one course on programming from the following: 1 program with biology, computer science or . The undergraduate APMA 0090 Introduction to Mathematical Modeling concentration guide is available here (http://www.brown.edu/academics/ APMA 0160 Introduction to Computing Sciences applied-mathematics/undergraduate/). CSCI 0040 Introduction to Scientific Computing and Both the A.B. and Sc.B. concentrations in Applied Mathematics require Problem Solving certain basic courses to be taken, but beyond this there is a great deal CSCI 0111 Computing Foundations: Data of flexibility as to which areas of application are pursued. Students are encouraged to take courses in applied mathematics, mathematics and one CSCI 0150 Introduction to Object-Oriented or more of the application areas in the natural sciences, social sciences Programming and Computer Science or engineering. Whichever areas are chosen should be studied in some CSCI 0170 Computer Science: An Integrated depth. Introduction Ten additional courses, of which six should be chosed from 10 Standard program for the A.B. degree. the 1000-level or higher level courses taught by the Division Prerequisites of Applied Mathematics. APMA 1910 cannot be used as an elective. MATH 0090 Introductory Calculus, Part I & MATH 0100 and Introductory Calculus, Part II Total Credits 18 Or their equivalent 1 Substitution of alternate courses for the specific requirements is Program subject to approval by the division. Ten additional semester courses approved by the Division of 2 1 Concentrators are urged to consider MATH 0540 as an alternative to Applied Mathematics. These classes must include: MATH 0520.

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3 APMA 0330, APMA 0340 will sometimes be accepted as substitutes to evolutionary biology has grown very rapidly in with the availability for APMA 0350, APMA 0360. of vast amounts of genomic sequence data. Required coursework in 4 Concentrators are urged to complete their introductory programming this program aims at ensuring expertise in mathematical and statistical course before the end of their sophomore year. sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior Professional Tracks capstone experience that pairs student and faculty in creative research collaborations. The requirements for the professional tracks include all those of each of the standard tracks, as well as the following: Standard program for the Sc.B. degree Students must complete full-time professional experiences doing work that Required coursework in this program aims at ensuring expertise in is related to their concentration programs, totaling 2-6 months, whereby mathematical and statistical sciences, and their application in biology. The each internship must be at least one month in duration in cases where students will focus in particular areas of biology. The program culminates students choose to do more than one internship experience. Such work in a senior capstone experience that pairs student and faculty in creative is normally done at a company, but may also be at a university under the research collaborations. Applied Math – Biology concentrators are supervision of a faculty member. Internships that take place between the prepared for careers in , public health, industry and academic end of the fall and the start of the spring semesters cannot be used to research. fulfill this requirement. On completion of each professional experience, the student must write and upload to ASK a reflective essay about the Required Courses: experience, to be approved by the student's concentration advisor. Students are required to take all of the following courses. On completion of each professional experience, the student must MATH 0090 Introductory Calculus, Part I 1 write and upload to ASK a reflective essay about the experience, to be MATH 0100 Introductory Calculus, Part II 1 approved by the student's concentration advisor: or MATH 0170 Advanced Placement Calculus • Which courses were put to use in your summer's work? Which topics, MATH 0180 Intermediate Calculus (or equivalent 1 in particular, were important? placement) • In retrospect, which courses should you have taken before embarking MATH 0520 Linear Algebra 1 on your summer experience? What are the topics from these courses or MATH 0540 Honors Linear Algebra that would have helped you over the summer if you had been more 1 familiar with them? CHEM 0330 Equilibrium, Rate, and Structure 1 • Are there topics you should have been familiar with in preparation PHYS 0030 Basic A 1 for your summer experience, but are not taught at Brown? What are or PHYS 0050 Foundations of Mechanics these topics? Select one of the following sequences: 2 • What did you learn from the experience that probably could not have APMA 0350 Applied Ordinary Differential Equations been picked up from course work? & APMA 0360 and Applied Partial Differential Equations I • Is the sort of work you did over the summer something you would like APMA 0330 Methods of Applied Mathematics I to continue doing once you graduate? Explain. & APMA 0340 and Methods of Applied Mathematics II • Would you recommend your summer experience to other Brown APMA 1650 Statistical Inference I 1 students? Explain. or APMA 1655 Honors Statistical Inference I APMA 1070 Quantitative Models of Biological Systems 1 Honors APMA 1080 Inference in Genomics and Molecular 1 The concentrations in Applied Math require that honors students Biology demonstrate excellence in grades for courses in the concentration. or NEUR 2110 Statistical Neuroscience Students must have earned grades of A or S-with-distinction in at least BIOL 0200 The Foundation of Living Systems (or 1 70% of the courses used for concentration credit, excluding calculus equivalent) and linear algebra, or be in the upper 20% of the student's cohort (as measured by the fraction of grades of A or S-with-distinction among Additional Courses courses used for concentration credit, excluding calculus and linear One additional course in Applied Math or Biology 1 algebra). Since S with distinctions do not appear on the internal academic We strongly recommend that Applied Mathematics-Biology record or the official transcript, the department will consult directly with concentrators take one of the programming courses on or before the Registrar’s Office to confirm a student’s grades in concentration their first semester as a concentrator. Those who do can use it to courses. Additional guidelines and requirements for honors are published satisfy this requirement: on the department website (https://www.brown.edu/academics/applied- APMA 0160 Introduction to Computing Sciences mathematics/undergraduate-program/honors/). CSCI 0040 Introduction to Scientific Computing and Problem Solving Applied Mathematics-Biology CSCI 0111 Computing Foundations: Data Concentration Requirements CSCI 0150 Introduction to Object-Oriented Programming and Computer Science The Applied Math - Biology concentration recognizes that mathematics is essential to address many modern biological problems in the post CSCI 0170 Computer Science: An Integrated genomic era. Specifically, high throughput technologies have rendered Introduction vast new biological data sets that require novel analytical skills for the CSCI 0190 Accelerated Introduction to Computer most basic analyses. These technologies are spawning a new "data- Science driven" paradigm in the biological sciences and the fields of bioinformatics CLPS 0950 Introduction to programming and systems biology. The foundations of these new fields are inherently One research-related course in Applied Math or Biology. For 1 mathematical, with a focus on probability, statistical inference, and example: systems dynamics. These mathematical methods apply very broadly in many biological fields including some like population growth, spread A senior seminar course from the APMA 193X, 194X series of disease, that predate the genomics revolution. Nevertheless, the A directed research/independent study course from the application of these methods in areas of biology from molecular genetics APMA 1970, 1971, or BIOL 1950, 1960, or NEUR 1970 series

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A directed research/independent study course in a related BIOL 1190 Synaptic Transmission and Plasticity discipline (i.e. STEM disciplines, ENVS, PHP, etc.) if the Total Credits 18 project is relevant to the student's learning goals in the 1 concentration and with approval from the concentration Students whose independent study is expected to be in an advisor. experimental field are strongly encouraged to take APMA 1660, which A pre-approved course that satisfies the Biology AB capstone covers experimental design and the analysis of variance (ANOVA), a requirement: BIOL 1100, 1250, 1515, 1555, 1565, 1575, 1600, method commonly used in the analysis of experimental data. 1820, 1970. Professional Tracks A relevant CURE (Course-based Undergraduate Research Experience) course: BIOL 0190R, 0190S, 0285, 0440, 0600, The requirements for the professional tracks include all those of each of 0940G, 1515, 1555; NEUR 1630, CLPS 1195. the standard tracks, as well as the following: A non-research course related to the concentration along Students must complete full-time professional experiences doing work that with a research experience equivalent in scope and scale is related to their concentration programs, totaling 2-6 months, whereby to work the student would pursue in an Applied Math or each internship must be at least one month in duration in cases where Biology independent study course. Examples include UTRAs, students choose to do more than one internship experience. Such work LINK awards, research programs at other institutions, etc. is normally done at a company, but may also be at a university under the This requires approval from the concentration advisor and supervision of a faculty member. Internships that take place between the appropriate documentation. end of the fall and the start of the spring semesters cannot be used to Other equivalent opportunities not listed - with approval from fulfill this requirement. On completion of each professional experience, the concentration advisor. the student must write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor. Four classes in the biological sciences agreed upon by the 4 student and advisor. These four courses should form a cohesive On completion of each professional experience, the student must grouping in a specific area of emphasis, at least two of which write and upload to ASK a reflective essay about the experience, to be should be at the 1000-level. Some example groupings are below: approved by the student's concentration advisor: Areas of Emphasis and Suggested Courses: • Which courses were put to use in your summer's work? Which topics, Some areas of possible emphasis for focusing of elective in particular, were important? courses are listed below. Given the large number of course • In retrospect, which courses should you have taken before embarking offerings in the biosciences and neuroscience, students are on your summer experience? What are the topics from these courses free to explore classes in these areas that are not listed below. that would have helped you over the summer if you had been more However, all classes must be approved by the concentration familiar with them? advisor. APMA 1910 cannot be used as an elective. • Are there topics you should have been familiar with in preparation Biochemistry for your summer experience, but are not taught at Brown? What are BIOL 0280 Biochemistry these topics? BIOL 1270 Advanced Biochemistry • What did you learn from the experience that probably could not have CHEM 0350/0360 Organic been picked up from course work? CHEM 1230 Chemical Biology • Is the sort of work you did over the summer something you would like Biotechnology and Physiology to continue doing once you graduate? Explain. BIOL 0800 Principles of Physiology • Would you recommend your summer experience to other Brown BIOL 1100 Cell Physiology and Biophysics students? Explain. and/or appropriate bioengineering courses, such as: Honors BIOL 1090 Polymer Science for Biomaterials Requirements and Process: Honors in the Applied Math-Biology BIOL 1120 Biomaterials concentration is based primarily upon an in-depth, original research project BIOL 1140 Tissue Engineering carried out under the guidance of a Brown (and usually Applied Math or BIOL 1150 Stem Cell Engineering BioMed) affiliated faculty advisor. Projects must be conducted for no less BIOL 1210 Synthetic Biological Systems than two full semesters, and students must register for two semesters of credit for the project via APMA 1970 or BIOL 1950/BIOL 1960 or Ecology, , and Genetics similar independent study courses. One of these courses can be used BIOL 0410 Invertebrate Zoology to fulfill the research-related course requirement, but the other cannot & BIOL 0480 and Evolutionary Biology be used elsewhere in the concentration. The project culminates in BIOL 0420 Principles of Ecology the writing of a thesis which is reviewed by the thesis advisor and a & BIOL 0430 and The Evolution of Plant Diversity second reader. It is essential that the student have one advisor from the BIOL 0470 Genetics biological sciences and one in Applied Mathematics. The thesis work BIOL 1420 Experimental Design in Ecology must be presented in the form of an oral presentation (arranged with the primary thesis advisor) or posted at the annual Undergraduate Research BIOL 1430 Foundations of Population Genetics Day in either Applied Mathematics or Biology. For information on BIOL 1465 Human Population Genomics registering for BIOL 1950/BIOL 1960, please see https://www.brown.edu/ BIOL 1540 Molecular Genetics academics/biology/undergraduate-education/undergraduate-research Neuroscience (https://www.brown.edu/academics/biology/undergraduate-education/ undergraduate-research/) APMA 0410 Mathematical Methods in the Brain Sciences The concentrations in Applied Math (including joint concentrations) require Neurosciences courses: See https://www.brown.edu/ that honors students demonstrate excellence in grades for courses in academics/neuroscience/undergraduate/neuroscience- the concentration. Students must have earned grades of A or S-with- concentration-requirements distinction in at least 70% of the courses used for concentration credit, excluding calculus and linear algebra, or be in the upper 20% of the BIOL 1100 Cell Physiology and Biophysics student's cohort (as measured by the fraction of grades of A or S-with- BIOL 1110 Topics in Signal Transduction distinction among courses used for concentration credit, excluding calculus and linear algebra). Since S with distinctions do not appear on

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the internal academic record or the official transcript, the department will CSCI 0190 Accelerated Introduction to Computer consult directly with the Registrar’s Office to confirm a student’s grades Science (and an additional CS course not in concentration courses. Additional guidelines and requirements for otherwise used to satisfy a concentration honors are published on the department website (https://www.brown.edu/ requirement; (this course may be CSCI academics/applied-mathematics/undergraduate-program/honors/) 0180, an intermediate-level CS course, or The deadline for applying to graduate with honors in the concentration a 1000-level course) ) are the same as those of the biology concentrations. However, students Series D 1 in the joint concentration must inform the undergraduate chair in Applied CSCI 0111 Computing Foundations: Data Mathematics of their intention to apply for honors by these dates. & CSCI 0112 and Computing Foundations: Program & CSCI 0180 Organization Applied Mathematics-Computer Science and Computer Science: An Integrated Introduction Concentration Requirements Select three of the following intermediate-level courses, one 3 The Sc.B. concentration in Applied Math-Computer Science provides a of which must be math-oriented and one systems-oriented. foundation of basic concepts and methodology of mathematical analysis The intermediate courses must cover the requirements of the and computation and prepares students for advanced work in computer pathway chosen under additional requirements for CS. science, applied mathematics, and scientific computation. Concentrators CSCI 0220 Introduction to Discrete Structures and must complete courses in mathematics, applied math, computer science, Probability (math) and an approved English writing course. While the concentration in CSCI 0320 Introduction to Software Engineering Applied Math-Computer Science allows students to develop the use of (systems) quantitative methods in thinking about and solving problems, knowledge that is valuable in all walks of life, students who have completed the CSCI 0330 Introduction to Computer Systems concentration have pursued graduate study, computer consulting and (systems) information industries, and scientific and statistical analysis careers or CSCI 0300 Fundamentals of Computer Systems in industry or government. This degree offers a standard track and a CSCI 1010 Theory of Computation (math) professional track. CSCI 1450 Advanced Introduction to Probability for 2 Requirements for the Standard Track of the Computing and Data Science (math) or APMA 1650 Statistical Inference I Sc.B. degree. Three 1000-level Computer Science courses. Two of these 3 Prerequisites - two semesters of Calculus, for example courses and the intermediate courses must satisfy one of the MATH 0090 Introductory Calculus, Part I CS pathways. Non-CSCI courses are not allowed, even if they & MATH 0100 and Introductory Calculus, Part II are approved as part of a CS pathway or allowed as part of a pure CS concentration. At most one of the three courses can MATH 0170 Advanced Placement Calculus be a designated arts, humanities, or social science oriented CS Concentration Requirements (17 courses) course (which includes CSCI 1250, 1280, 1370, 1800, 1805, Core-Math: 1870, but see here [ https://cs.brown.edu/degrees/undergrad/ MATH 0180 Intermediate Calculus 1 concentrating-in-cs/concentration-requirements-2020/new-scb- 3 or MATH 0350 Honors Calculus requirements/] for the current list). MATH 0520 Linear Algebra 1 Three 1000-level Applied Mathematics courses approved by the 3 concentration advisor, of which two should constitute a standard or MATH 0540 Honors Linear Algebra sequence or address a common theme. Typical sequences or CSCI 0530 Coding the Matrix: An Introduction to Linear include: APMA 1200/1210 and APMA 1650 or 1655/1660. Algebra for Computer Science APMA 1910 cannot be used as an elective. Core-Applied Mathematics: A capstone course: a one-semester course, taken in the 1 APMA 0350 Applied Ordinary Differential Equations 1 student's last undergraduate year, in which the student (or group APMA 0360 Applied Partial Differential Equations I 1 of students) use a significant portion of their undergraduate education, broadly interpreted, in studying some current topic APMA 1170 Introduction to Computational Linear 1 in depth, to produce a culminating artifact such as a paper or Algebra software project. The title and abstract of the artifact, along with or APMA 1180 Introduction to Numerical Solution of Differential the student's and faculty-sponsor's names, will be placed in the Equations CS website. The inclusion of a relevant image or system diagram Core-Computer Science: is strongly encouraged. The complete text of the best artifacts of Select one of the following Series: 2 each class will be featured on the CS website. A senior thesis, which involves two semesters of work, may count as a capstone. Series A 4 CSCI 0150 Introduction to Object-Oriented & CSCI 0160 Programming and Computer Science Note: CSCI 1010 and 1450 may be used either as a math- and Introduction to Algorithms and Data oriented intermediate courses or as advanced courses. Structures CSCI 1010 was formerly known as CSCI 510: they are the same course and hence only one may be taken for credit. CSCI 1450 Series B was formerly known as CSCI 450: they are the same course CSCI 0170 Computer Science: An Integrated and hence only one may be taken for credit. Applied Math & CSCI 0180 Introduction 1650 or 1655 may be used in place of CSCI 1450. However, and Computer Science: An Integrated concentration credit will be given for only one of Applied Math Introduction 1650, 1655, and CSCI 1450. Series C Total Credits 17

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1 Students wishing to go directly (a) 1 from CSCI 0111 to CSCI 0180 (without CSCI 0112) will need to APMA 0350 Applied Ordinary Differential Equations 2 successfully complete additional exercises to receive an instructor & APMA 0360 and Applied Partial Differential Equations override code for CSCI 0180. In 2020-21, these exercises will be I 2 offered within CSCI 0111. Students from prior CSCI 0111 offerings Select one of the following: 1 should contact Professor Fisler to arrange to do this work. 2 APMA 0160 Introduction to Computing Sciences APMA 1650 may only be used if not being used as an Applied Math (preferred) course. 3 APMA 0200 Introduction to Modelling Pathways may be viewed here: https://cs.brown.edu/degrees/ undergrad/concentrating-in-cs/concentration-requirements-2020/ CSCI 0111 Computing Foundations: Data pathways-for-undergraduate-and-masters-students/ CSCI 0150 Introduction to Object-Oriented 4 Capstone Options may be found here: https://cs.brown.edu/degrees/ Programming and Computer Science undergrad/concentrating-in-cs/concentration-requirements-2020/ CSCI 0170 Computer Science: An Integrated capstone/ Introduction CSCI 0190 Accelerated Introduction to Computer Requirements for the Professional Track of Science the Sc.B. degree. Select one of the following: 1 The requirements for the professional tracks include all those of each of APMA 1200 Operations Analysis: Probabilistic Models the standard tracks, as well as the following: or APMA 1210 : Deterministic Models Students must complete full-time professional experiences doing work that Select one of the following: 1 is related to their concentration programs, totaling 2-6 months, whereby APMA 1650 Statistical Inference I each internship must be at least one month in duration in cases where or APMA 1655 Honors Statistical Inference I students choose to do more than one internship experience. Such work 1 is normally done at a company, but may also be at a university under the (b) supervision of a faculty member. Internships that take place between the Select one of the following: 1 end of the fall and the start of the spring semesters cannot be used to fulfill APMA 1160 An Introduction to Numerical Optimization this requirement. APMA 1180 Introduction to Numerical Solution of On completion of each professional experience, the student must Differential Equations write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor: APMA 1200 Operations Analysis: Probabilistic Models - Which courses were put to use in your summer's work? Which topics, in APMA 1210 Operations Research: Deterministic particular, were important? Models - In retrospect, which courses should you have taken before embarking on APMA 1330 Applied Partial Differential Equations II your summer experience? What are the topics from these courses that APMA 1360 Applied Dynamical Systems would have helped you over the summer if you had been more familiar with them? APMA 1660 Statistical Inference II - Are there topics you should have been familiar with in preparation for APMA 1690 Computational Probability and Statistics your summer experience, but are not taught at Brown? What are these APMA 1670 Statistical Analysis of Time Series topics? APMA 1680 Nonparametric Statistics - What did you learn from the experience that probably could not have APMA 1690 Computational Probability and Statistics been picked up from course work? - Is the sort of work you did over the summer something you would like to APMA 1710 Information Theory continue doing once you graduate? Explain. APMA 1720 Monte Carlo Simulation with Applications - Would you recommend your summer experience to other Brown to Finance students? Explain. APMA 1740 Recent Applications of Probability and Statistics APMA 1860 Graphs and Networks Applied Mathematics-Economics MATH 1010 Analysis: Functions of One Variable APMA 193X, 194X Senior Seminar series, depending on topic Concentration Requirements 6 The Applied Mathematics-Economics concentration is designed to reflect the mathematical and statistical nature of modern economic theory and Economics Requirements: 3 empirical research. This concentration has two tracks. The first is the ECON 1130 Intermediate Microeconomics 3 advanced economics track, which is intended to prepare students for (Mathematical) graduate study in economics. The second is the mathematical finance ECON 1210 Intermediate Macroeconomics track, which is intended to prepare students for graduate study in finance, ECON 1630 Mathematical Econometrics I or for careers in finance or financial engineering. Both tracks have A.B. Two 1000-level courses from the "mathematical-economics" 2 degree versions and Sc.B. degree versions, as well as a Professional group: 4 track option. ECON 1170 Welfare Economics and Social Choice Standard Program for the A.B. degree Theory (Advanced Economics track): ECON 1225 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies Prerequisites: ECON 1255 Unemployment: Models and Policies MATH 0100 Introductory Calculus, Part II ECON 1460 Industrial Organization (Mathematical) MATH 0520 Linear Algebra ECON 1470 Bargaining Theory and Applications Course Requirements: ECON 1490 Designing Internet Marketplaces Applied Mathematics Requirements

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ECON 1545 Topics in Macroeconomics, Development APMA 0350 Applied Ordinary Differential Equations 2 and International Economics & APMA 0360 and Applied Partial Differential Equations 2 ECON 1640 Mathematical Econometrics II I ECON 1660 Big Data Select one of the following: 1 ECON 1670 Advanced Topics in Econometrics APMA 0160 Introduction to Computing Sciences ECON 1750 Investments II (preferred) ECON 1805 Economics in the Laboratory APMA 0200 Introduction to Modelling ECON 1820 Theory of Behavioral Economics CSCI 0111 Computing Foundations: Data ECON 1850 Theory of Economic Growth CSCI 0190 Accelerated Introduction to Computer Science ECON 1860 The Theory of General Equilibrium CSCI 0150 Introduction to Object-Oriented ECON 1870 Game Theory and Applications to Programming and Computer Science Economics 4 CSCI 0170 Computer Science: An Integrated One 1000-level course from the "data methods" group: 1 Introduction ECON 1301 Economics of Education I Select one of the following: 1 ECON 1310 Labor Economics APMA 1200 Operations Analysis: Probabilistic Models ECON 1315 Health, Education, and Social Policy or APMA 1210 Operations Research: Deterministic Models ECON 1340 Economics of Global Warming APMA 1650 Statistical Inference I 1 ECON 1355 Environmental Issues in Development or APMA 1655 Honors Statistical Inference I Economics (b) 1 ECON 1360 Health Economics Select two of the following: 2 ECON 1375 Inequality of Opportunity in the US APMA 1160 An Introduction to Numerical Optimization ECON 1400 The Economics of Mass Media APMA 1180 Introduction to Numerical Solution of ECON 1430 The Economics of Social Policy Differential Equations ECON 1480 Public Economics APMA 1200 Operations Analysis: Probabilistic Models ECON 1510 Economic Development APMA 1210 Operations Research: Deterministic ECON 1530 Health, Hunger and the Household in Models Developing Countries APMA 1330 Applied Partial Differential Equations II ECON 1629 Applied Research Methods for Economists APMA 1360 Applied Dynamical Systems ECON 1640 Mathematical Econometrics II APMA 1660 Statistical Inference II ECON 1660 Big Data APMA 1670 Statistical Analysis of Time Series ECON 1670 Advanced Topics in Econometrics APMA 1680 Nonparametric Statistics ECON 1765 Finance, Regulation, and the Economy APMA 1690 Computational Probability and Statistics ECON 1825 Behavioral Economics and Public Policy APMA 1710 Information Theory ECON 1830 Behavioral Finance APMA 1720 Monte Carlo Simulation with Applications 5 One additional 1000-level economics course. 1 to Finance Total Credits 13 APMA 1740 Recent Applications of Probability and 1 Statistics No course may be used to simultaneously satisfy (a) and (b). 2 APMA 1860 Graphs and Networks APMA 0330 and APMA 0340 may be substituted with advisor approval, but these are no longer being offered. MATH 1010 Analysis: Functions of One Variable 3 APMA 193X, 194X Senior Seminar series, depending on topic Or ECON 1110 with permission. For students matriculating at Brown in 6 Fall 2021 or later, note that if ECON 1110 is used, then one additional course from the mathematical-economics group will be required Economics Requirements: 4 ECON 1130 Intermediate Microeconomics 1 No course may be used to simultaneously satisfy the "mathematical 3 economics" and the "data methods" requirements. (Mathematical) 5 Note that ECON 1620, ECON 1960, and ECON 1970 (independent ECON 1210 Intermediate Macroeconomics 1 study) cannot be used for concentration credit. However, 1620 and ECON 1630 Mathematical Econometrics I 1 1960 can be used for university credit and up to two 1970s may be Three 1000-level courses from the "mathematical-economics" 3 used for university credit. 4 6 group: Requires written approval of the Director of Undergraduate Studies in ECON 1170 Welfare Economics and Social Choice Economics. APMA 1910 is not permitted. Theory Standard program for the Sc.B. degree ECON 1225 Advanced Macroeconomics: Monetary, Fiscal, and Stabilization Policies (Advanced Economics track): ECON 1255 Unemployment: Models and Policies Prerequisites: ECON 1460 Industrial Organization (Mathematical) MATH 0100 Introductory Calculus, Part II ECON 1470 Bargaining Theory and Applications MATH 0520 Linear Algebra ECON 1490 Designing Internet Marketplaces Course Requirements: ECON 1545 Topics in Macroeconomics, Development Applied Mathematics Requirements and International Economics (a) 1 ECON 1640 Mathematical Econometrics II ECON 1660 Big Data

6 Applied Mathematics Applied Mathematics 7

ECON 1670 Advanced Topics in Econometrics APMA 0160 Introduction to Computing Sciences ECON 1750 Investments II (preferred) ECON 1805 Economics in the Laboratory APMA 0200 Introduction to Modelling ECON 1820 Theory of Behavioral Economics CSCI 0111 Computing Foundations: Data ECON 1850 Theory of Economic Growth CSCI 0150 Introduction to Object-Oriented ECON 1860 The Theory of General Equilibrium Programming and Computer Science ECON 1870 Game Theory and Applications to CSCI 0170 Computer Science: An Integrated Economics Introduction One 1000-level course from the "data methods" group: 4 1 CSCI 0190 Accelerated Introduction to Computer Science ECON 1301 Economics of Education I APMA 1200 Operations Analysis: Probabilistic Models 1 ECON 1310 Labor Economics APMA 1650 Statistical Inference I 1 ECON 1315 Health, Education, and Social Policy or APMA 1655 Honors Statistical Inference I ECON 1340 Economics of Global Warming (b) ECON 1355 Environmental Issues in Development Select one of the following: 1 Economics ECON 1360 Health Economics APMA 1160 An Introduction to Numerical Optimization ECON 1375 Inequality of Opportunity in the US APMA 1180 Introduction to Numerical Solution of Differential Equations ECON 1400 The Economics of Mass Media APMA 1210 Operations Research: Deterministic ECON 1430 The Economics of Social Policy Models ECON 1480 Public Economics APMA 1330 Applied Partial Differential Equations II ECON 1510 Economic Development APMA 1360 Applied Dynamical Systems ECON 1530 Health, Hunger and the Household in APMA 1660 Statistical Inference II Developing Countries APMA 1670 Statistical Analysis of Time Series ECON 1629 Applied Research Methods for Economists APMA 1680 Nonparametric Statistics ECON 1640 Mathematical Econometrics II APMA 1690 Computational Probability and Statistics ECON 1660 Big Data APMA 1710 Information Theory ECON 1765 Finance, Regulation, and the Economy APMA 1720 Monte Carlo Simulation with Applications ECON 1825 Behavioral Economics and Public Policy to Finance (preferred) ECON 1830 Behavioral Finance APMA 1740 Recent Applications of Probability and 5 Two additional 1000-level economics courses 2 Statistics Total Credits 16 APMA 1860 Graphs and Networks 1 No course may be used to simultaneously satisfy (a) and (b). MATH 1010 Analysis: Functions of One Variable 2 APMA 193X, 194X Senior Seminar series, depending on topic APMA 0330 and APMA 0340 may be substituted with advisor 5 approval, but these are no longer being offered. 3 Or ECON 1110 with permission. For students matriculating at Brown in Economics Requirements: ECON 1130 Intermediate Microeconomics 1 Fall 2021 or later, note that if ECON 1110 is used, then one additional 3 course from the mathematical-economics group will be required (Mathematical) 4 No course may be used to simultaneously satisfy the "mathematical ECON 1210 Intermediate Macroeconomics 1 economics" and the "data methods" requirements. ECON 1630 Mathematical Econometrics I 1 5 Students may use either ECON 1070 or ECON 1090 toward the Select two 1000-level courses from the "financial economics" 2 concentration, but not both. Note that ECON 1620, ECON 1960, and group: 2 ECON 1970 (independent study) cannot be used for concentration ECON 1710 Investments I credit. However, 1620 and 1960 can be used for university credit and ECON 1720 Corporate Finance up to two 1970s may be used for university credit. 6 ECON 1730 Venture Capital, Private Equity, and Requires written approval of the Director of Undergraduate Studies in Entrepreneurship Economics. APMA 1910 is not permitted. ECON 1750 Investments II Standard program for the A.B. degree ECON 1760 Financial Institutions (Mathematical Finance track): ECON 1780 Advanced Topics in Corporate Finance ECON 1830 Behavioral Finance Prerequisites: Select one 1000-level course from the "mathematical economics" 1 MATH 0100 Introductory Calculus, Part II group: 2 MATH 0520 Linear Algebra ECON 1170 Welfare Economics and Social Choice Course Requirements: 13 Courses: 6 Applied Math and 7 Theory Economics ECON 1225 Advanced Macroeconomics: Monetary, Applied Mathematics Requirements Fiscal, and Stabilization Policies (a) ECON 1255 Unemployment: Models and Policies APMA 0350 Applied Ordinary Differential Equations 2 ECON 1460 Industrial Organization (Mathematical) & APMA 0360 and Applied Partial Differential Equations ECON 1470 Bargaining Theory and Applications I 1 ECON 1490 Designing Internet Marketplaces Select one of the following: 1

Applied Mathematics 7 8 Applied Mathematics

ECON 1545 Topics in Macroeconomics, Development APMA 0160 Introduction to Computing Sciences and International Economics (preferred) ECON 1640 Mathematical Econometrics II APMA 0200 Introduction to Modelling ECON 1660 Big Data CSCI 0111 Computing Foundations: Data ECON 1670 Advanced Topics in Econometrics CSCI 0150 Introduction to Object-Oriented ECON 1750 Investments II Programming and Computer Science ECON 1805 Economics in the Laboratory CSCI 0170 Computer Science: An Integrated ECON 1820 Theory of Behavioral Economics Introduction ECON 1850 Theory of Economic Growth CSCI 0190 Accelerated Introduction to Computer Science ECON 1860 The Theory of General Equilibrium APMA 1200 Operations Analysis: Probabilistic Models 1 ECON 1870 Game Theory and Applications to APMA 1650 Statistical Inference I 1 Economics Select one 1000-level course from the "data methods" group: 2 1 or APMA 1655 Honors Statistical Inference I ECON 1301 Economics of Education I (b) ECON 1310 Labor Economics Select two of the following: 2 ECON 1315 Health, Education, and Social Policy APMA 1160 An Introduction to Numerical Optimization ECON 1340 Economics of Global Warming APMA 1180 Introduction to Numerical Solution of Differential Equations ECON 1355 Environmental Issues in Development APMA 1210 Operations Research: Deterministic Economics Models ECON 1360 Health Economics APMA 1330 Applied Partial Differential Equations II ECON 1375 Inequality of Opportunity in the US APMA 1360 Applied Dynamical Systems ECON 1400 The Economics of Mass Media APMA 1660 Statistical Inference II ECON 1430 The Economics of Social Policy APMA 1670 Statistical Analysis of Time Series ECON 1480 Public Economics APMA 1680 Nonparametric Statistics ECON 1510 Economic Development APMA 1690 Computational Probability and Statistics ECON 1530 Health, Hunger and the Household in APMA 1710 Information Theory Developing Countries ECON 1629 Applied Research Methods for Economists APMA 1720 Monte Carlo Simulation with Applications to Finance (preferred) ECON 1640 Mathematical Econometrics II APMA 1740 Recent Applications of Probability and ECON 1660 Big Data Statistics ECON 1825 Behavioral Economics and Public Policy APMA 1860 Graphs and Networks ECON 1830 Behavioral Finance MATH 1010 Analysis: Functions of One Variable Total Credits 13 APMA 193X, 194X Senior Seminar series, depending on topic 1 5 APMA 0330 and APMA 0340 may be substituted with advisor approval, but these are no longer being offered. Economics Requirements: 2 ECON 1130 Intermediate Microeconomics 1 No course may be used to simultaneously satisfy the "mathematical 3 economics" and the "data methods" requirements. (Mathematical) 3 Or ECON 1110 with permission. For students matriculating at Brown in ECON 1210 Intermediate Macroeconomics 1 Fall 2021 or later, note that if ECON 1110 is used, then one additional ECON 1630 Mathematical Econometrics I 1 course from the mathematical-economics group will be required 4 Select three 1000-level courses from the "financial economics" 3 Note that ECON 1620, ECON 1960, and ECON 1970 (independent group: 2 study) cannot be used for concentration credit. However, 1620 and ECON 1710 Investments I 1960 can be used for university credit and up to two 1970s may be ECON 1720 Corporate Finance used for university credit. 5 ECON 1730 Venture Capital, Private Equity, and Requires written approval of the Director of Undergraduate Studies in Entrepreneurship Economics. APMA 1910 is not permitted. ECON 1750 Investments II Standard program for the Sc.B. degree ECON 1760 Financial Institutions (Mathematical Finance track): ECON 1780 Advanced Topics in Corporate Finance ECON 1830 Behavioral Finance Prerequisites: Select two 1000-level courses from the "mathematical 2 MATH 0100 Introductory Calculus, Part II economics" group: 2 MATH 0520 Linear Algebra ECON 1170 Welfare Economics and Social Choice Course Requirements: 16 courses: 7 Applied Math and 9 Theory Economics ECON 1225 Advanced Macroeconomics: Monetary, Applied Mathematics requirements: Fiscal, and Stabilization Policies (a) ECON 1255 Unemployment: Models and Policies APMA 0350 Applied Ordinary Differential Equations 2 ECON 1460 Industrial Organization (Mathematical) & APMA 0360 and Applied Partial Differential Equations ECON 1470 Bargaining Theory and Applications I 1 ECON 1490 Designing Internet Marketplaces Select one of the following: 1

8 Applied Mathematics Applied Mathematics 9

ECON 1545 Topics in Macroeconomics, Development Students must complete full-time professional experiences doing work that and International Economics is related to their concentration programs, totaling 2-6 months, whereby ECON 1640 Mathematical Econometrics II each internship must be at least one month in duration in cases where students choose to do more than one internship experience. Such work ECON 1660 Big Data is normally done at a company, but may also be at a university under the ECON 1670 Advanced Topics in Econometrics supervision of a faculty member. Internships that take place between the ECON 1750 Investments II end of the fall and the start of the spring semesters cannot be used to fulfill ECON 1805 Economics in the Laboratory this requirement. ECON 1820 Theory of Behavioral Economics On completion of each professional experience, the student must ECON 1850 Theory of Economic Growth write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor. ECON 1860 The Theory of General Equilibrium • Which courses were put to use in your summer's work? Which topics, ECON 1870 Game Theory and Applications to in particular, were important? Economics • In retrospect, which courses should you have taken before embarking Select one 1000-level course from the "data methods" group: 2 1 on your summer experience? What are the topics from these courses ECON 1301 Economics of Education I that would have helped you over the summer if you had been more ECON 1310 Labor Economics familiar with them? ECON 1315 Health, Education, and Social Policy • Are there topics you should have been familiar with in preparation ECON 1340 Economics of Global Warming for your summer experience, but are not taught at Brown? What are these topics? ECON 1355 Environmental Issues in Development Economics • What did you learn from the experience that probably could not have been picked up from course work? ECON 1360 Health Economics • Is the sort of work you did over the summer something you would like ECON 1375 Inequality of Opportunity in the US to continue doing once you graduate? Explain. ECON 1400 The Economics of Mass Media • Would you recommend your summer experience to other Brown ECON 1430 The Economics of Social Policy students? Explain. ECON 1480 Public Economics ECON 1510 Economic Development Data Fluency Concentration Requirements ECON 1530 Health, Hunger and the Household in The Certificate in Data Fluency provides a curricular structure to Developing Countries undergraduates in concentrations other than applied mathematics, ECON 1629 Applied Research Methods for Economists computational biology, computer science, math, and statistics who wish to gain fluency and facility with the tools of data analysis and its conceptual ECON 1640 Mathematical Econometrics II framework. The driving intellectual question is how we can infer meaning ECON 1660 Big Data from data whilst avoiding false predictions. The required experiential ECON 1670 Advanced Topics in Econometrics learning component provides students with the opportunity to apply their ECON 1825 Behavioral Economics and Public Policy data-science skills in their concentration, engage in research that uses ECON 1830 Behavioral Finance data science, teach data science as an undergraduate teaching assistant, or undertake an internship that has a substantive data-science component. Total Credits 16 As with all undergraduate certificates (https://www.brown.edu/academics/ 1 APMA 0330 and APMA 0340 may be substituted with advisor college/degree/undergraduatecertificates/), students must be enrolled approval, but these are no longer being offered. in or have completed at least two courses toward the certificate at the 2 No course may be used to simultaneously satisfy the "mathematical time they declare in ASK, which must be no earlier than the beginning economics" and the "data methods" requirements. of the fifth semester and no later than the last day of classes of the 3 antepenultimate (typically the sixth) semester, in order to facilitate planning Or ECON 1110 with permission. For students matriculating at Brown in for the experiential learning opportunity. Students must submit a proposal Fall 2021 or later, note that if ECON 1110 is used, then one additional for their experiential learning opportunity by the end of the sixth semester. course from the mathematical-economics group will be required 4 Note that ECON 1620, ECON 1960, and ECON 1970 (independent Excluded Concentrations: Applied Mathematics, Computational study) cannot be used for concentration credit. However, 1620 and Biology, Computer Science, Mathematics, and Statistics (including joint 1960 can be used for university credit and up to two 1970s may be concentrations in these areas) used for university credit. 5 Certificate Requirements Requires written approval of the Director of Undergraduate Studies in Economics. APMA 1910 is not permitted. Core Courses: DATA 0080 Data, Ethics and Society 1 Honors CSCI 0111 Computing Foundations: Data 1 Applied Math-Economics concentrators who wish to pursue honors must or CSCI 0150 Introduction to Object-Oriented Programming and find a primary faculty thesis advisor in either Economics or Applied Math. Computer Science They will be held to the Honors requirements of their advisor’s department. or CSCI 0170 Computer Science: An Integrated Introduction Joint concentrators in Applied Mathematics-Economics with an Economics or CSCI 0190 Accelerated Introduction to Computer Science thesis advisor should follow the requirements published here (https:// DATA 0200 Data Science Fluency 1 economics.brown.edu/academics/undergraduate/honors-and-capstones/ thesis/), while concentrators with an Applied Math thesis advisor Elective Course: Select one follow-up Applied Math, Biostatistics, 1 should follow the requirements published here (https://www.brown.edu/ Computer Science or domain-specific course with a significant academics/applied-mathematics/undergraduate-program/honors/). data component from the following list (or another course with approval from the certificate advisor): Professional Track ANTH 1201 Introduction to Geographic Information The requirements for the professional track include all those of the Systems and Spatial Analysis standard track, as well as the following: APMA 1650 Statistical Inference I

Applied Mathematics 9 10 Applied Mathematics

BIOL 0495 Statistical Analysis of Biological Data APMA 0090. Introduction to Mathematical Modeling. BIOL 1555 Methods in Informatics and Data Science We will explore issues of mathematical modeling and analysis. Five to for Health six self-contained topics will be discussed and developed. The course will include seminars in which modeling issues are discussed, lectures BIOL 1565 Survey of Biomedical Informatics to provide mathematical background, and computational experiments. CLPS 0900 Statistical Methods Required mathematical background is knowledge of one-variable calculus, CLPS 1291 Computational Methods for Mind, Brain and no prior computing experience will be assumed. and Behavior APMA 0100. Elementary Probability for Applications. CSCI 1450 Advanced Introduction to Probability for This course serves as an introduction to probability and stochastic Computing and Data Science processes with applications to practical problems. It will cover basic CSCI 1470 Deep Learning probability and stochastic processes such as basic concepts of probability CSCI 1951A Data Science and conditional probability, simple random walk, Markov chains, continuous distributions, Brownian motion and option pricing. Enrollment DATA 1150 Data Science Fellows limited to 19 first year students. ECON 1620 Introduction to Econometrics ECON 1660 Big Data APMA 0110. What’s the big deal with Data Science?. This seminar serves as a practical introduction to the interdisciplinary ENVS 1105 Introduction to Environmental GIS field of data science. Over the course of the semester, students will be EEPS 1320 Introduction to Geographic Information exposed to the diversity of questions that data science can address by Systems for Environmental Applications reading current scholarly works from leading researchers. Through hands- EEPS 1330 Global Environmental Remote Sensing on labs and experiences, students will gain facility with computational and MATH 1610 Probability visualization techniques for uncovering meaning from large numerical and text-based data sets. Ultimately, students will gain fluency with data PHP 1501 Essentials of Data Analysis science vocabulary and ideas. There are no prerequisites for this course. PHP 1510 Principles of Biostatistics and Data Analysis APMA 0111. Data Science and Social Justice. This first-year seminar explores the impact of data and algorithms on SOC 1020 Methods of Social Research equity and justice. Data analysis and visualization can help identify SOC 1100 Introductory Statistics for Social Research inequities and advocate for social justice. At the same time, data-based SOC 1340 Principles and Methods of Geographic algorithms can institutionalize and rationalize unfair practices and Information Systems injustices. In this course, we will engage with fundamental questions about Capstone: 0-1 the role of data to work towards social justice. We will gain introductory data science skills, discuss algorithmic bias, equity, and fairness, and work The required experiential learning component provides with real-world data sets and algorithms to examine inequities in health, students with the opportunity to apply their data-science policing, prisons, and welfare. Everybody is welcome - no prerequisites skills in their concentration, engage in research that uses are needed. data science, teach data science as UTAs, or undertake an internship that has a data-science component. The capstone APMA 0120. Mathematics of Finance. may be completed for credit via an independent study course The current volatility in international financial markets makes it imparative or not for credit. 1 for us to become competent in financial calculations early in our liberal Total Credits 4-5 arts and scientific career paths. This course is designed to prepare the 1 student with those elements of mathematics of finance appropriate for the Students must submit a proposal for their practicum project by the end calculations necessary in financial transactions. of the sixth semester. APMA 0160. Introduction to Computing Sciences. For students in any discipline that may involve numerical computations. Applied Mathematics Graduate Program Includes instruction for programming in MATLAB. Applications discussed The department of Applied Mathematics offers graduate programs leading include solution of linear equations (with vectors and matrices) and to the Master of Science (Sc.M.) degree and the Doctor of Philosophy nonlinear equations (by bisection, iteration, and Newton's method), (Ph.D.) degree. interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. For more information on admission and program requirements, please visit Prerequisites: MATH 0100 or equivalent. the following website: Spr APMA0160 S01 24764 MWF 9:00-9:50(02) ’To Be Arranged' http://www.brown.edu/academics/gradschool/programs/applied- mathematics (http://www.brown.edu/academics/gradschool/programs/ APMA 0180. Modeling the World with Mathematics: An Introduction applied-mathematics/) for Non-Mathematicians. Mathematics is the foundation of our technological society and most of Courses its powerful ideas are quite accessible. This course will explain some of these using historical texts and Excel. Topics include the predictive power APMA 0070. Introduction to Applied Complex Variables. of 'differential equations' from the planets to epidemics, oscillations and Applications of complex analysis that do not require calculus as a music, chaotic systems, randomness and the atomic bomb. Prerequisite: prerequisite. Topics include algebra of complex numbers, plane geometry some knowledge of calculus. by means of complex coordinates, complex exponentials, and logarithms and their relation to trigonometry, polynomials, and roots of polynomials, conformal mappings, rational functions and their applications, finite Fourier series and the FFT, iterations and fractals. Uses MATLAB, which has easy and comprehensive complex variable capabilities.

10 Applied Mathematics Applied Mathematics 11

APMA 0200. Introduction to Modelling. APMA 0650. Essential Statistics. This course provides an introduction to the mathematical modeling of A first course in probability and statistics emphasizing statistical reasoning selected biological, chemical, engineering, and physical processes. and basic concepts. Topics include visual and numerical summaries of The goal is to illustrate the typical way in which applied mathematicians data, representative and non-representative samples, elementary discrete approach practical applications, from understanding the underlying , the normal distribution, sampling variability, elementary problem, creating a model, analyzing the model using mathematical statistical inference, measures of association. Examples and applications techniques, and interpreting the findings in terms of the original problem. from the popular press and the life, social and physical sciences. Not Single-variable calculus is the only requirement; all other techniques from calculus-based. No prerequisites. differential equations, linear algebra, and numerical methods, to probability Spr APMA0650 S01 24775 TTh 9:00-10:20(01) (W. Zhao) and statistics will be introduced in class. Prerequisites: Math 0100 or equivalent. APMA 1070. Quantitative Models of Biological Systems. Quantitative dynamic models help understand problems in biology and Fall APMA0200 S01 16934 TTh 9:00-10:20(05) (Y. Guo) there has been rapid progress in recent years. This course provides APMA 0330. Methods of Applied Mathematics I. an introduction to the concepts and techniques, with applications to This course will cover mathematical techniques involving ordinary population dynamics, infectious diseases, enzyme kinetics, and cellular differential equations used in the analysis of physical, biological, and biology. Additional topics covered will vary. Mathematical techniques economic phenomena. The course emphasizes the use of established will be discussed as they arise in the context of biological problems. methods in applications rather than rigorous foundation. Topics include: Prerequisites: APMA 0350 or equivalent. first and second order differential equations, an introduction to numerical Spr APMA1070 S01 24776 MWF 1:00-1:50(06) (L. Bienenstock) methods, series solutions, and Laplace transformations. Prerequisites: MATH 0100 or equivalent. APMA 1080. Inference in Genomics and Molecular Biology. Massive quantities of fundamental biological and geological sequence APMA 0340. Methods of Applied Mathematics II. data have emerged. The goal of this course is to enable students to This course will cover mathematical techniques involving ordinary and construct and apply probabilistic models to draw inferences from sequence partial differential equations and statistics used in the analysis of physical, data on problems novel to them. Statistical topics: Bayesian inferences; biological, and economic phenomena. The course emphasizes the use estimation; hypothesis testing and false discovery rates; statistical decision of established methods rather than rigorous foundations. Topics include: theory; change point algorithm; hidden Markov models; Kalman filters; and applications of linear algebra to systems of equations; numerical methods; significances in high dimensions. Prerequisites: APMA 1650 or equivalent; nonlinear problems and stability; introduction to partial differential APMA 0160 or CSCI 0111 or equivalent. equations; introduction to statistics. Prerequisites: APMA 0330 or Fall APMA1080 S01 16039 MWF 1:00-1:50(06) (M. Harrison) equivalent. Fall APMA0340 S01 15989 MWF 12:00-12:50(15) (V. Dobrushkin) APMA 1150. Machine Learning for Scientific Modeling: Data-Driven Discovery of Differential Equations(MATH 1150). Spr APMA0340 S01 24770 MWF 12:00-12:50(05) (V. Dobrushkin) Interested students must register for MATH 1150. . APMA 0350. Applied Ordinary Differential Equations APMA 1160. An Introduction to Numerical Optimization. This course provides a comprehensive introduction to ordinary differential This course provides a thorough introduction to numerical methods and equations and their applications. During the course, we will see how algorithms for solving non-linear continuous optimization problems. A applied mathematicians use ordinary differential equations to solve particular attention will be given to the mathematical underpinnings to practical applications, from understanding the underlying problem, understand the theoretical properties of the optimization problems and creating a differential-equations model, solving the model using analytical, the algorithms designed to solve them. Topics will include: line search numerical, or qualitative methods, and interpreting the findings in terms methods, trust-region methods, nonlinear conjugate gradient methods, an of the original problem. We will also learn about the underlying rigorous introduction to constrained optimization (Karush-Kuhn-Tucker conditions, theoretical foundations of differential equations. Format: lectures and mini-maximization, saddle-points of Lagrangians). Some applications in problem-solving workshops. Prerequisites: MATH 0100 or equivalent; signal and image processing will be explored. Prerequisites: MATH 0180 knowledge of matrix-vector operations, determinants, and linear systems. or equivalent; MATH 0520 or equivalent; APMA 0160 or CSCI 0111 or Fall APMA0350 S01 15995 MWF 12:00-12:50(15) (B. Sandstede) equivalent. APMA 1170 or equivalent is recommended. Spr APMA0350 S01 24771 MWF 12:00-12:50(05) (T. Yastrzhembskiy) Fall APMA1160 S01 16028 MWF 2:00-2:50(01) (Y. Shin) APMA 0360. Applied Partial Differential Equations I. APMA 1170. Introduction to Computational Linear Algebra. This course provides an introduction to partial differential equations and Focuses on fundamental algorithms in computational linear algebra with their applications. We will learn how to use partial differential equations to relevance to all science concentrators. Basic linear algebra and matrix solve problems that arise in practical applications, formulating questions decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical about a real-world problem, creating a partial differential equation model analysis of errors and convergence. Iterative methods and conjugate that can help answer these questions, solving the resulting system using gradient techniques. Computation of eigenvalues and eigenvectors, and analytical, numerical, and qualitative methods, and interpreting the results an introduction to least squares methods. Prerequisites: MATH 0100 or in terms of the original application. To help us support and justify our equivalent; MATH 0520 or equivalent. Experience with a programming approaches and solutions, we will also learn about theoretical foundations language is strongly recommended. of partial differential equations. Prerequisites: APMA 0350 or equivalent. Fall APMA1170 S01 16025 TTh 9:00-10:20(05) (M. Fabien) Fall APMA0360 S01 15991 MWF 10:00-10:50(14) (M. Maxey) Spr APMA0360 S01 24773 MWF 12:00-12:50(05) (Y. Shin) APMA 1180. Introduction to Numerical Solution of Differential Equations. APMA 0410. Mathematical Methods in the Brain Sciences. Fundamental numerical techniques for solving ordinary and partial Basic mathematical methods commonly used in the neural and cognitive differential equations. Overview of techniques for approximation and sciences. Topics include: introduction to probability and statistics, integration of functions. Development of multi-step and multi-stage emphasizing hypothesis testing and modern nonparametric methods; methods, error analysis, step-size control for ordinary differential introduction to differential equations and systems of differential equations, equations. Solution of two-point boundary value problems, introduction emphasizing qualitative behavior and simple phase-plane analysis. to methods for solving linear partial differential equations. Students will Examples from neuroscience, cognitive science, and other sciences. be required to use Matlab (or other computer languages) to implement Prerequisite: MATH 0100 or equivalent. the mathematical algorithms under consideration: experience with a programming language is therefore strongly recommended. Prerequisites: APMA 0350 or equivalent; APMA 0360 or equivalent. Spr APMA1180 S01 24778 MWF 2:00-2:50(07) (M. Fabien)

Applied Mathematics 11 12 Applied Mathematics

APMA 1190. Finite Volume Method for CFD: A Survey. APMA 1655. Honors Statistical Inference I. This course will provide students with an overview of the subjects Students may opt to enroll in APMA 1655 for more in depth coverage of necessary to perform robust simulations of computational fluid dynamics APMA 1650. Enrollment in 1655 will include an optional recitation section (CFD) problems. After an initial overview of the finite volume method and and required additional individual work. Applied Math concentrators are fluid mechanics, students will use the finite volume library OpenFOAM encouraged to take 1655. Prerequisites: MATH 0180 or equivalent. to explore the different components that make up a modern CFD code Fall APMA1655 S01 16030 MWF 1:00-1:50(06) (H. Wang) (discretization, linear algebra, timestepping, boundary conditions, splitting Spr APMA1655 S01 24782 MWF 10:00-10:50(03) (H. Wang) schemes, and multiphysics) and learn how to navigate a production scale software library. APMA 1660. Statistical Inference II. APMA 1660 is designed as a sequel to APMA 1650 to form one of APMA 1200. Operations Analysis: Probabilistic Models. the alternative tracks for an integrated year's course in mathematical Basic probabilistic problems and methods in operations research and statistics. The main topic is linear models in statistics. Specific topics management science. Methods of problem formulation and solution. include likelihood-ratio tests, nonparametric tests, introduction to statistical Markov chains, birth-death processes, stochastic service and queueing computing, matrix approach to simple-linear and multiple regression, systems, the theory of sequential decisions under , dynamic analysis of variance, and design of experiments. Prerequisites: APMA programming. Applications. Prerequisites: APMA 1650 or equivalent; 1650 or equivalent; MATH 0520 or equivalent. MATH 520 or equivalent. Spr APMA1660 S01 24783 TTh 1:00-2:20(08) (B. Gidas) Spr APMA1200 S01 24779 TTh 2:30-3:50(11) (P. Lopatto) APMA 1670. Statistical Analysis of Time Series. APMA 1210. Operations Research: Deterministic Models. Time series analysis is an important branch of mathematical statistics An introduction to the basic mathematical ideas and computational with many applications to signal processing, econometrics, geology, etc. methods of optimizing allocation of effort or resources, with or without The course emphasizes methods for analysis in the frequency domain, constraints. Linear programming, network models, dynamic programming, in particular, estimation of the spectrum of time-series, but time domain and integer programming. Prerequisites: MATH 0100 or equivalent; MATH methods are also covered. Prerequisites: elementary probability and 520 or equivalent; APMA 0160 or CSCI 0111 or equivalent. statistics on the level of APMA 1650-1660. Fall APMA1210 S01 16026 TTh 10:30-11:50(13) (C. Klivans) APMA 1680. Nonparametric Statistics. APMA 1250. Advanced Engineering Mechanics (ENGN 1370). A systematic treatment of distribution-free alternatives to classical Interested students must register for ENGN 1370. statistical tests. These nonparametric tests make minimum assumptions APMA 1260. Introduction to the Mechanics of Solids and Fluids. about distributions governing the generation of observations, yet are of An introduction to the dynamics of fluid flow and deforming elastic solids nearly equal power to the classical alternatives. Prerequisite: APMA 1650 for students in the physical or mathematical sciences. Topics in fluid or equivalent. Offered in alternate years. mechanics include statics, simple viscous flows, inviscid flows, potential APMA 1681. Computational Neuroscience (NEUR 1680). flow, linear water waves, and acoustics. Topics in solid mechanics include Interested students must register for NEUR 1680. elastic/plastic deformation, strain and stress, simple elastostatics, and elastic waves with reference to seismology. Offered in alternate years. APMA 1690. Computational Probability and Statistics. Examination of probability theory and mathematical statistics from APMA 1330. Applied Partial Differential Equations II. the perspective of computing. Topics selected from random number Review of vector calculus and curvilinear coordinates. Partial differential generation, Monte Carlo methods, limit theorems, stochastic dependence, equations. Heat conduction and diffusion equations, the wave equation, Bayesian networks, dimensionality reduction. Prerequisites: APMA 1650 Laplace and Poisson equations. Separation of variables, special functions, or equivalent; programming experience is recommended. Fourier series and power series solution of differential equations. Sturm- Fall APMA1690 S01 16031 MWF 11:00-11:50(16) ’To Be Arranged' Liouville problem and eigenfunction expansions. Prerequisites: APMA 0360 or equivalent. APMA 1700. The Mathematics of Insurance. Fall APMA1330 S01 16027 TTh 2:30-3:50(12) (T. Yastrzhembskiy) The course consists of two parts: the first treats life contingencies, i.e. the construction of models for individual life insurance contracts. The second APMA 1340. Methods of Applied Mathematics III, IV. treats the Collective Theory of Risk, which constructs mathematical See Methods Of Applied Mathematics III, IV (APMA 1330) for course models for the insurance company and its portfolio of policies as a description. whole. Suitable also for students proceeding to the Institute of Actuaries APMA 1360. Applied Dynamical Systems. examinations. Prerequisites: Probability Theory to the level of APMA 1650 This course gives an overview of the theory and applications of dynamical or MATH 1610. systems modeled by differential equations and maps. We will discuss APMA 1710. Information Theory. changes of the dynamics when parameters are varied, investigate periodic Information theory is the study of the fundamental limits of information and homoclinic solutions that arise in applications, and study the impact transmission and storage. This course, intended primarily for advanced of additional structures such as time reversibility and conserved quantities undergraduates and beginning graduate students, offers a broad on the dynamics. We will also study systems with complicated "chaotic" introduction to information theory and its applications: Entropy and dynamics that possess attracting sets which do not have an integer information, lossless data compression, communication in the presence of dimension. Applications to chemical reactions, climate, epidemiology, noise, channel capacity, channel coding, source-channel separation, lossy and phase transitions will be discussed. Prerequisites: APMA 0350 or data compression. Prerequisites: APMA 1650 or equivalent. equivalent. Spr APMA1710 S01 24784 TTh 9:00-10:20(01) (P. Lopatto) Spr APMA1360 S01 24780 MWF 11:00-11:50(04) ’To Be Arranged' APMA 1720. Monte Carlo Simulation with Applications to Finance. APMA 1650. Statistical Inference I. The course will cover the basics of Monte Carlo and its applications APMA 1650 is an integrated first course in mathematical statistics. The to financial engineering: generating random variables and simulating first half of APMA 1650 covers probability and the last half is statistics, stochastic processes; analysis of simulated data; variance reduction integrated with its probabilistic foundation. Specific topics include techniques; binomial trees and option pricing; Black-Scholes formula; probability spaces, discrete and continuous random variables, methods portfolio optimization; interest rate models. The course will use MATLAB for parameter estimation, confidence intervals, and hypothesis testing. as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610 Prerequisites: MATH 0100 or equivalent. Fall APMA1650 S01 16029 MWF 11:00-11:50(16) (W. Zhao) Spr APMA1650 S01 24781 MWF 2:00-2:50(07) (N. Kaihnsa)

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APMA 1740. Recent Applications of Probability and Statistics. APMA 1930C. Information Theory. This course develops the mathematical foundations of modern Information theory is the mathematical study of the fundamental limits of applications of statistics to the computational, cognitive, engineering, information transmission (or coding) and storage (or compression). This and neural sciences. The course is rigorous, but the emphasis is on course offers a broad introduction to information theory and its real-world application. Topics include: Gibbs ensembles and their relation to applications. A subset of the following is covered: entropy and information; maximum entropy, large deviations, exponential models, and information the asymptotic equipartition property; theoretical limits of lossless data theory; statistical estimation and classification; graphical models, dynamic compression and practical algorithms; communication in the presence programming, MCMC, parameter estimation, and the EM algorithm. of noise-channel coding, channel capacity; source-channel separation; Graduate version: 2610; Undergraduate version: 1740. Prerequisites: Gaussian channels; Lossy data compression. APMA 1650 or equivalent; programming experience; strong mathematics APMA 1930D. Mixing and Transport in Dynamical Systems. background. APMA 1200 or APMA 1690 or similar courses recommended. Mixing and transport are important in several areas of applied science, MATH 1010 or equivalent is recommended for APMA 2610. including fluid mechanics, atmospheric science, chemistry, and particle Spr APMA1740 S01 24785 MWF 1:00-1:50(06) (S. Geman) dynamics. In many cases, mixing seems highly complicated and APMA 1850. Introdution to High Performance Parallel Computing. unpredictable. We use the modern theory of dynamical systems to No description available. understand and predict mixing and transport from the differential equations describing the physical process in question. Prerequisites: APMA 0330, APMA 1860. Graphs and Networks. 0340; or APMA 0350, 0360. Selected topics about the mathematics of graphs and networks with an emphasis on random graph models and the dynamics of processes APMA 1930E. Ocean Dynamics. operating on these graphs. Topics include: empirical properties of Works through the popular book by Henry Stommel entitled A View of biological, social, and technological networks (small-world effects, scale- the Sea. Introduces the appropriate mathematics to match the physical free properties, transitivity, community structure); mathematical and concepts introduced in the book. statistical models of random graphs and their properties (Bernoulli random APMA 1930G. The Mathematics of Sports. graphs, preferential attachment models, stochastic block models, phase Topics to be discussed will range from the determination of who transitions); dynamical processes on graphs and networks (percolation, won the match, through biomechanics, free-fall of flexible bodies cascades, epidemics, queuing, synchronization). Prerequisites: MATH and aerodynamics, to the flight of ski jumpers and similar unnatural 520 or equivalent; APMA 0350 or equivalent; APMA 1650 or equivalent; phenomena. Prerequisite: APMA 0340 or equivalent, or permission of the programming experience. APMA 1200 or APMA 1690 or similar courses instructor. recommended. Spr APMA1860 S01 24786 TTh 10:30-11:50(09) (O. Nguyen) APMA 1930H. Scaling and Self-Similarity. The themes of scaling and self-similarity provide the simplest, and yet APMA 1880. Advanced Matrix Theory. the most fruitful description of complicated forms in nature such as Canonical forms of orthogonal, Hermitian and normal matrices: Rayleigh the branching of trees, the structure of human lungs, rugged natural quotients. Norms, eigenvalues, matrix equations, generalized inverses. landscapes, and turbulent fluid flows. This seminar is an investigation of Banded, sparse, non-negative and circulant matrices. Prerequisite: APMA some of these phenomena in a self-contained setting requiring little more 0340 or 0360, or MATH 0520 or 0540, or permission of the instructor. mathematical background than high school algebra. APMA 1910. Race and Gender in the Scientific Community. Topics to be covered: Dimensional analysis; empirical laws in biology, This course examines the (1) disparities in representation in the scientific geosciences, and physics and the interplay between scaling and community, (2) issues facing different groups in the sciences, and (3) function; an introduction to fractals; social networks and the "small world" paths towards a more inclusive scientific environment. We will delve phenomenon. into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the APMA 1930I. Random Matrix Theory. history, philosophy, and sociology of science. We will also explore the In the past few years, random matrices have become extremely important specific problems faced by underrepresented and well-represented racial in a variety of fields such as computer science, physics and statistics. minorities, women, and LGBTQ community members. The course is They are also of basic importance in various areas of mathematics. This reading intensive and discussion based. To be added to the waitlist for this class will serve as an introduction to this area. The focus is on the basic course, please go to https://goo.gl/forms/foK0fyGxm5Eu2irA2 matrix ensembles and their limiting distributions, but several applications will be considered. Prerequisites: MATH 0200 or 0350; and MATH 0520 Fall APMA1910 S01 16032 TTh 1:00-2:20(08) (K. Ramanan) or 0540; and APMA 0350, 0360, 1650, and 1660. APMA 1170 and MATH APMA 1911. Race and Gender in the Scientific Community (MATH 1010 are recommended, but not required. . 1910) APMA 1930J. Mathematics of Random Networks. Interested students must register for MATH 1910. An intro to the emerging field of random networks and a glimpse of APMA 1930A. Actuarial Mathematics. some of the latest developments. Random networks arise in a variety A seminar considering selected topics from two fields: (1) life of applications including statistics, communications, physics, biology contingencies-the study of the valuation of life insurance contracts; and and social networks. They are studied using methods from a variety of (2) collective risk theory, which is concerned with the random process disciplines ranging from probability, graph theory and to that generates claims for a portfolio of policies. Topics are chosen from nonlinear dynamical systems. Describes elements of these theories and Actuarial Mathematics, 2nd ed., by Bowers, Gerber, Hickman, Jones, and shows how they can be used to gain practical insight into various aspects Nesbitt. Prerequisite: knowledge of probability theory to the level of APMA of these networks including their structure, design, distributed control 1650 or MATH 1610. Particularly appropriate for students planning to take and self-organizing properties. Prerequisites: Advanced calculus, basic the examinations of the Society of Actuaries. knowledge of probability. Enrollment limited to 40. APMA 1930B. Computational Probability and Statistics. APMA 1930M. Applied Asymptotic Analysis. Examination of probability theory and mathematical statistics from Many problems in applied mathematics and physics are nonlinear and the perspective of computing. Topics selected from: random number are intractable to solve using elementary methods. In this course we will generation, Monte Carlo methods, limit theorems, stochastic dependence, systematically develop techniques for obtaining quantitative information Bayesian networks, probabilistic grammars. from nonlinear systems by exploiting small scale parameters. Topics will include: regular and singular perturbations, boundary layer theory, multiscale and averaging methods and asymptotic expansions of integrals. Along the way, we will discuss many applications including nonlinear waves, coupled oscillators, nonlinear optics, fluid dynamics and pattern formation.

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APMA 1930P. Mathematics and Climate. APMA 1940E. Mathematical Biology. The study of Earth’s climate involves many scientific components; This course is designed for undergraduate students in mathematics who mathematical tools play an important role in relating these through have an interest in the life sciences. No biological experience is necessary, quantitative models, computational experiments and data analysis. The as we begin by a review of the relevant topics. We then examine a number course aims to introduce students in applied mathematics to several of of case studies where mathematical tools have been successfully applied the conceptual models, the underlying physical principles and some of the to biological systems. Mathematical subjects include differential equations, ways data is analyzed and incorporated. Students will develop individual topology and geometry. projects later in the semester. Prerequisites: APMA 0360, or APMA 0340, APMA 1940F. Mathematics of Physical Plasmas. or written permission; APMA 1650 is recommended. Plasmas can be big, as in the solar wind, or small, as in fluorescent bulbs. APMA 1930U. Introduction to Stochastic Differential Equations. Both kinds are described by the same mathematics. Similar mathematics This seminar course serves as an introduction to stochastic differential describes semiconducting materials, the movement of galaxies, and equations at the senior undergraduate level. Topics covered include the re-entry of satellites. We consider how all of these physical systems Brownian motion and white noise, stochastic integrals, the are described by certain partial differential equations. Then we invoke Itô calculus, existence and uniqueness of solutions to Itô stochastic the power of mathematics. The course is primarily mathematical. differential equations, and the Feynman-Kac formula. More advanced Prerequisites: APMA 0340 or 0360, MATH 0180 or 0200 or 0350, and topics, such as fractional Brownian motion, Lévy processes, and PHYS 0060 or PHYS 0080 or ENGN 0510. stochastic control theory, may be addressed depending on the interests of APMA 1940G. Multigrid Methods. the class and time restrictions. Mulitgrid methods are a very active area of research in Applied APMA 1930V. Randomized Algorithms for Counting, Integration and Mathematics. An introduction to these techniques will expose the student Optimization. to cutting-edge mathematics and perhaps pique further interest in the field We will investigate Monte Carlo methods for computing expected values, of scientific computation. evaluating , and counting in discrete spaces. The emphasis APMA 1940H. Numerical Linear Algebra. is on the (common) problem setting where straightforward Monte This course will deal with advanced concepts in numerical linear algebra. Carlo is inadequate, and we must therefore resort to more complicated Among the topics covered: Singular Value Decompositions (SVD) QR (and mathematically more interesting) methods such as importance factorization, Conditioning and Stability and Iterative Methods. sampling, particle splitting, Monte Carlo and systems of interacting particles. Applications include counting the solutions for APMA 1940I. The Mathematics of Finance. combinatorial problems, rare event estimation and approximation of The mathematics of speculation as reflected in the securities and stationary distributions for random processes commodities markets. Particular emphasis placed on the evaluation of Fall APMA1930VS01 16098 TTh 10:30-11:50(13) (P. Dupuis) risk and its role in decision-making under uncertainty. Prerequisite: basic probability. APMA 1940A. Coding and Information Theory. In a host of applications, from satellite communication to compact disc APMA 1940J. The Mathematics of Speculation. technology, the storage, retrieval, and transmission of digital data relies The course will deal with the mathematics of speculation as reflected upon the theory of coding and information for efficient and error-free in the securities and commodities markets. Particular emphasis will be performance. This course is about choosing representations that minimize placed on the evaluation of risk and its role in decision making under the amount of data (compression) and the probability of an error in data uncertainty. Prerequisite: basic probability. handling (error-correcting codes). Prerequisite: A knowledge of basic APMA 1940K. Fluid Dynamics and Physical Oceanography. probability theory at the level of APMA 1650 or MATH 1610. Introduction to fluid dynamics as applied to the mathematical modeling and APMA 1940B. Information and Coding Theory. simulation of ocean dynamics and near-shore processes. Oceanography Originally developed by C.E. Shannon in the 1940s for describing bounds topics include: overview of atmospheric and thermal forcing of the oceans, on information rates across telecommunication channels, information ocean circulation, effects of topography and Earth's rotation, wind-driven and coding theory is now employed in a large number of disciplines for currents in upper ocean, coastal upwelling, the Gulf Stream, tidal flows, modeling and analysis of problems that are statistical in nature. This wave propagation, tsunamis. course provides a general introduction to the field. Main topics include APMA 1940L. Mathematical Models in Biophysics. entropy, error correcting codes, source coding, data compression. Of Development mathematical descriptions of biological systems aid in special interest will be the connection to problems in pattern recognition. understanding cell function and physiology. The course will explore a Includes a number of projects relevant to neuroscience, cognitive and range of topics including: biomechanics of blood flow in arteries and linguistic sciences, and computer vision. Prerequisites: High school capillaries, motile cells and chemotaxis, cell signaling and quorum algebra, calculus. MATLAB or other computer experience helpful. Prior sensing, and additional topics. Formulating and using numerical exposure to probability theory/statistics helpful. simulations will be a further component. Students will develop individual APMA 1940C. Introduction to Mathematics of Fluids. projects. Prerequisites: APMA 0360, or APMA 0340, or written permission. Equations that arise from the description of fluid motion are born in APMA 1940M. The History of Mathematics. physics, yet are interesting from a more mathematical point of view as The course will not be a systematic survey but will focus on specific well. Selected topics from fluid dynamics introduce various problems topics in the history of mathematics such as Archimedes and integration. and techniques in the analysis of partial differential equations. Possible Oresme and graphing, Newton and infinitesimals, simple harmonic topics include stability, existence and uniqueness of solutions, variational motion, the discovery of 'Fourier' series, the Monte Carlo method, reading problems, and active scalar equations. No prior knowledge of fluid and analyzing the original texts. A basic knowledge of calculus will be dynamics is necessary. assumed. APMA 1940D. Iterative Methods. Large, sparse systems of equations arise in many areas of mathematical application and in this course we explore the popular numerical solution techniques being used to efficiently solve these problems. Throughout the course we will study preconditioning strategies, Krylov subspace acceleration methods, and other projection methods. In particular, we will develop a working knowledge of the Conjugate Gradient and Minimum Residual (and Generalized Minimum Residual) algorithms. Multigrid and Domain Decomposition Methods will also be studied as well as parallel implementation, if time permits.

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APMA 1940N. Introduction to Mathematical Models in Computational APMA 1941D. Pattern Theory. Biology. This course is an introduction to some probabilistic models and numerical This course is designed to introduce students to the use of mathematical algorithms that model some aspects of human cognition. The class models in biology as well as some more recent topics in computational begins with stochastic models of language introduced by Shannon and biology. Mathematical techniques will involve difference equations and develops related models for speech and vision. The classes stresses dynamical systems theory, ordinary differential equations and some mathematical foundations, in particular the role of information theory in partial differential equations. These techinques will be applied in the developing Bayesian models and the increasing importance of dynamics study of many biological applications such as: (i) Difference Equations: in several algorithms, especially in optimization and deep learning. Student population dynamics, red blood cell production, population genetics; (ii) assessment will be based on computational projects that implement the Ordinary Differential Equations: predator/prey models, Lotka/Volterra principles discussed in lecture. model, modeling the evolution of the genome, heart beat model/cycle, Spr APMA1941DS01 24787 TTh 10:30-11:50(09) (G. Menon) tranmission dynamics of HIV and gonorrhea; (iii) Partial Differential Equations: tumor growth, modeling evolution of the genome, pattern APMA 1970. Independent Study. formation. Prerequisites: APMA 0330 and 0340. Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course. APMA 1940O. Approaches to Problem Solving in Applied . Mathematics. APMA 1971. Independent Study - WRIT The aim of the course is to illustrate through the examination of unsolved Section numbers vary by instructor. Please check Banner for the correct (but elementary) problems the ways in which professional applied section number and CRN to use when registering for this course. This mathematicians approach the solution of such questions. Ideas considered course should be taken in place of APMA 1970 if it is to be used to satisfy include: choosing the "simplest" nontrivial example; generalization; and the WRIT requirement. specification. Ways to think outside convention. Some knowledge of APMA 2050. Mathematical Methods of Applied Science. probability and linear algebra helpful. Introduces science and engineering graduate students to a variety of Suggested reading. fundamental mathematical methods. Topics include linear algebra, "How to solve it", G. Polya complex variables, Fourier series, Fourier and Laplace transforms and "Nonplussed", Julian Havil their applications, ordinary differential equations, tensors, curvilinear coordinates, partial differential equations, and calculus of variations. APMA 1940P. Biodynamics of Block Flow and Cell Locomotion. APMA 2060. Mathematical Methods of Applied Science. APMA 1940Q. Filtering Theory. Introduces science and engineering graduate students to a variety of Filtering (estimation of a "state process" from noisy data) is an important fundamental mathematical methods. Topics include linear algebra, area of modern statistics. It is of central importance in navigation, signal complex variables, Fourier series, Fourier and Laplace transforms and and image processing, control theory and other areas of engineering and their applications, ordinary differential equations, tensors, curvilinear science. Filtering is one of the exemplary areas where the application coordinates, partial differential equations, and calculus of variations. of modern mathematics and statistics leads to substantial advances in engineering. This course will provide a student with the working knowledge APMA 2080. Inference in Genomics and Molecular Biology. sufficient for cutting edge research in the field of nonlinear filtering and its Massive quantities of fundamental biological and geological sequence practical applications. Topics will include: hidden Markov models, Kalman data have emerged. The goal of this course is to enable students to and Wiener filters, optimal nonlinear filtering, elements of Ito calculus construct and apply probabilistic models to draw inferences from sequence and Wiener chaos, Zakai and Kushner equations, spectral separating data on problems novel to them. Statistical topics: Bayesian inferences; filters and wavelet based filters, numerical implementation of filters. We estimation; hypothesis testing and false discovery rates; statistical decision will consider numerous applications of filtering to speech recognition, theory; change point algorithm; hidden Markov models; Kalman filters; and analysis of financial data, target tracking and image processing. No prior significances in high dimensions. Prerequisites: APMA 1650 or equivalent; knowledge in the field is required but a good understanding of the basic APMA 0160 or CSCI 0111 or equivalent. Probability Theory (APMA1200 or APMA2630) is important. Fall APMA2080 S01 16096 MWF 1:00-1:50(06) (M. Harrison) APMA 1940R. Linear and Nonlinear Waves. APMA 2110. Real Analysis. From sound and light waves to water waves and traffic jams, wave Provides the basis of real analysis which is fundamental to many of the phenomena are everywhere around us. In this seminar, we will discuss other courses in the program: metric spaces, measure theory, and the linear and nonlinear waves as well as the propagation of wave packets. theory of integration and differentiation. Among the tools we shall use and learn about are numerical simulations APMA 2110A. Real Function Theory (MATH 2210). in Matlab and analytical techniques from ordinary and partial differential Interested students must register for MATH 2210. equations. We will also explore applications in nonlinear optics and to traffic flow problems. Prerequisites: MATH 0180 and either APMA APMA 2120. Hilbert Spaces and Their Applications. 0330-0340 or APMA 0350-0360. No background in partial differential A continuation of APMA 2110: metric spaces, Banach spaces, Hilbert equations is required. spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential APMA 1940Y. Wavelets and Applications. equations. The aim of the course is to introduce you to: the relatively new and interdisciplinary area of wavelets; the efficient and elegants algorithms to APMA 2120A. Real Function Theory (MATH 2210). which they give rise including the wavelet transform; and the mathematical Interested students must register for MATH 2220. tools that can be used to gain a rigorous understanding of wavelets. APMA 2130. Methods of Applied Mathematics: Partial Differential We will also cover some of the applications of these tools including Equations. the compression of video streams, approximation of solution of partial Solution methods and basic theory for first and second order partial differential equations, and signal analysis. differential equations. Geometrical interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.

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APMA 2140. Methods of Applied Mathematics: Integral Equations. APMA 2410. Fluid Dynamics I. Integral equations. Fredholm and Volterra theory, expansions in orthogonal Formulation of the basic conservation laws for a viscous, heat conducting, functions, theory of Hilbert-Schmidt. Singular integral equations, method of compressible fluid. Molecular basis for thermodynamic and transport Wiener-Hopf. Calculus of variations and direct methods. properties. Kinematics of vorticity and its transport and diffusion. Introduction to potential flow theory. Viscous flow theory; the application of APMA 2160. Methods of Applied Mathematics: Asymptotics. Calculus of asymptotic expansions, evaluation of integrals. Solution of dimensional analysis and scaling to obtain low and high Reynolds number linear ordinary differential equations in the complex plane, WKB method, limits. special functions. APMA 2420. Fluid Mechanics II. Introduction to concepts basic to current fluid mechanics research: APMA 2170. Functional Analysis and Applications. Topics vary according to interest of instructor and class. hydrodynamic stability, the concept of average fluid mechanics, introduction to turbulence and to multiphase flow, wave motion, and topics APMA 2190. Nonlinear Dynamical Systems I. in inviscid and compressible flow. Basic theory of ordinary differential equations, flows, and maps. Two- dimensional systems. Linear systems. Hamiltonian and integrable APMA 2450. Exchange Scholar Program. systems. Lyapunov functions and stability. Invariant manifolds, including Fall APMA2450 S01 15671 Arranged ’To Be Arranged' stable, unstable, and center manifolds. Bifurcation theory and normal Spr APMA2450 S01 24557 Arranged ’To Be Arranged' forms. Nonlinear oscillations and the method of averaging. Chaotic motion, APMA 2470. Topics in Fluid Dynamics. including horseshoe maps and the Melnikov method. Applications in the Initial review of topics selected from flow stability, turbulence, turbulent physical and biological sciences. mixing, surface tension effects, and thermal convection. Followed by Fall APMA2190 S01 16033 TTh 2:30-3:50(12) (J. Mallet-Paret) focussed attention on the dynamics of dispersed two-phase flow and APMA 2200. Nonlinear Dynamical Systems: Theory and Applications. complex fluids. Basic theory of ordinary differential equations, flows, and maps. Two- APMA 2480. Topics in Fluid Dynamics. dimensional systems. Linear systems. Hamiltonian and integrable No description available. systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal APMA 2550. Numerical Solution of Partial Differential Equations I. forms. Nonlinear oscillations and the method of averaging. Chaotic motion, Finite difference methods for solving time-dependent initial value problems including horseshoe maps and the Melnikov method. Applications in the of partial differential equations. Fundamental concepts of consistency, physical and biological sciences. accuracy, stability and convergence of finite difference methods will be covered. Associated well-posedness theory for linear time-dependent Spr APMA2200 S01 24704 TTh 2:30-3:50(11) (J. Mallet-Paret) PDEs will also be covered. Some knowledge of computer programming APMA 2210. Topics in Nonlinear Dynamical Systems. expected. Topics to be covered in this course may vary depending on the audiences. Fall APMA2550 S01 15966 W 3:00-5:30(10) (J. Guzman) One of the goals that is planned for this course is to discuss the boundary layers and/or the boundary value problems that appear and play a very APMA 2560. Numerical Solution of Partial Differential Equations II. important role in the kinetic theory of gases; in particular, in the theory An introduction to weighted residual methods, specifically spectral, of the Boltzmann equations. Students are encouraged to attend and finite element and spectral element methods. Topics include a review of participate in the kinetic theory program offered by the ICERM institute in variational calculus, the Rayleigh-Ritz method, approximation properties of the Fall 2011 semester. This course may be taken twice for credit. spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems. APMA 2230. Partial Differential Equations I. Spr APMA2560 S01 24706 W 3:00-5:30(10) (M. Ainsworth) The theory of the classical partial differential equations, as well as the method of characteristics and general first order theory. Basic analytic APMA 2570A. Numerical Solution of Partial Differential Equations III. tools include the Fourier transform, the theory of distributions, Sobolev We will cover spectral methods for partial differential equations. spaces, and techniques of harmonic and functional analysis. More general Algorithm formulation, analysis, and efficient implementation issues will linear and nonlinear elliptic, hyperbolic, and parabolic equations and be addressed. Prerequisite: APMA 2550 or equivalent knowledge in properties of their solutions, with examples drawn from physics, differential numerical methods. geometry, and the applied sciences. Generally, semester II of this course Fall APMA2570AS01 15967 M 3:00-5:30(03) (M. Ainsworth) concentrates in depth on several special topics chosen by the instructor. APMA 2570B. Numerical Solution of Partial Differential Equations III. Fall APMA2230 S01 16034 MWF 10:00-10:50(14) (C. Dafermos) We will cover finite element methods for ordinary differential equations APMA 2240. Partial Differential Equations. and for elliptic, parabolic and hyperbolic partial differential equations. The theory of the classical partial differential equations, as well as the Algorithm development, analysis, and computer implementation issues method of characteristics and general first order theory. Basic analytic will be addressed. In particular, we will discuss in depth the discontinuous tools include the Fourier transform, the theory of distributions, Sobolev Galerkin finite element method. Prerequisite: APMA 2550 or equivalent spaces, and techniques of harmonic and functional analysis. More general knowledge in numerical methods. linear and nonlinear elliptic, hyperbolic, and parabolic equations and APMA 2580A. Computational Fluid Dynamics. properties of their solutions, with examples drawn from physics, differential The course will focus primarily on finite difference methods for viscous geometry, and the applied sciences. Generally, semester II of this course incompressible flows. Other topics will include multiscale methods, e.g. concentrates in depth on several special topics chosen by the instructor. molecular dynamics, dissipative particle dynamics and lattice Boltzmann Spr APMA2240 S01 24705 MWF 10:00-10:50(03) (C. Dafermos) methods. We will start with the mathematical nature of the Navier- Stokes equations and their simplified models, learn about high-order APMA 2260. Introduction to Stochastic Control Theory. The course serves as an introduction to the theory of stochastic control explicit and implicit methods, time stepping, and fast solvers. We will and dynamic programming technique. Optimal stopping, total expected then cover advection-diffusion equations and various forms of the Navier- (discounted) cost problems, and long-run average cost problems will be Stokes equations in primitive variables and in vorticity/streamfunction discussed in discrete time setting. The last part of the course deals with formulations. In addition to the homeworks the students are required to continuous time determinstic control and game problems. The course develop a Navier-Stokes solver as a final project. requires some familiarity with the probability theory.

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APMA 2580B. Computational Fluid Dynamics for Compressible APMA 2670. Mathematical Statistics I. Flows. This course presents advanced statistical inference methods. Topics An introduction to computational fluid dynamics with emphasis on include: foundations of statistical inference and comparison of classical, compressible flows. We will cover finite difference, finite volume and finite Bayesian, and minimax approaches, point and set estimation, hypothesis element methods for compressible Euler and Navier-Stokes equations testing, linear regression, linear classification and principal component and for general hyperbolic conservation laws. Background material in analysis, MRF, consistency and asymptotic normality of Maximum hyperbolic partial differential equations will also be covered. Algorithm Likelihood and estimators, statistical inference from noisy or degraded development, analysis, implementation and application issues will data, and computational methods (E-M Algorithm, Markov Chain Monte be addressed. Prerequisite: APMA 2550 or equivalent knowledge in Carlo, Bootstrap). Prerequisite: APMA 2630 or equivalent. numerical methods. APMA 2680. Mathematical Statistics II. APMA 2580C. Optimization. The course covers modern nonparametric statistical methods. Topics This course provides a solid presentation of modern convex analysis include: density estimation, multiple regression, adaptive smoothing, and convex optimization algorithms for large scale problems. Topics cross-validation, bootstrap, classification and regression trees, nonlinear include: Ekeland's variational principle, subdifferential calculus, duality and discriminant analysis, projection pursuit, the ACE algorithm for time series Fenchel-Legendre transform, projection on a closed convex set, proximal prediction, support vector machines, and neural networks. The course will operators and Moreau's regularization, accelerated and optimal first-order provide the mathematical underpinnings, but it will also touch upon some methods, Hopf/Lax-Oleinik representation formulas for Hamilton-Jacobi applications in computer vision/speech recognition, and biological, neural, equations. The course will provide the mathematical and algorithmic and cognitive sciences. Prerequisite: APMA 2670. underpinnings. It will also explore some applications in signal and image APMA 2720. Information Theory. processing, optimal control and machine learning. Information theory and its relationship with probability, statistics, APMA 2610. Recent Applications of Probability and Statistics. and data compression. Entropy. The Shannon-McMillan-Breiman This course develops the mathematical foundations of modern theorem. Shannon's source coding theorems. Statistical inference; applications of statistics to the computational, cognitive, engineering, hypothesis testing; model selection; the minimum description length and neural sciences. The course is rigorous, but the emphasis is on principle. Information-theoretic proofs of limit theorems in probability: application. Topics include: Gibbs ensembles and their relation to Law of large numbers, central limit theorem, large deviations, Markov maximum entropy, large deviations, exponential models and information chain convergence, Poisson approximation, Hewitt-Savage 0-1 law. theory; statistical estimation and classification; graphical models, dynamic Prerequisites: APMA 2630, 1710. programming, MCMC, parameter estimation, and the EM algorithm. APMA 2810A. Computational Biology. Graduate version: 2610; Undergraduate version: 1740. Prerequisites: Provides an up-to-date presentation of the main problems and algorithms APMA 1650 or equivalent, programming experience, strong mathematics in bioinformatics. Emphasis is given to statistical/ probabilistic methods background. APMA 1200 or APMA 1690 or similar courses recommended. for various molecular biology tasks, including: comparison of genomes of MATH 1010 or equivalent is recommended for APMA 2610. different species, finding genes and motifs, understanding transcription Spr APMA2610 S01 24707 MWF 1:00-1:50(06) (S. Geman) control mechanisms, analyzing microarray data for gene clustering, and APMA 2630. Theory of Probability I. predicting RNA structure. Part one of a two semester course that provides an introduction to APMA 2810B. Computational Molecular Biology. probability theory based on measure theory. The first semester (APMA Provides an up-to-date presentation of problems and algorithms in 2630) covers the following topics: countable state Markov chains, review bioinformatics, beginning with an introduction to biochemistry and of real analysis and metric spaces, probability spaces, random variables molecular genetics. Topics include: proteins and nucleic acids, the genetic and measurable functions, Borel-Cantelli lemmas, weak and strong laws code, the central dogma, the genome, gene expression, metabolic of large numbers, conditional expectation and beginning of discrete time transformations, and experimental methods (gel electrophoresis, X-ray martingale theory. Prerequisites—undergraduate probability and analysis, crystallography, NMR). Also, algorithms for DNA sequence alignment, co-requisite—graduate real analysis. database search tools (BLAST), and DNA sequencing. Fall APMA2630 S01 15990 TTh 1:00-2:20(08) (O. Nguyen) APMA 2810C. Elements of High Performance Scientific Computing. APMA 2640. Theory of Probability II. No description available. Part two of a two semester course that provides an introduction to probability theory based on measure theory. Standard topics covered APMA 2810D. Elements of High Performance Scientific Computing II. in the second-semester (APMA 2640) include the following: discrete No description available. time martingale theory, weak convergence (also called convergence in APMA 2810E. Far Field Boundary Conditions for Hyperbolic distribution) and the central limit theorem, and a study of Brownian motion. Equations. Optional topics include the ergodic theorem and large deviation theory. No description available. Prerequisites—undergraduate probability and analysis, co-requisite— graduate real analysis. APMA 2810F. Introduction to Non-linear Optics. No description available. Spr APMA2640 S01 24708 TTh 1:00-2:20(08) (P. Dupuis) APMA 2810G. Large Deviations. APMA 2660. Stochastic Processes. No description available. Review of the theory of stochastic differential equations and reflected SDEs, and of the ergodic and stability theory of these processes. APMA 2810H. Math of Finance. Introduction to the theory of weak convergence of probability measures No description available. and processes. Concentrates on applications to the probabilistic modeling, APMA 2810I. Mathematical Models and Numerical Analysis in control, and approximation of modern communications and queuing Computational Quantum Chemistry. networks; emphasizes the basic methods, which are fundamental tools We shall present on some models in the quantum chemistry field (Thomas throughout applications of probability. Fermi and related, Hartree Fock, Kohn Sham) the basic tools of functional analysis for the study of their solutions. Then some of the discretization methods and iterative algorithms to solve these problems will be presented and analyzed. Some of the open problems that flourish in this field will also be presented all along the lectures. APMA 2810J. Mathematical Techniques for Neural Modeling. No description available.

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APMA 2810K. Methods of Algebraic Geometry in Control Theory I. APMA 2810W. Advanced Topics in High Order Numerical Methods for Develops the ideas of algebraic geometry in the context of control theory. Convection Dominated Problems. The first semester examines scalar linear systems and affine algebraic This is an advanced seminar course. We will cover several topics in high geometry while the second semester addresses multivariable linear order numercial methods for convection dominated problems, including systems and projective algebraic geometry. methods for solving Boltzman type equations, methods for solving unsteady and steady Hamilton-Jacobi equations, and methods for solving APMA 2810L. Numerical Solution of Hyperbolic PDE's. No description available. moment models in semi-conductor device simulations. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis. APMA 2810M. Some Topics in Kinetic Theory. Nonlinear instabilities as well as boundary effects in a collisionless APMA 2810X. Introduction to the Theory of Large Deviations. plasmas; Stable galaxy configurations; A nonlinear energy method in The theory of large deviations attempts to estimate the probability of the Boltzmann theory will also be introduced. Self-contained solutions to rare events and identify the most likely way they happen. The course specific concrete problems. Focus on ideas but not on technical aspects. will begin with a review of the general framework, standard techniques Open problems and possible future research directions will then be (change-of-measure, PDE, weak convergence, etc.), and basic examples (e.g., Sanov's and Cramer's Theorems). We then will cover the Wentsel- discussed so that students can gain a broader perspective. Prerequisite: One semester of PDE (graduate level) is required. Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control and Monte APMA 2810N. Topics in Nonlinear PDEs. Carlo methods. Prerequisites: APMA 2630 and 2640. Aspects of the theory on nonlinear evolution equations, which includes kinetic theory, nonlinear wave equations, variational problems, and APMA 2810Y. Discrete high-D Inferences in Genomics. dynamical stability. Genomics is revolutionizing biology and biomedicine and generated a mass of clearly relevant high-D data along with many important high-D APMA 2810O. Stochastic Differential Equations. discreet inference problems. Topics: special characteristics of discrete This course develops the theory and some applications of stochastic high-D inference including Bayesian posterior inference; point estimation; differential equations. Topics include: stochastic integral with respect interval estimation; hypothesis tests; model selection; and statistical to Brownian motion, existence and uniqueness for solutions of SDEs, decision theory. Markov property of solutions, sample path properties, Girsanov's Theorem, weak existence and uniqueness, and connections with partial differential APMA 2810Z. An Introduction to the Theory of Large Deviations. equations. Possible additional topics include stochastic stability, reflected The theory of large deviations attempts to estimate the probability of rare diffusions, numerical approximation, and stochastic control. Prerequisite: events and identify the most likely way they happen. The course will begin APMA 2630 and 2640. with a review of the general framework, standard techniques (change- of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's APMA 2810P. Perturbation Methods. and Cramer's Theorems). We then will cover large deviations for diffusion Basic concepts of asymptotic approximations with examples with processes and the Wentsel-Freidlin theory. The last part of the course will examples such as evaluation of integrals and functions. Regular and be one or two related topics, possibly drawn from (but not limited to) risk- singular perturbation problems for differential equations arising in fluid sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman mechanics, wave propagation or nonlinear oscillators. Methods include equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640. matched asymptotic expansions and multiple scales. Methods and results will be discussed in the context of applications to physical problems. APMA 2811A. Directed Methods in Control and System Theory. Various general techniques have been developed for control and system APMA 2810Q. Discontinous Galerkin Methods. problems. Many of the methods are indirect. For example, control In this seminar course we will cover the algorithm formulation, stability problems are reduced to a problem involving a differential equation analysis and error estimates, and implementation and applications of (such as the partial differential equation of Dynamic Programming) or to discontinuous Galerkin finite element methods for solving hyperbolic a system of differential equations (such as the canonical system of the conservation laws, convection diffusion equations, dispersive wave Maximum Principle). Since these indirect methods are not always effective equations, and other linear and nonlinear partial differential equations. alternative approaches are necessary. In particular, direct methods are of Prerequisite: APMA 2550. interest. APMA 2810R. Computational Biology Methods for Gene/Protein We deal with two general classes, namely: 1.) Integration Methods; and, Networks and Structural Proteomics. 2.) Representation Methods. Integration methods deal with the integration The course presents computational and statistical methods for gene of function space differential equations. Perhaps the most familiar is and protein networks and structural proteomics; it emphasizes: (1) the so-called Gradient Method or curve of steepest descent approach. Probablistic models for gene regulatory networks via microarray, chromatin Representation methods utilize approximation in function spaces and immunoprecipitation, and cis-regulatory data; (2) Signal transduction include both deterministic and stochastic finite element methods. Our pathways via tandem mass spectrometry data; (3) Molecular Modeling concentration will be on the theoretical development and less on specific forligand-receptor coupling and docking. The course is recommended for numerical procedures. The material on representation methods for Levy graduate students. processes is new. APMA 2810S. Topics in Control. APMA 2811B. Computational Methods for Signaling Pathways and No description available. Protein Interactions. The course will provide presentation of the biology and mathematical APMA 2810T. Nonlinear Partial Differential Equations. This course introduces techniques useful for solving many nonlinear partial models/algorithms for a variety of topics, including: (1) The analysis differential equations, with emphasis on elliptic problems. PDE from a and interpretation of tandem mass spectrometry data for protein variety of applications will be discussed. Contact the instructor about identification and determination of signaling pathways, (2) Identification of prerequisites. Phosphorylation sites and motifs and structural aspects of protein docking problems. Prerequisites: The course is recommended for graduate APMA 2810U. Topics in Differnetial Equations. students. It will be self-contained; students will be able to fill in knowledge No description available. by reading material to be indicated by the instructor. APMA 2810V. Topics in Partial Differential Equations. The course will cover an introduction of the L_p theory of second order elliptic and parabolic equations, finite difference approximations of elliptic and parabolic equations, and some recent developments in the Navier- Stokes equations and quasi-geostrophic equations. Some knowledge of real analysis will be expected.

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APMA 2811C. Stochastic Partial Differential Equations. APMA 2811H. Survival Analysis. SPDEs is an interdisciplinary area at the crossroads of stochastic APMA 2811I. An Introduction to Turbulence Modeling. processes and partial differential equations (PDEs). The topics of the Turbulence is the last mystery of classical physics. It surrounds us course include: geneses of SPDEs in real life applications, mathematical everywhere – in the air, in the ocean, in pipes carrying fluids and foundations and analysis of SPDEs, numerical and computational aspects even in human body arteries. The course helps to understand what of SPDEs, applications of SPDEs to fluid dynamics, population biology, makes modeling the turbulence so difficult and challenging. The course hidden Markov models, etc. Prerequisites: familiarity with stochastic covers the following issues: The nature of turbulence, characteristics calculus and PDEs (graduate level). of turbulence and classical constants of turbulence; Turbulent scales; APMA 2811D. Asymptotic Problems For Differential Equations And Navier-Stokes equations, Reynolds stresses and Reynolds-Averaged Stochastic Processes. Navier-Stokes (RANS) equations; RANS turbulence models: algebraic Topics that will be covered include: WKB method: zeroth and first models, one-equation models, two-equation models; Low-Reynolds orders; turning points; Perturbation theory: regular perturbation, singular number turbulence models; Renormalization Group (RNG) turbulence perturbation and boundary layers; Homogenization methods for ODE's, model; Large-Eddy Simulation (LES); Students will be provided with user- elliptic and parabolic PDE's; Homogenization for SDE's, diffusion friendly computer codes to run different benchmark cases. The final grade processes in periodic and random media; Averaging principle for ODE's is based on two take home projects - computing or published papers and SDE's. Applications will be discussed in class and in homework analysis, optionally. problems. APMA 2811K. Computational/Statistical Methods for Signaling APMA 2811E. A Posteriori Estimates for Finite Element Methods. Pathways and Protein Interactions. This course gives an introduction to the the basic concepts of a posteriori The course will cover the main mathematical/computational models/ estimates of finite element methods. After an overview of different algorithms for a variety of tasks in proteomics and structural proteomics, techniques the main focus will be shed on residual based estimates including: (1) The analysis and interpretation of tandem mass where as a starting point the Laplace operator is analyzed. Effectivity spectrometry data for protein identification and determination of signaling and reliability of the error estimator will be proven. In a second part of the pathways, (2) Identification of Phosphorylation sites and motifs, and (3) course, students will either study research articles and present them or structural aspects of protein docking problems on the basis of NMR data. implement the error estimates for some specific problem and present their Open to graduate students only. numerical results. Recommended prerequisites: basic knowledge in finite APMA 2811L. Topics in Homogenization: Theory and Computation. elements, APMA 2550, 2560, 2570. Topics that will be covered include: Homogenization methods for ODE's, APMA 2811F. Numerical Solution of Ordinary Differential Equations: for elliptic and parabolic PDE's and for stochastic differential Equations IVP Problems and PDE Related Issues. (SDE's) in both periodic and random media; Averaging principle for ODE's The purpose of the course is to lay the foundation for the development and SDE's. Both theoretical and computational aspects will be studied. and analysis of numerical methods for solving systems of ordinary Applications will be discussed in class and in homework problems. differential equations. With a dual emphasis on analysis and efficient Prerequisites: Some background in PDE's and probability will be helpful, implementations, we shall develop the theory for multistage methods even though the class will be largely self contained. (Runge-Kutta type) and multi-step methods (Adams/BDF methods). APMA 2811O. Dynamics and Stochastics. We shall also discuss efficient implementation strategies using Newton- This course provides a synthesis of mathematical problems at the type methods and hybrid techniques such as Rosenbruck methods. The interface between stochastic problems and dynamical systems that arise discussion includes definitions of different notions of stability, stiffness in systems biology. For instance, in some biological systems some species and stability regions, global/local error estimation, and error control. may be modeled stochastically while other species can be modeled using Time permitting, we shall also discuss more specialized topics such deterministic dynamics. Topics will include an introduction to biological as symplectic integration methods and parallel-in-time methods. A key networks, multiscale analysis, analysis of network structure, among other component of the course shall be the discussion of problems and methods topics. Prerequisites: probability theory (APMA 2630/2640, concurrent designed with the discretization of ODE systems originating from PDE's in enrollment in APMA 2640 is acceptable). mind. Topics include splitting methods, methods for differential-algebraic equations (DAE),deferred correction methods. and order reduction APMA 2811Q. Calculus of Variations. problems for IBVP, TVD and IMEX methods. Part of the class will consist An introduction to modern techniques in the calculus of variations. of student presentations on more advanced topics, summarizing properties Topics covered will include: existence of solutions and the direct method, and known results based on reading journal papers. Euler-Lagrange equations and necessary and sufficient conditions, one- dimensional problems, multidimensional nonconvex problems, relaxation APMA 2811G. Topics in Averaging and Metastability with and quasiconvexity, Young's measures, and singular perturbations. The Applications. emphasis of the course will be equal parts theory and applications with Topics that will be covered include: the averaging principle for stochastic numerous examples drawn from topics in nonlinear elasticity, pattern dynamical systems and in particular for Hamiltonian systems; metastability formation, wrinkling thin elastic sheets, martensitic phase transitions, and stochastic resonance. We will also discuss applications in class minimal surfaces, differential geometry and optimal control. and in homework problems. In particular we will consider metastability issues arising in chemistry and biology, e.g. in the dynamical behavior APMA 2811S. Levy Processes. of proteins. The course will be largely self contained, but a course in Lévy processes are the continuous-time analogues of random walks, and graduate probability theory and/or stochastic calculus will definitely help. include Brownian motion, compound Poisson processes, and square- integrable pure-jump martingales with many small jumps. In this course we will develop the basic theory of general Lévy processes and subordinators, and discuss topics including local time, excursions, and fluctuations. Time permitting we will finish with selected applications which are of mutual interest to the instructor and students enrolled in the class. Prerequisite: APMA 2640 or equivalent.

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APMA 2811Z. Stochastic Partial Differential Equations: Theory and APMA 2820H. Kinetic Theory. Numerics. We will focus on two main topics in mathematical study of the kinetic This course introduces basic theory and numerics of stochastic partial theory: (1) The new goal method to study the trend to Maxwellians; differential equations (SPDEs). Topics include Brownian motion and (2) various hydrodynamical (fluids) limits to Euler and Navier-Stokes stochastic calculus in Hilbert spaces, classification of SPDEs and equations. Main emphasis will be on the ideas behind proofs, but not on solutions, stochastic elliptic, hyperbolic and parabolic equations, regularity technical details. of solutions, linear and nonlinear equations, analytic and numerical APMA 2820I. Multiscale Methods and Computer Vision. methods for SPDEs. Topics of particular interest will also be discussed Course will address some basic multiscale computational methods such upon agreements between the instructor and audience. as: multigrid solvers for physical systems, including both geometric and All three courses APMA 2630, APMA 2640, APMA 2550 are background algebraic multigrid, fast integral transforms of various kinds (including recommended but not required. a fast Radon transform), and fast inverse integral transforms. Basic APMA 2812B. An Introduction to SPDE's. problems in computer vision such as global contour detection and their An introduction to the basic theory of Stochastic PDE's. Topics will likely completion over gaps, image segmentation for textural images and include (time permitting) Gaussian measure theory, stochastic integration, perceptual grouping tasks in general will be explained in more details. stochastic convolutions, stochastic evolution equations in Hilbert spaces, APMA 2820J. Numerical Linear Algebra. Ito's formula, local well-posedness for semi-linear SPDE with additive Solving large systems of linear equations: The course will use the text of noise, weak Martingale solutions to 3D Navier-Stokes, Markov processes Treften and BAO that includes all the modern concepts of solving linear on Polish spaces, the Krylov–Bogolyubov theorem, the Doob-Khasminskii equations. theorem, and Bismut-Elworthy-Li formula for a class of non-degenerate SPDE. The presentation will be largely self contained, but will assume APMA 2820K. Numerical Solution of Ordinary Differential Equations. some basic knowledge in measure theory, functional analysis, and We discuss the construction and general theory of multistep and probability theory. Some familiarity with SDE and PDE is also very helpful, multistage methods for numerically solving systems of ODE's, including but not required. stiff and nonlinear problems. Different notions to stability and error estimation and control. As time permits we shall discuss more advanced APMA 2812D. Topics in Modern Learning Theory and its Applications. topics such as order reduction, general linear and additive methods, The course will provide an up-to-date presentation of modern Learning symplectic methods, and methods for DAE. Prerequisites: APMA 2190 Theory. Its focus will be on currently forefront methods such as Neural and APMA 2550 or equivalent. Some programming experience is Nets/Deep Learning, Support Vector Machine, Boosting, Classification expected. and Regression Trees, Random Forest, and Projection Pursuit. Model complexity (VC – dimension, Rademacher complexity, effective number of APMA 2820L. Random Processes in Mechanics. degrees of freedom), and issues related to the curse of dimensionality will No description available. be central parts of the course. The course will also cover some real-world APMA 2820M. Singularities in Eliptic Problems and their Treatment applications to biological and artificial intelligence problems. by High-Order Finite Element Methods. Fall APMA2812DS01 16035 Th 4:00-6:30(04) (B. Gidas) Singular solutions for elliptic problems (elasticity and heat transfer) are APMA 2812E. Semidefinite and Combinatorial Optimization.. discussed. These may arise around corners in 2-D and along edges and The course provides an introduction to basic mathematical theory vertices in 3-D domains. Derivation of singular solutions, charactized by and computational methods for semidefinite optimization with a eigenpairs and generalized stress/flux intensity factors (GSIF/GFIFs) particular focus on polynomial optimization problems. Topics include: are a major engineering importance (because of failure initiation and duality, semialgebraic sets, polynomial optimization, sum of squares, propagation). High-order FE methods are introduced, and special spectrahedron, semidefinite relaxation, combinatorial optimization. algorithms for extracting eigenpairs and GSIF/GFIFs are studied (Steklov, Prerequisites: Math 0520, 0540, or equivalent, basic programming skills. dual-function, ERR method, and others). Math 1530 is recommended. APMA 2820N. Topics in Scientific Computing. Fall APMA2812ES01 17961 MWF 2:00-2:50(01) (N. Kaihnsa) No description available. APMA 2820A. A Tutorial on Particle Methods. APMA 2820O. The Mathematics of Shape with Applications to No description available. Computer Vision. APMA 2820B. Advanced Topics in Information Theory. Methods of representing shape, the geometry of the space of shapes, Explores classical and recent results in information theory. Topics warping and matching of shapes, and some applications to problems chosen from: multi-terminal/network information theory; communication in computer vision and medical imaging. Prerequsite: See instructor for under channel uncertainty; side information problems (channel, source, prerequisites. and the duality between them); identification via channels; and multi- APMA 2820P. Foundations in Statistical Inference in Molecular antenna fading channels. Prerequisite: APMA 1710 or basic knowledge of Biology. information theory. In molecular biology, inferences in high dimensions with missing data are APMA 2820C. Computational Electromagnetics. common. A conceptual framework for Bayesian and frequentist inferrnces No description available. in this setting including: sequence alignment, RNA secondary structure prediction, database search, and functional genomics. Statistical topics: APMA 2820D. Conventional, Real and Quantum Computing with parameter estimation, hypothesis testing, and characterization of posterior Applications to Factoring and Root Finding. spaces. Core course in proposed PhD program in computational molecular No description available. biology. APMA 2820E. Geophysical Fluid Dynamics. APMA 2820Q. Topics in Kinetic Theory. No description available. This course will introduce current mathematical study for Boltzmann APMA 2820F. Information Theory and Networks. equation and Vlasov equation. We will study large time behavior and No description available. hydrodynamic limits for Boltzmann theory and instabilities in the Vlasov theory. Graduate PDE course is required. APMA 2820G. Information Theory, Statistics and Probability. No description available. APMA 2820R. Structure Theory of Control Systems. The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S' ? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

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APMA 2820S. Topics in Differential Equations. APMA 2821F. Computational Linear Algebra. A sequel to APMA 2210 concentrating on similar material. The course will cover basic and advanced algorithms for solution of linear and nonlinear systems as well as eigenvalue problems. APMA 2820T. Foundations in Statistical Inference in Molecular Biology. APMA 2821G. High-Performance Discontinuous Galerkin Solvers. In molecular biology, inferences in high dimensions with missing data are Addresses strategies and algorithms in devising efficient discontinuous common. A conceptual framework for Bayesian and frequentist inferences Galerkin solvers for fluid flow equations such as Euler and Navier-Stokes. in this setting including: sequence alignment. RNA secondary structure The course starts with an introduction to discontinuous Galerkin methods prediction, database search, and tiled arrays. Statistical topics: parameter for elliptic and hyperbolic equations and then focuses on the following estimation, hypothesis testing, recursions, and characterization of topics: 1) Serial and parallel implementations of various discontinuous posterior spaces. Core course in proposed PhD program in computational Galerkin operators for curvilinear ele- ments in multiple space dimensions. molecular biology. 2) Explicit, semi-explicit and implicit time discretizations. 3) Multigrid (multi- level) solvers and preconditioners for systems arising from discontinuous APMA 2820U. Structure Theory of Control Systems. The course deals with the following problems: given a family of control Galerkin approximations of the partial differential equations. systems S and a family of control systems S', when does there exist an APMA 2821H. Introduction to High Performance Computing: Tools appropriate embedding of S into S' ? Most of the course will deal with and Algorithms. the families of linear control systems. Knowledge of control theory and This course will cover fundamental concepts of parallel computing: shared mathematical sophistication are required. and distributed memory models; metrics for performance measuring; roof-line model for analysis of computational kernels, prediction and APMA 2820V. Progress in the Theory of Shock Waves. Course begins with self-contained introduction to theory of "hyperbolic improving their performance on different processors; code optimization. conservation laws", that is quasilinear first order systems of partial We will analyze algorithms maximizing data reuse, and memory bandwidth differential equations whose solutions spontaneously develop singularities utilization. Prior experience in coding is a plus. One course meeting will that propagate as shock waves. A number of recent developments will take place at IBM/Research, students will interact with experts in areas be discussed. Aim is to familiarize the students with current status of the of HPC, visualization, social media and more. There will be bi-weekly theory as well as with the expanding areas of applications of the subject. homework assignments and a final project. Students are encouraged to suggest final project relevant to their research and level of expertise. APMA 2820W. An Introduction to the Theory of Large Deviations. The theory of large deviations attempts to estimate the probability of rare APMA 2821I. Formulation and Approximation of Linear and Non- events and identify the most likely way they happen. The course will begin linear Problems of Solid Mechanics. with a review of the general framework, standard techniques (change- Presents the formulation and approximation by the Finite Element of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's Method (FEM) of linear and non-linear problems of Solid Mechanics. The and Cramer's Theorems). We then will cover large deviations for diffusion formulation of problems is based on the Virtual Work Principle (VWP). processes and the Wentsel¿Freidlin theory. The last part of the course Increasing complexity problems will be considered such as simple bar will be one or two related topics, possibly drawn from (but not limited to) under traction, beams, plates, plane problems and solids with linear risk¿sensitive control; weak convergence methods; Hamilton-Jacobi- and hyperelastic materials. All problems are formulated using the same Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and sequence of presentation which includes kinematics, strain measure, rigid 2640. body deformation, internal work, external work, VWP and constitutive equations. The approximation of the given problems is based on the High- APMA 2820X. Boundary Conditions for Hyperbolic Systems: order FEM. Examples will be presented using a Matlab code. Numerical and Far Field. APMA 2821J. Some Topics in Kinetic Theory. APMA 2820Y. Approaches to Problem Solving in Applied In this advanced topic course, we will go over several aspects of recent Mathematics. mathematical work on kinetic theory. Graduate level PDE is required. TBA APMA 2821K. Probabilistic and Statistical Models for Graphs and APMA 2820Z. Topics in Discontinuous Galerkin Methods. Networks. We will cover discontinuous Galerkin methods for time-dependent Many modern data sets involve observations about a network of and steady state problems. Stability and error estimates of different interacting components. Probabilistic and statistical models for graphs and discontinuous Galerkin methods will be discussed. In particular, we will networks play a central role in understanding these data sets. This is an discuss in depth the local discontinuous Galerkin method. Prerequisite: area of active research across many disciplines. Students will read and APMA 2550 or equivalent knowledge of numerical analysis. discuss primary research papers and complete a final project. APMA 2821A. Parallel Scientific Computing: Algorithms and Tools. APMA 2821L. Introduction to Malliavin Calculus. No description available. The Malliavin calculus is a stochastic calculus for random variables on APMA 2821B. To Be Determined. Gaussian probability spaces, in particular the classical Wiener space. It was originally introduced in the 1970s by the French mathematician Paul APMA 2821C. Topics in Partial Differential Equations. Malliavin as a probabilistic approach to the regularity theory of second- The course will start by reviewing the theory of elliptic and parabolic order deterministic partial differential equations. Since its introduction, equations in Holder spaces. Then we will discuss several topics in Malliavin's calculus has been extended beyond its original scope and nonlinear elliptic and parabolic equations, for instance, the Navier-Stokes has found applications in many branches of stochastic analysis; e.g. equation and Monge-Ampere type equations. This course is a sequel to filtering and optimal control, mathematical finance, numerical methods for APMA 2810V, but APMA 2810V is not a prerequisite. stochastic differential equations. This course will introduce, starting in a APMA 2821D. Random Processes and Random Variables. simple setting, the basic concepts and operations of the Malliavin calculus, which will then be applied to the study of regularity of stochastic differential APMA 2821E. Topics in Variational Methods. equations and their associated partial differential equations. In addition, This course consists of two parts: a general introduction to variational applications from optimal control and finance, including the Clark-Ocone methods in PDE, and a more focused foray into some special topics. For foruma and its connection with hedging, will be presented. the former we will cover the direct method in the calculus of variations, various notions of convexity, Noether's theorem, minimax methods, index theory, and gamma-convergence. For the latter we will focus on several specific problems of recent interest, with emphasis on the Ginzburg- Landau energy functional.

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APMA 2821M. Some Mathematical Problems in Materials Science. APMA 2821T. Theory of Large Deviations. We will study a variety of mathematical models for problems in materials The theory of large deviations is concerned with the probabilities of very science. Mainly we will consider models of phase transformation, static rare events. There are many applications where a rare event can have a and dynamic. Some of the topics to be treated are: (1) models of phase significant impact (think of the lottery) and it is of interest to know when transformation; (2) gradient flows; (3) kinetic theories of domain growth; (4) and how these events occur. The course will begin with a review of the stochastic models; (5) free boundary problems. A working familiarity with general framework, standard techniques, and elementary examples (e.g., partial differential equations is required. Cramer's and Sanov's Theorems) before proceeding with general theory and applications. If time permits, the course will end with a study of large APMA 2821N. Numerical Solution of Ordinary Differential Equations: deviations for diffusion processes. IVP Problems and PDE Related Issues. The course seeks to lay the foundation for the development and APMA 2821U. Kinetic Theory. analysis of numerical methods for solving systems of ordinary Topics in kinetic theory, particularly concerning Boltzmann equations and differential equations. With a dual emphasis on analysis and efficient related but simpler models (e.g.the Kac model). Key issues include the implementations, we shall develop the theory for multistage methods mathematical derivation of the Boltzmann equation, the Cauchy problem, (Runge-Kutta type) and multi-step methods (Adams/BDF methods). The Boltzmann's $H$ theorem, and hydrodynamic limits yielding the equations discussion includes definitions of different notions of stability, stiffness of fluid mechanics. We will be most interested in rigorous results, but and stability regions, global/local error estimation, and error control. will not turn away from formal calculations when these are the only We also discuss more specialized topics such as symplectic integration things available. A probabilistic viewpoint will be emphasized. In addition methods, parallel-in-time methods, include splitting methods, methods for to these "traditional'' topics, we will also introduce the Smoluchowski differential-algebraic equations (DAE), deferred correction methods, and coagulation equation and a similar equation, and some microscopic order reduction problems for IBVP, TVD and IMEX methods. models described by these in the kinetic limit or exactly. Students should have PDE background equivalent to or exceeding MATH 2370/APMA APMA 2821O. Topics in Posteriori Error Estimations: Finite Element 2230. Familiarity with probability will be helpful, but we will review this and Reduced Basis Methods. The course will contain two related parts. An introduction of different according to the audience's needs. types of a posteriori error estimations for various finite element methods, APMA 2821V. Neural Dynamics: Theory and Modeling. certified reduced basis method, where a posteriori error estimations play Our thoughts and actions are mediated by the dynamic activity of the an important role. Emphasize both the theory and implementation. Related brain’s neurons. This course will use mathematics and computational Matlab programs. Residual-type, local-problem type, and recovery-type modeling as a tool to study neural dynamics at the level of signal neurons error estimators for conforming, mixed, non-conforming, and discontinuous and in more complicated networks. We will focus on relevance to modern galerkin finite element methods for different types of equations. Reduced day neuroscience problems with a goal of linking dynamics to function. basis methods, offline-online procedure, greedy algorithm, error estimator, Topics will include biophysically detailed and reduced representations of empirical interpolation method, and successive constraint method will be neurons, bifurcation and phase plane analysis of neural activity, neural discussed. Goal-Oriented primal-dual approach for both FEM and RBM rhythms and coupled oscillator theory. Audience: advanced undergraduate will be covered. Objective: To learn various theoretical and practical results or graduate students. Prerequisite: APMA 0350-0360 and Matlab of adaptive finite element methods and reduced basis methods. programming course. Instructor permission required. APMA 2821P. Topics in the Atomistic-to-Continuum Coupling APMA 2822B. Introduction to Parallel Computing on Heterogeneous Methods for Material Science. (CPU+GPU) Systems. Atomistic-to-continuum coupling methods (a/c methods) have been This course we will learn fundamental aspects of parallel computing proposed to increase the computational efficiency of atomistic on heterogeneous systems composed of multi-core CPUs and GPUs. computations involving the interaction between local crystal defects with This course will contain lectures and hands-on parts. We will cover the long-range elastic fields. This course provides an introduction to the following topics: shared memory and distributed memory programming fundamentals required to understand modeling and computer simulation models, parallelization strategies and nested parallelism. We will of material behavior. This course will first briefly review material from also learn techniques for managing memory and data on systems continuum mechanics, materials science including crystals and defects with heterogeneous memories (DDR for CPUs and HBM for GPUs); and then move on to advanced topics in development and analysis of a/c parallelization strategies using OpenMP and MPI; programming GPUs coupling methods both in static and dynamic cases. I will also select topics using OpenMP4.5 directives and CUDA. We will focus on programming from statistical mechanics and temporal multiscale accelerated molecular strategies and application performance. Grading will be based on home- dynamics methods (hyperdynamics, parallel replica dynamics). work assignments, and final project. APMA 2821R. Topics in the Atomistic-to-Continuum Coupling APMA 2822E. Topic Course in Kinetic Theory. Methods for Material Science. This is an advanced graduate topic course for PDE techniques in the Atomistic-to-continuum coupling methods (a/c methods) have been study of kinetic theory. The course will cover Boltzmann equations and proposed to increase the computational efficiency of atomistic Vlasov type of systems arising in mathematical physics. We will study computations involving the interaction between local crystal defects with mathematical tools such as the averaging lemma, solutions near vacuum long-range elastic fields. and Maxwellians, Hilebrt expansion and hydrodynamic limits, as well as This is an advanced topics course for graduate students. Provides an stability and instability in the Vlasov theory. introduction to the fundamentals required to understand modeling and Spr APMA2822ES01 25627 MWF 11:00-11:50(04) (Y. Guo) computer simulation of material behavior. First briefly review material from APMA 2980. Research in Applied Mathematics. continuum mechanics, materials science including crystals and defects Section numbers vary by instructor. Please check Banner for the correct and then move on to advanced topics in development and analysis of section number and CRN to use when registering for this course. a/c coupling methods both in static and dynamic cases. select topics from statistical mechanics and temporal multiscale accelerated molecular APMA 2990. Thesis Preparation. dynamics methods (hyperdynamics, parallel replica dynamics). For graduate students who have met the residency requirement and are continuing research on a full time basis. Fall APMA2990 S01 15672 Arranged ’To Be Arranged' Spr APMA2990 S01 24558 Arranged ’To Be Arranged'

22 Applied Mathematics