MATHEMATICS Spring 2008: updates since Spring 2007 are in red

Mathematics (MAT)

Major and Minor in Mathematics Department of Mathematics, College of Arts and Sciences CHAIRPERSON: David Ebin DIRECTOR OF UNDERGRADUATE STUDIES: Scott Sutherland ASSISTANT TO THE CHAIR: Lucille Meci OFFICE: Mathematics P143 PHONE: (631) 632-8250 E-MAIL: [email protected] WEB ADDRESS: http://www.math.sunysb.edu Minors of particular interest to students majoring in Mathematics: Applied Mathematics and Statistics (AMS), Computer Science (CSE), Economics (ECO), Physics (PHY)

Faculty Jeremy Kahn, Lecturer, Ph.D., University of Bernard Maskit, Professor, Ph.D., New York Michael Anderson, Professor, Ph.D., University California, Berkeley: Dynamical systems, University: Riemann surfaces; Kleinian groups of California, Berkeley: differential geometry. complex analysis. and deformation spaces. William Barcus, Professor Emeritus and Director Ljudmilla Kamenova, Simons Instructor, Ph.D., Dusa McDuff, Distinguished Professor, Ph.D., of Mathematics Learning Center, D. Phil, Massachusetts Institute of Technology: Cambridge University, England: Symplectic University of Oxford, England: Algebraic topology. Complex geometry. topology. Christopher Bishop, Professor, Ph.D., University Nadia Kennedy, Assistant Professor, Ed.D., Marie-Louise Michelsohn, Professor, Ph.D., of Chicago: Complex analysis. Montclair State University: Mathmatics University of Chicago: Differential geometry. education. Kingshook Biswas, Lecturer, Ph.D., University John Milnor, Distinguished Professor and of California Los Angeles: Holomorphic dynam- Marcus Khuri, Assistant Professor, Ph.D., Director of the Institute for Mathematical ics, Riemann surfaces. University of Pennsylvania: Differential Sciences, Ph.D., Princeton University: geometry, partial differential equations, and Sylvain Bonnot, Lecturer, Ph.D., Universite de Dynamical systems. general relativity. Provence, Marseille: Complex dynamics, Anthony Phillips, Professor, Ph.D., Princeton Holomorphic dynamics, several complex Alexander Kirillov, Jr., Associate Professor, University: Differential topology and applica- variables. Ph.D., Yale University: Representation theory; tions to mathematical physics. low dimensional topology; mathematical Sebastian Casalaina-Martin, Simons Instructor, physics. Olga Plamenevskaya, Assistant Professor, Ph.D., Columbia University: Algebraic geometry. Ph.D., : Contact and sym- Valentina Kiritchenko, Simons Instructor, Ph.D., Mark de Cataldo, Assistant Professor, Ph.D., plectic geometry; low-dimensional topology. : Algebraic geometry. University of Notre Dame: Higher dimensional Sorin Popescu, Associate Professor, Ph.D., Irwin Kra, Distinguished Service Professor, geometry. University of Saarland, Germany: Algebraic Ph.D., Columbia University: Complex analysis; Reza Chamanara, Lecturer, Ph.D., Graduate geometry; computational algebraic geometry. Kleinian groups, Reimann surfaces; Teichmuller Center, CUNY: Moduli spaces, Kleinian groups, theory; applications to mathematical physics Corbett Redden, Simons Instructor, Ph.D., complex analysis. and number theory. Notre Dame University: Riemannian geometry; Moira Chas, Lecturer, Ph.D., Universitat algebraic topology. Paul Kumpel, Professor Emeritus, Ph.D., Autonoma de Barcelona: Topology and dynami- : Algebraic topology. Alexander Retakh, Visiting Assistant Professor, cal systems. Recipient of the State University Chancellor’s Ph.D., Yale University: Algebra; mathematical David Ebin, Professor, Ph.D., Massachusetts Award for Excellence in Teaching, 1990, physics. Institute of Technology: Global analysis; mathe- and the President’s Award for Excellence Frederic Rochon, Simons Instructor, Ph.D., matics of continuum mechanics; partial differ- in Teaching, 1990. Massachusetts Institute of Technology: Index ential equations. H. Blaine Lawson, Jr., Distinguished Professor, theory; pseudodifferential equations; K-theory. Daryl Geller, Professor, Ph.D., Princeton Ph.D., Stanford University: Differential geome- Scott Simon, Simons Instructor, Ph.D., Purdue University: Partial differential equations; har- try; topology; algebraic geometry. University: Infinite-dimensional complex analy- monic analysis; several complex variables; Claude LeBrun, Professor, D. Phil, University of Lie groups. sis; several complex variables. Oxford, England: Complex analysis; mathemati- James Glimm, Distinguished Professor, Ph.D., Jason Starr, Assistant Professor, Ph.D., Harvard cal physics; differential geometry; algebraic University: Algebraic geometry. Columbia University: Applied mathematics; geometry. numerical analysis; mathematical physics. Dennis Sullivan, Distinguished Professor, Ph.D., Krastio Lilov, Lecturer, Ph.D, University of Princeton University: Dynamical systems; Detlef Gromoll, Professor, Ph.D., University Michigan: Complex dynamics; complex geometry; partial differential equations. of Bonn, Germany: Differential geometry. variables. Scott Sutherland, Associate Professor, Ph.D., Eric Harrelson, RTG Fellow, Ph.D., University of William Linch III, RTG Fellow, Ph.D., University Boston University: Dynamical systems; root Minnesota: Algebraic topology, String field theory. of Maryland: Theoretical physics. finding algorithms; computing. Xuhua He, Simons Instructor, Ph.D., Mikhail Lyubich, Professor and Co-director of Leon Takhtajan, Professor, Ph.D., Leningrad Massachusetts Institute of Technology: the Institute for Mathematical Sciences, Ph.D., Branch of the Steklov Mathematical Institute, Representation theory, Algebraic geometry. Tashkent State University, former Soviet Union: Russia: Mathematical physics. C. Denson Hill, Professor, Ph.D., New York Dynamical systems. University: Partial differential equations; , Associate Professor, Ph.D., Vladlen Timorin, Lecturer, Ph.D, University of several complex variables. Belgrade University: Teichmuller theory and Toronto: Differential geometry. Jerome Jenquin, RTG Fellow, Ph.D, University hyperbolic geometry; harmonic maps between Dror Varolin, Assistant Professor, Ph.D., of Texas at Austin: Differential geometry. manifolds. University of Wisconsin, Madison: Several Lowell Jones, Professor, Ph.D., Yale University: Marco Martens, Associate Professor, Ph.D., complex variables; algebraic geometry; Topology; geometry. Delft University: Dynamics. complex geometry; dynamical systems.

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Judith Wiegand, Lecturer and Coordinator of er field, such as physics, computer science, MAT 205 Calculus III Field Experience and Student Teaching, M.A., applied mathematics and statistics, or eco- MAT 211 Introduction to Linear Polytechnic University of Brooklyn. nomics, are common and are encouraged. Algebra Aleksey Zinger, Assistant Professor, Ph.D., Massachusetts Institute of Technology: The secondary teacher education option MAT 260 Problem Solving in Symplectic topology and enumerative algebraic is designed for students planning a Mathematics geometry. career teaching mathematics in a sec- MAT 303 Calculus IV with Applications ondary school. This option is described in Teaching Assistants detail in the “Education and Teacher MAT 305 Calculus IV Estimated number: 60 Certification” entry in the alphabetical MAT 310 Linear Algebra listings of Approved Majors, Minors, and MAT 311 Number Theory Programs. athematics is an essential ele- MAT 312 Applied Algebra ment in a wide range of human The Department of Mathematics offers MAT 313 Abstract Algebra Mactivities. It is the language of tutorial help to all undergraduate stu- MAT 316 Invitation to Modern the physical sciences, and as such is an dents in its 100-level courses in the Mathematics indispensable tool in the formulation of Mathematics Learning Center. Since the the laws of nature. In the social and bio- Center’s staff consists of faculty and MAT 318 Classical Algebra logical sciences, it plays an increasingly graduate students in mathematics as MAT 319 Foundations of Analysis important role in modeling complicated, well as undergraduate tutors, students in MAT 320 Introduction to Analysis large-scale phenomena. In addition, ma- more advanced courses can also find thematics has an aesthetic side: aware- assistance there. MAT 322 Analysis in Several Dimensions ness of the possibility of elegance and MAT 324 Real Analysis beauty in mathematical arguments has The Department encourages students to seek information and advice on appropri- MAT 331 Computer-Assisted been a significant feature of human cul- Mathematical Problem Solving ture throughout history. Today more ate mathematics courses, programs, and mathematics is being done, and more career goals. Professors in mathematics MAT 336-H History of Mathematics needs to be done, than ever before. are available as advisors in the Under- MAT 341 Applied Real Analysis graduate Mathematics Office to help MAT 342 Applied Complex Analysis The undergraduate course offerings in with these matters. Advising hours can Mathematics allow students to set up be obtained by calling the Department of MAT 351 Differential Equations: individualized programs of study consis- Mathematics. Dynamics and Chaos tent with their academic interests and MAT 360 Geometric Structures career plans. Students should consider Courses Offered in Mathematics MAT 362 Differential Geometry of majoring in Mathematics even if they do See the Course Descriptions listing in Surfaces not plan to become mathematicians or this Bulletin for complete information. teachers of mathematics. The training in MAT 364 Topology and Geometry abstract reasoning and problem-solving MAP 101 Fundamentals of Arithmetic MAT 371 Logic and Algebra is an excellent foundation for many dif- MAT 373 Analysis of Algorithms ferent careers, such as law, graduate MAP 103 Proficiency Algebra MAT 401 Seminar in Mathematics health professions, and business. Com- MAT 118-C Mathematical Thinking pletion of a major in Mathematics points MAT 402 Seminar in Mathematics MAT 122-C Overview of Calculus with to a thinking person. MAT 475 Undergraduate Teaching Applications Students are encouraged to explore the Practicum MAT 123-C Introduction to Calculus various branches of pure and applied MAT 487 Independent Study in mathematics, as well as other mathemati- MAT 125-C Calculus A Special Topics cally oriented disciplines, to gain both MAT 126-C Calculus B MAT 495 Honors Thesis breadth of knowledge and insight into MAT 127-C Calculus C career options. Mathematics majors can MAT 129 Introduction to Integration Courses Offered in use their training as the foundation for Mathematics Education advanced professional study, leading to MAT 130 Functions See the Course Descriptions listing in research and teaching in universities or MAT 131-C Calculus I this Bulletin for complete information. research in industrial research laborato- MAT 132-C Calculus II ries; they can use it also in secondary MAE 301 Foundations of Secondary school teaching. In industry, undergradu- MAT 141-C Honors Calculus I School Mathematics ate training in mathematics is excellent MAT 142 Honors Calculus II MAE 302 Methods and Materials for preparation for the important task of liai- MAT 160 Mathematical Problems Teaching Secondary School Mathematics son work between the technological arm of and Games MAE 311 Introduction to Methods of a company and its marketing arm. A major MAT 171 Accelerated Single Variable Teaching Secondary School Mathematics in Mathematics is particularly appropriate Mathematics for work in computer applications, opera- MAE 312 Micro-Teaching tions research, and actuarial science. MAT 200 Logic, Language and Proof MAE 330 Technology in Mathematics Double majors in Mathematics and anoth- MAT 203 Calculus III with Applications Education

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MAE 412 Issues in Teaching and Sample Course Sequence for the Learning in Collegiate Mathematics Major in Mathematics MAE 447 Directed Readings in Mathematics Education Freshman Fall Credits Spring Credits MAE 451 Supervised Teaching–– First Year Seminar 101 1 First Year Seminar 102 1 Middle School Level Grades 7-9 D.E.C. A 3 D.E.C. A 3 MAT 131 or 141 or 125* 3-4 MAT 132 or 142 or 171 or 126* 3-4 MAE 452 Supervised Teaching–– D.E.C. 3 D.E.C. 3 High School Grades 10-12 D.E.C. 3 D.E.C. 3 MAE 454 Student Teaching Seminar Elective 3 MAT 200 or Elective 3 Total 16-17 Total 16-17 Requirements for the Major in Mathematics (MAT) Sophomore Fall Credits Spring Credits The major in Mathematics leads to the MAT 203 or 205 or AMS 261 3 MAT 303 or 305 or AMS 361 3 Bachelor of Science degree. Every stu- MAT 211 or AMS 210 3 MAT 331 3 dent majoring in Mathematics is expected D.E.C. 3 D.E.C. 3 to complete some form of a one-variable D.E.C. 3 D.E.C. 3 calculus sequence, which is a prerequisite Elective 3 Elective 3 for some of the courses listed below. Total 15 Total 15 Appropriate sequences at Stony Brook total 8 to 12 credits. Completion of the major requires 33 to 37 Junior Fall Credits Spring Credits credits. MAT 312 or 313 3 MAT 322 or 341 or 342 or 324 3 MAT 319 or 320 3 MAT 310 3 A. Mathematics and D.E.C. 3 D.E.C. 3 Mathematics-Related Courses D.E.C. 3 Upper-Division electives 6 1. One course in multivariate calculus: Elective 3 Total 15 MAT 203 or AMS 261 or MAT 205 Total 15 and one course in linear algebra: MAT 211 or AMS 210 2. Preparation in the language and Senior Fall Credits Spring Credits logic of mathematics: this require- Upper-Division MAT electives 9 Upper-Division MAT electives 9 ment can be met by either passing D.E.C. 3 Electives 6 MAT 200 or by passing the MAT Elective 3 Total 15 200 challenge examination. Total 15 (Note: the writing intensive course MAT 200 is a requirement for stu- dents in the Secondary Teacher Education Program.) * Students who take MAT 125, 126 must also complete MAT 127. 3. One course in differential equations: MAT 303 or AMS 361 or MAT 305 b. For students graduating with the in the Undergraduate Mathematics 4. One course in computer literacy: Secondary Teacher Education Office. Students in the Secondary MAT 331 or MEC 111 or CSE 114 option: MAT 319 or 320 Teacher Education Program must or (for students graduating 7. Five mathematics-related courses fufill a modified version of this with the Secondary Teacher beyond those taken to satisfy requirement, consisting of AMS 310, Education option) MAE 330. Requirements 5 and 6 (four will suf- MAT 336, MAT 360, and MAE Note: MAT 331 and MAE 330 fice if all of them are MAT courses), courses. may be used both here and to be chosen from the following: B. Upper-Division Writing Requirement in Requirement 7. MAE 301 To satisfy the Departmental writing 5. Two courses in algebra: MAT courses numbered 310 or above requirement, each student majoring MAT 310 and except 475 in Mathematics, including double MAT 312 or 313 or 318 majors, must submit an acceptable AMS courses numbered 301 or above 6. Analysis portfolio of three pieces of writing except 361 and 475 Students must satisfy from upper-division MAT or MAE CSE courses numbered 301 or above either a or b: coursework. Students should aim for except 475 completion of the portfolio early in a. Two courses in analysis: A list of acceptable upper-division their next-to-last semester to allow MAT 319 or 320 and courses in chemistry, economics, time to resolve any difficulties. Late MAT 322 or 324 or 341 or 342 philosophy, and physics is available completion may delay graduation.

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Each portfolio must be submitted no at the discretion of the undergraduate Beginning Mathematics Courses later than the beginning of the final director. Conferral of honors is contin- semester, and each piece in it must gent upon: have been approved by a Departmental High School 1. Completion of the set of seven MAP faculty member as being mathematical- Level Courses courses with a grade point average 101 ly correct and well written. of at least 3.50; Notes: 2. Approval for honors by the faculty 1. Under special circumstances a student member or members who supervise MAP may request the director of undergrad- MAT 495. 103 uate studies to allow substitution of an equivalent individual program for some Mathematics Secondary Teacher or all of these requirements. Education Program 2. All courses used to fulfill the require- See the Education and Teacher Certi- First- Statis MAT ments for the major must be taken for MAT fication entry in the alphabetical list- Year -tics 118 a letter grade and must be completed 122 ings of Approved Majors, Minors, and Courses with a grade of C or higher. Programs. MAT 3. Students whose scores on the College 123 Entrance Examination Board (CEEB) Requirements for the MAT Advanced Placement Examination are Minor in Mathematics (MAT) 130 documented earn credits as follows: The minor in Mathematics is available MAT 4 or 5 on BC examination: credit for for those students who want their formal 125 MAT 131, 132 (8 credits); university records to emphasize a serious MAT 131 4 or 5 on AB examination: credit for amount of upper-division work in mathe- or 141 MAT 131 (4 credits); matics. Although a one-variable calculus MAT 126 3 on either examination: 3 credits sequence is not a requirement, it is a pre- applicable to graduation but not the requisite for some of the courses listed below. The requirements listed below do major. MAT MAT MAT 132 not include single variable calculus or 127 211 or 142 4. Students who learned some linear alge- MAT 200 Logic, Language, and Proof; bra or multivariate calculus before or 171 these are prerequisites for some of the entering Stony Brook should see an courses listed below. advisor in the Undergraduate MAT 203 or AMS 261 Mathematics Office. For a student who 1. MAT 211 or AMS 210 Second-Year or MAT 205 has had some linear algebra, it may be 2. MAT 203 or AMS 261 or MAT 205 Courses appropriate to skip MAT 211 and to 3. MAT 310 or 312 or 313 or 318 MAT 200 enroll directly in MAT 310. 4. MAT 319 or 320 or 341 or 342 5. Six credits of graduate MAT courses MAT 211 5. Three additional MAT courses num- MAT 303 or AMS 361 may be used in place of undergraduate bered 300 or higher (excluding 475) or MAT 305 courses in Requirement A7. MAT 260 All courses used to fulfill the require- Honors Program in Mathematics ments for the minor must be passed with 161 rather than MAT 125, 126, 127 The honors program is open to junior a letter grade of C or higher. and senior Mathematics majors who because of the many requirements they have completed at least two upper-divi- Beginning Mathematics Courses must meet. MAT 141, 142 is an enriched sion MAT courses with grades of B or The Mathematics curriculum begins with version of MAT 131, 132. MAT 171 is a higher and who have maintained a 3.00 a choice of calculus sequences, some version of MAT 142 for students who overall grade point average. A prospec- including preparatory material from have not taken MAT 141; offered only in tive honors major must declare to the 12th-year mathematics in high school and the fall semester. director of undergraduate studies an some not. The three first-term calculus MAT 122 and MAT 123 combine precalcu- intention to participate in the program courses that assume knowledge of 12th- lus and calculus for students who have not before registering for the senior year. year mathematics are MAT 125, MAT had a precalculus course in high school. A 131, MAT 141 and AMS 151. A student The program consists of a set of seven student who completes MAT 122 will have may start any of these with the same MAT courses, at least three of which are learned some precalculus material and background. not used to fulfill the MAT major will have a good idea of what calculus is requirements. These courses must The three-semester sequence of one- and how it is used. MAT 123 is designed to include: MAT 322 or 324; MAT 401 or 402; variable calculus, MAT 125, 126, 127, is lead into MAT 125 or MAT 131. Although a course in algebra other than MAT 310 academically equivalent to the two- MAT 122 is not designed as preparation or 318; and MAT 495. Substitution of semester sequence MAT 131, 132. for further calculus courses, students may appropriate graduate courses is permit- Engineering students normally take the follow that course with MAT 125 or MAT ted, and other substitutions are possible faster-paced MAT 131, 132, or AMS 151, 131 if they take the one-credit course

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MAT 130 in the same semester as MAT of the exam. Part I covers high school Examination. In some cases, a course 125 or MAT 131. algebra, Part II deals with 12th year designator ending in PQ (such as MAT high school mathmatics (precalculus), 131PQ) may be placed on the student’s MAT 118 is a non-calculus course that and Part III covers single-variable transcript. In addition, transferred math- surveys various topics in mathematics calculus. The outcome of the test is one ematics courses are automatically evaluat- that do not require a background in pre- of nine levels: ed for applicability to the entry skill in calculus or calculus; it is designed for stu- mathematics requirement and the D.E.C. dents who do not intend to take further Outcome Placement category C requirement; this evaluation courses in mathematics. Level 1 MAP 101 does not depend on the result of the For students whose high school prepara- Level 2 MAP 103 placement examination. tion is insufficient to begin the MAT cur- Level 2+ MAT 118 riculum, or to enroll in another course and Skill 1 or statistics applicable to the D.E.C. category C requirement, Mathematical and Statis- Level 3 MAT 118, 122, tical Reasoning, there are two review 123 or statistics courses numbered MAP 101 and 103. Level 4 MAT 125 These courses do not carry graduation Level 5 MAT 131 or 141 credit. MAP 103, a skills course, is for or AMS 151 students who need further work in high Level 6 MAT 126 school algebra and related topics before Level 7 MAT 132 or 142 or 171 continuing with calculus or other mathe- or AMS 161 matics. Some students, upon completing MAP 103, are able to pass the Mathe- Level 8 MAT 127 or 132 or 171 matics Placement Examination at a level or 142 or AMS 161 that allows them to go directly into MAT Level 9 Beyond 100- 125 or 131. level calculus Levels 1-3 can be achieved by a suffi- Placement ciently high score on Part I, and levels 4- The Department of Mathematics offers a 5 can be achieved by a sufficiently high placement examination which indicates score on Part II, and attaining levels 6-9 the level of mathematical preparation of requires sufficiently high scores on Parts each student. The score on the examina- II and III. The entry skill in mathemat- tion is used to place the student in appro- ics requirement may be satisfied by priate courses in mathematics, applied attaining a score of level 3 or higher on mathematics and statistics, biology, the proctored exam. The general educa- chemistry, and physics. It tests the stu- tion requirement for Mathematics dent’s skills at the time the test is taken; (D.E.C. category C) may be satisfied by students are advised to study before- attaining a score of level 6 or higher on hand. There is a preliminary version of the proctored exam. Certain majors will the examination given prior to orienta- also accept a sufficiently high score on tion; all incoming students, including the proctored exam in lieu of required transfers, should take the preliminary math courses. A student who achieves a placement examination. This exam is particular level is free to begin with a used only for registration purposes and mathematics course corresponding to a cannot be used to fulfill graduation lower level, so long as taking the course requirements. The preliminary score does not mean that credit is given for the becomes invalid after two semesters. same material twice. A student wishing to use the placement Transfer Credit examination to fulfill D.E.C. Category C When they enter, transfer students auto- or other graduation-related require- matically receive credit toward gradua- ments or Skill 1, or if they have been or tion at Stony Brook for any courses they wish to be accepted into a major in the have already successfully completed at College of Engineering and Applied accredited institutions of higher educa- Sciences, must take a proctored version tion and that count toward graduation at of the examination. This examination is that institution. The number of credits given several times during the academic transferred appears on the Stony Brook year, and by appointment with the transcript with no courses or grades Mathematics Department. indicated, and the number of transferred The placement exam consists of several credits is unaffected by the student’s parts; not all students will take all parts score on the Mathematics Placement

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