Math Times, Summer 2021
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
An Analog of the Minimax Theorem for Vector Payoffs
Pacific Journal of Mathematics AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL Vol. 6, No. 1 November 1956 AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL 1. Introduction* The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\\ with real elements there exist a number v and vectors P=(Pi, •••, Pr)f Q={QU •••> Qs)f Pi, Qj>β, such that i> 3) for all i, j. Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any ε>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — ε and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question. -
Blackwell, David; Diaconis, Persi a Non-Measurable Tail Set
Blackwell, David; Diaconis, Persi A non-measurable tail set. Statistics, probability and game theory, 1--5, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, 1996. Blackwell, David Operator solution of infinite $G\sb \delta$ games of imperfect information. Probability, statistics, and mathematics, 83--87, Academic Press, Boston, MA, 1989. 60040 Blackwell, David; Mauldin, R. Daniel Ulam's redistribution of energy problem: collision transformations. Lett. Math. Phys. 10 (1985), no. 2-3, 149--153. Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691--692. 60B10 Blackwell, David; Maitra, Ashok Factorization of probability measures and absolutely measurable sets. Proc. Amer. Math. Soc. 92 (1984), no. 2, 251--254. Blackwell, David; Girshick, M. A. Theory of games and statistical decisions. Reprint of the 1954 edition. Dover Publications, Inc., New York, 1979. xi+355 pp. ISBN: 0-486-63831-6 90D35 (62Cxx) Blackwell, David On stationary policies. With discussion. J. Roy. Statist. Soc. Ser. 133 (1970), no. 1, 33--37. 90C40 Blackwell, David The stochastic processes of Borel gambling and dynamic programming. Ann. Statist. 4 (1976), no. 2, 370--374. Blackwell, David; Dubins, Lester E. On existence and non-existence of proper, regular, conditional distributions. Ann. Probability 3 (1975), no. 5, 741--752. Blackwell, David; MacQueen, James B. Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 (1973), 353--355. Blackwell, David; Freedman, David On the amount of variance needed to escape from a strip. Ann. Probability 1 (1973), 772--787. Blackwell, David Discreteness of Ferguson selections. -
Rational Inattention with Sequential Information Sampling∗
Rational Inattention with Sequential Information Sampling∗ Benjamin Hébert y Michael Woodford z Stanford University Columbia University July 1, 2016 Preliminary and Incomplete. WARNING: Some results in this draft are not corrected as stated. Please contact the authors before citing. Comments welcome. Abstract We generalize the rationalize inattention framework proposed by Sims [2010] to allow for cost functions other than Shannon’s mutual information. We study a prob- lem in which a large number of successive samples of information about the decision situation, each only minimally informative by itself, can be taken before a decision must be made. We assume that the cost required for each individual increment of in- formation satisfies standard assumptions about continuity, differentiability, convexity, and monotonicity with respect to the Blackwell ordering of experiments, but need not correspond to mutual information. We show that any information cost function sat- isfying these axioms exhibits an invariance property which allows us to approximate it by a particular form of Taylor expansion, which is accurate whenever the signals acquired by an agent are not very informative. We give particular attention to the case in which the cost function for an individual increment to information satisfies the ad- ditional condition of “prior invariance,” so that it depends only on the way in which the probabilities of different observations depend on the state of the world, and not on the prior probabilities of different states. This case leads to a cost function for the overall quantity of information gathered that is not prior-invariant, but “sequentially prior-invariant.” We offer a complete characterization of the family of sequentially prior-invariant cost functions in the continuous-time limit of our dynamic model of in- formation sampling, and show how the resulting theory of rationally inattentive choice differs from both static and dynamic versions of a theory based on mutual information. -
An Annotated Bibliography on the Foundations of Statistical Inference
• AN ANNOTATED BIBLIOGRAPHY ON THE FOUNDATIONS OF STATISTICAL INFERENCE • by Stephen L. George A cooperative effort between the Department of Experimental • Statistics and the Southeastern Forest Experiment Station Forest Service U.S.D.A. Institute of Statistics Mimeograph Series No. 572 March 1968 The Foundations of Statistical Inference--A Bibliography During the past two hundred years there have been many differences of opinion on the validity of certain statistical methods and no evidence that ,. there will be any general agreement in the near future. Also, despite attempts at classification of particular approaches, there appears to be a spectrum of ideas rather than the existence of any clear-cut "schools of thought. " The following bibliography is concerned with the continuing discussion in the statistical literature on what may be loosely termed ''the foundations of statistical inference." A major emphasis is placed on the more recent works in this area and in particular on recent developments in Bayesian analysis. Invariably, a discussion on the foundations of statistical inference leads one to the more general area of scientific inference and eventually to the much more general question of inductive inference. Since this bibliography is intended mainly for those statisticians interested in the philosophical foundations of their chosen field, and not for practicing philosophers, the more general discussion of inductive inference was deliberately de-emphasized with the exception of several distinctive works of particular relevance to the statistical problem. Throughout, the temptation to gather papers in the sense of a collector was resisted and most of the papers listed are of immediate relevance to the problem at hand. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
David Blackwell Instance for Activities That Could Run Sometime Between Now and the End of June 2012
Volume 39 • Issue 8 IMS1935–2010 Bulletin October 2010 IMS Short Courses Proposal New IMS President Peter Hall outlines a proposed IMS program of short courses and Contents short meetings: 1 IMS Short Courses The IMS is considering developing a program of short courses and short meetings, reflecting all of the research areas encompassed by our journals. The motivation is at 2–3 Members’ News: Marta Sanz- Solé; David Brillinger; PK Sen; Elizaveta least two-fold. Firstly, we are not very widely involved in specialist meetings, and from Levina; Jerry Friedman; Samuel Kou; this point of view our scientific leadership can probably be enhanced. Secondly, in the Emily Berg; Wayne Fuller; Xuming He; area of short courses, there is an opportunity for both the IMS and its members to earn Jun Liu; Yajun Mei, Nicoleta Serban, a little extra income, by sharing the profits earned by delivering those courses. Roshan Joseph Vengazihiyil, Ming Yuan The best way in which to operate our short courses and short meetings will prob- ably be learned over time, but a few principles can be predicted in advance. In particu- 4 IMS 2010 Gothenburg meeting report & photos lar, the short courses and meetings should be held in places not in direct competition with relevant, major existing conferences, for example the Joint Statistical Meetings. Of 6 Peking University’s Center course, there are many opportunities for avoiding clashes such as these. The concept of for Statistical Science “IMS Winter Workshops” in North America, perhaps held in early January in southern 7 Conference in memory of US, is only one suggestion. -
Math Matters
MATH MATTERS CORNELL UNIVERSITY JANUARY 2020 Letter From The Chair Ravi Ramakrishna ’88 s is the case every year, our beginning of their first year with Department (along with all others studentsA and faculty are participat- their advisor, in the College) recently reclassi- ing a rich tapestry of mathematical and then as fied all its courses under the new events and scholarship. From needed. This system, with two falling under running conferences and summer often meant the Statistics and Data Science schools to supervising undergradu- never. The new (SDS) heading and the rest under ate research to (of course) proving model has a the Symbolic and Mathematical wonderful theorems, Cornell math- group of 10 first Reasoning (SMR) heading. We are ematicians at all levels are doing semester stu- curious to see how the new clas- great things. dents meeting sifications and requirements will Ravi Ramakrishna As you will read throughout with their advisor affect our enrollments, especially this issue of Math Matters, they once a week for 9 weeks. The at the first-year and sophomore are also being recognized for their intent is to orient students to the levels. outstanding accomplishments. College and break down the barri- Speaking of enrollments, we Our faculty have in the last year ers between faculty and students. will this year graduate about 95 won two College-wide teaching 2019 was the first year where the math majors. We now have the awards, three National Science seminar served all incoming A&S fifth most majors in the College. Foundation CAREER awards, a students and seven Math faculty Our enrollments at junior and Sloan Fellowship, two Simons participated. -
NEW TRENDS in SYZYGIES: 18W5133
NEW TRENDS IN SYZYGIES: 18w5133 Giulio Caviglia (Purdue University, Jason McCullough (Iowa State University) June 24, 2018 – June 29, 2018 1 Overview of the Field Since the pioneering work of David Hilbert in the 1890s, syzygies have been a central area of research in commutative algebra. The idea is to approximate arbitrary modules by free modules. Let R be a Noetherian, commutative ring and let M be a finitely generated R-module. A free resolution is an exact sequence of free ∼ R-modules ···! Fi+1 ! Fi !··· F0 such that H0(F•) = M. Elements of Fi are called ith syzygies of M. When R is local or graded, we can choose a minimal free resolution of M, meaning that a basis for Fi is mapped onto a minimal set of generators of Ker(Fi−1 ! Fi−2). In this case, we get uniqueness of minimal free resolutions up to isomorphism of complexes. Therefore, invariants defined by minimal free resolutions yield invariants of the module being resolved. In most cases, minimal resolutions are infinite, but over regular rings, like polynomial and power series rings, all finitely generated modules have finite minimal free resolutions. Beyond the study of invariants of modules, syzygies have connections to algebraic geometry, algebraic topology, and combinatorics. Statements like the Total Rank Conjecture connect algebraic topology with free resolutions. Bounds on Castelnuovo-Mumford regularity and projective dimension, as with the Eisenbud- Goto Conjecture and Stillman’s Conjecture, have implications for the computational complexity of Grobner¨ bases and computational algebra systems. Green’s Conjecture provides a link between graded free resolutions and the geometry of canonical curves. -
David Blackwell, 1919–2010: an Explorer In
RETROSPECTIVE David Blackwell, 1919–2010: An explorer in mathematics and statistics RETROSPECTIVE Peter J. Bickela,1 David Blackwell, a pioneering explorer who made foundational contributions to several branches of mathematics and statistics, passed away on July 8, 2010. He was born in Centralia, Illinois, on April 24, 1919, and, as his mathematical talents were recog- nized early, he entered the University of Illinois at Urbana–Champaign at age 16. Although racial dis- crimination affected his life and career in painful ways, his accomplishments were eventually rewarded with the honors they deserved, including election to the National Academy of Sciences in 1965 as the first Black member and the American Academy of Arts and Sciences in 1968. Nevertheless, his love of math- ematics, science, people, and his sunny personality prevailed. Blackwell left contributions that bear his name and other major ideas in five quite different areas of mathematics, statistics, and operations research. With limited opportunities available to him, Black- well initially thought of becoming an elementary school teacher. However, his professors at the Univer- sity of Illinois soon recognized his talent for mathe- matics and encouraged him to pursue graduate studies in the Illinois Mathematics program instead. During his graduate studies, Blackwell worked with Joseph Doob, one of the founding figures of modern David Blackwell. Image credit: The Blackwell family. probability theory, a National Academy of Sciences member, and National Medal of Science winner. In sequential analysis, game theory, and decision the- 1941, at age 22, he completed a doctoral thesis in the ory. They included: what is now known as the Rao- theory of Markov chains, a set of ideas to which he Blackwell theorem, a fundamental improvement frequently returned in his later work. -
Revised August 2014 DAVID EISENBUD VITA Born April 8, 1947
Revised August 2014 DAVID EISENBUD VITA Born April 8, 1947, New York City US Citizen Married, with two children EDUCATION B. S. University of Chicago 1966 M. S. University of Chicago 1967 Ph. D. University of Chicago 1970 Advisors: Saunders MacLane, J. C. Robson Thesis: Torsion Modules over Dedekind Prime Rings POSITIONS HELD Lecturer, Brandeis University 1970{72 Assistant Professor, Brandeis University 1972{73 Sloan Foundation Fellow 1973{75 Visiting scholar, Harvard University 1973{74 Fellow, I. H. E. S. (Bures-Sur-Yvette) 1974{75 Associate Professor, Brandeis University 1976{80 Visiting Researcher, University of Bonn (SFB 40) 1979{80 Professor, Brandeis University 1980{1998 Research Professor, Mathematical Sciences Research Institute, Berkeley 1986{87 Visiting Professor, Harvard University 1987{88 and Fall 1994 Chercheur Associ´e`al'Institut Henri Poincar´e(CNRS), Paris, Spring 1995. Professor, University of California at Berkeley, 1997{ Director, Mathematical Sciences Research Institute, 1997{ 2007 Director for Mathematics and the Physical Sciences, Simons Foundation, 2010{2012 HONORS, PRIZES Elected Fellow of the American Academy of Arts and Sciences, 2006 Leroy P. Steele Prize for Exposition, American Mathematical Society, 2010 CURRENT RESEARCH INTERESTS Algebraic Geometry Commutative Algebra Computational Methods 1 OTHER LONG-TERM MATHEMATICAL INTERESTS • Noncommutative Rings • Singularity Theory • Knot Theory and Topology 2 Professional Activities American Mathematical Society Council 1978{1982 (as member of the editorial board -
Download a PDF of the Program
Department of Mathematics 2021 Awards Tonight’s Program Welcome + Opening Remarks Professor Jeremy Tyson Chair, Department of Mathematics Land-Grant Statement + College of LAS Remarks Professor Matt Ando Associate Dean for Life and Physical Sciences, College of LAS Alumni Award Recognition + Remarks Steven Armstrong (BS ’92) 2020 Actuarial Science Program Alumnus of the Year Undergraduate Awards Professor Randy McCarthy Director of Undergraduate Studies Alumni Award Recognition + Remarks Danyel Graves Larsen (BA ’04) 2021 Alumni Humanitarian Award Illinois Geometry Lab Awards Professor Philipp Hieronymi Director of Illinois Geometry Lab Graduate Awards Professor Lee DeVille Director of Graduate Studies Alumni Award Recognition + Remarks Michael Stillman (BA ’78) 2021 Outstanding Achievement Award Faculty + Staff Awards Professor Jeremy Tyson Alumni Award Recognition + Remarks Judy Leavitt Walker (MS ’92, Ph.D. ’96) 2021 Outstanding Achievement Award Closing Remarks Professor Jeremy Tyson Undergraduate Awards H. Roy Brahana Prize Salma Wanna Memorial Award The H. Roy Brahana Prize is named for a distinguished The Salma Wanna Memorial Award honors the member of our mathematics faculty and recognizes memory of Salma Wanna, who received her PhD the student with “the most exceptional undergraduate from the University of Illinois in 1976. It is given for mathematics career.” “exceptional performance in mathematics to the Presented with honor to David Brewster, a senior most outstanding continuing student.” mathematics major. Presented with honor to Ariel Lerman, a sophomore mathematics major. Ariel received the Elsie Thomas Most Outstanding Major Awards Fraser Award in 2020 and was recently named a Barry In 1996, the department established the Major Awards M. Goldwater Scholar. He enjoys playing the cello, to recognize the most outstanding undergraduate creative writing and gaming. -
The First One Hundred Years
The Maryland‐District of Columbia‐Virginia Section of the Mathematical Association of America: The First One Hundred Years Caren Diefenderfer Betty Mayfield Jon Scott November 2016 v. 1.3 The Beginnings Jon Scott, Montgomery College The Maryland‐District of Columbia‐Virginia Section of the Mathematical Association of America (MAA) was established, just one year after the MAA itself, on December 29, 1916 at the Second Annual Meeting of the Association held at Columbia University in New York City. In the minutes of the Council Meeting, we find the following: A section of the Association was established for Maryland and the District of Columbia, with the possible inclusion of Virginia. Professor Abraham Cohen, of Johns Hopkins University, is the secretary. We also find, in “Notes on the Annual Meeting of the Association” published in the February, 1917 Monthly, The Maryland Section has just been organized and was admitted by the council at the New York meeting. Hearty cooperation and much enthusiasm were reported in connection with this section. The phrase “with the possible inclusion of Virginia” is curious, as members from all three jurisdictions were present at the New York meeting: seven from Maryland, one from DC, and three from Virginia. However, the report, “Organization of the Maryland‐Virginia‐District of Columbia Section of the Association” (note the order!) begins As a result of preliminary correspondence, a group of Maryland mathematicians held a meeting in New York at the time of the December meeting of the Association and presented a petition to the Council for authority to organize a section of the Association in Maryland, Virginia, and the District of Columbia.