Showing papers in "Bulletin of the American Mathematical Society in 2009"

Topology and data

Abstract: An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology It is also clear that the nature of the data we are obtaining is significantly different For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it The data obtained is also often much noisier than in the past and has more missing information (missing data) This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, ie finite sets of points equipped with a distance function It then adapts tools from the various branches of geometry to the study of point clouds The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise Here are some of the key points which come up when applying these geometric methods to data analysis • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, ie to understand how it is organized on a large scale For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Abstract: We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices

Abstract: The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.

Representation theory of symmetric groups and related Hecke algebras

Abstract: We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category O, W -algebras, . . .

On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)

Abstract: Assume that, in the near future, someone can prove resolution of singularities in arbitrary characteristic and dimension. Then one may want to know why the case of positive characteristic is so much harder than the classical characteristic zero case. Our intention here is to provide this piece of information for people who are not necessarily working in the field. A singularity of an algebraic variety in positive characteristic is called wild if the resolution invariant from characteristic zero, defined suitably without reference to hypersurfaces of maximal contact, increases under blowup when passing to the transformed singularity at a selected point of the exceptional divisor (a so called kangaroo point). This phenomenon represents one of the main obstructions for the still unsolved problem of resolution in positive characteristic. In the present article, we will try to understand it.

Frontiers of reality in Schubert calculus

Abstract: The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifica- tions in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and gen- eralizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.

Taubes’s proof of the Weinstein conjecture in dimension three

Abstract: Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No, according to counterexamples by K Kuperberg and others On the other hand there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit This conjecture was recently proved by Taubes using Seiberg-Witten theory We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits Taubes’s proof of the Weinstein conjecture is the culmination of a large body of work, both by Taubes and by others In an attempt to make this story accessible to nonspecialists, much of the present article is devoted to background and context, and Taubes’s proof itself is only partially explained Hopefully this article will help prepare the reader to learn the full story from Taubes’s paper [62] More exposition of this subject (which was invaluable in the preparation of this article) can be found in the online video archive from the June 2008 MSRI hot topics workshop [44], and in the article by Auroux [5] Below, in §1–§3 we introduce the statement of the Weinstein conjecture and discuss some examples In §4–§6 we discuss a natural strategy for approaching the Weinstein conjecture, which proves it in many but not all cases, and provides background for Taubes’s work In §7 we give an overview of the big picture surrounding Taubes’s proof of the Weinstein conjecture Readers who already have some familiarity with the Weinstein conjecture may wish to start here In §8–§9 we recall necessary material from Seiberg-Witten theory In §10 we give an outline of Taubes’s proof, and in §11 we explain some more details of it To conclude, in §12 we discuss some further results and open problems related to the Weinstein conjecture 1 Statement of the Weinstein conjecture The Weinstein conjecture asserts that certain vector fields must have closed orbits Before stating the conjecture at the end of this section, we first outline its origins This is discussion is only semi-historical, because only a sample of the relevant works will be cited, and not always in chronological order 11 Closed orbits of vector fields Let Y be a closed manifold (in this article all manifolds and all objects defined on them are smooth unless otherwise stated), 2000 Mathematics Subject Classification 57R17,57R57,53D40 Partially supported by NSF grant DMS-0806037

Remarks on Chern-Simons theory

Abstract: In the late 1980s Witten used the Chern-Simons form of a connection to construct new invariants of 3-manifolds and knots, recovering in particular the Jones invariants. Since then the associated topological quantum field theory (TQFT) has served as a key example in understanding the structure of TQFTs in general. We survey some of that structure with a particular focus on the "multi-tier" aspects. We discuss general axioms, generators-and-relations theorems, a priori constructions, dimensional reduction and K-theory, and Chern-Simons as a 0-1-2-3 theory. An appendix gives a lightening treatment of the Chern-Simons-Weil theory of connections. The paper concludes with general remarks about the Geometry-QFT-Strings interaction.

Lang-Trotter revisited

Abstract: Dedicated to the memory of Serge Lang Table of contents 0. Preface

Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

Abstract: This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin $s=0, 1/2, 1, 2$, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations

Abstract: von Neumann: I would like to make some remarks about the general hydrodynamical discussion on motions in one dimension, following Riemann’s theory, which was expounded by Dr. Burgers and subjected to a critical analysis by Dr. McVittie. The question as to whether a solution which one has found by mathematical reasoning really occurs in nature and whether the existence of several solutions with certain good or bad features can be excluded beforehand, is a quite difficult and ambiguous one. This subject has been considered in the classical literature as well as in the more recent literature, on widely varying levels of rigor and of its opposite. In summa, it is quite difficult ever to be sure of anything in this domain. Mathematically, one is in a continuous state of uncertainty, because the usual theorems of existence and uniqueness of a solution, that one would like to have, have never been demonstrated and are probably not true in their obvious forms. To this day, the only thing of any degree of generality that we possess is the classical discussion by Riemann, and this very strictly in one dimension and very strictly in the isentropic case. In this case at least, Riemann proved that there are no discontinuities. He also gave the exact conditions under which there can be a solution at all and he proved that in those cases there is only one. So he proved that the number of solutions is either zero or one. He also showed that it is zero in general, i.e., unless certain (infinitely many) very stringent conditions are satisfied. Thus, unless the initial state of the gas fulfills some very particular conditions, the (continuous) solution will cease to exist after some definite finite time. Riemann also inferred, essentially by physical insight, what happens when the continuous solution ceases to exist. He made it very plausible that a discontinuity of a certain type, a “shock wave”, develops. This was subsequently independently rediscovered, and further developed, by Hugoniot. It is also true that in the entire literature up to 1910, i.e., up to the time of the work of Rayleigh and G. I. Taylor, there was a considerable confusion

Von Neumann’s comments about existence and uniqueness for the initial-boundary value problem in gas dynamics

Abstract: The context is that of a conference, Problems of Cosmical Aerodynamics, held in Paris, France, on August 16–19, 1949. It was organized at the initiative of the newly founded IUTAM and of the IAU. It gathered fifty-two scientists, among which there were thirty-four astronomers and eighteen physicists or fluid mechanicists. The chairs were J. M. Burgers for IUTAM and H. C. van de Hulst for IAU. Among the participants were W. Heisenberg, J. H. Oort, E. Schatzman, T. von Karman, C. F. von Weizsacker and, of course, J. von Neumann. G. I. Taylor could not attend the conference because of sickness. The conference was managed in a rather informal style, the program being decided during the first day, a little bit like our Oberwolfach workshops. Discussion sessions happened every day in the afternoon. I wish to comment here on the session chaired by von Neumann, devoted to the Existence and uniqueness or multiplicity of solutions of the aerodynamical equations. The records of the discussion are reprinted in this issue after these comments. They are taken from pp. 75–85 of the conference proceedings, published by the Central Air Documents Office, Dayton (Ohio), 1951. Instead of a discussion, the session was actually a long monologue by von Neumann, after which he apologize(s) for having taken up so much of the limited time. He then answers the questions of a handful of scientists. The session ends with a comment by Burgers about the consequences of this discussion for astronomical problems. In order to make the context as clear as possible, it is necessary to recall that during WWII, von Neumann had been involved in the Manhattan Project, dedicated to the construction of the American nuclear bomb, while Heisenberg led the German nuclear weapon program, to which Weizsacker collaborated. This was the origin of their common, though independent, interest in fluid mechanics and shock waves. However, the 1949 conference had a peaceful target, and the conversations remained courteous. In addition, Germany was no longer a threat, thus it became meaningful to involve German scientists in such a conference. The

Marshall Stone and the internationalization of the American mathematical research community

Abstract: The American mathematical research community celebrated, symbolically at least, its fiftieth anniversary in 1938. Many of those fifty years had marked a period of consolidation and growth at home of programs in mathematics at institutions of higher education supportive of high-level research as well as of a corps of talented researchers capable of making seminal contributions in a variety of mathematical areas. By the middle decades of the twentieth century—the 1930s, 1940s, and 1950s—members of that community, like members of the broader American public, began increasingly to look outward beyond the national boundaries of the United States and toward a larger international arena. This paper explores the contexts within which the American mathematical research community, in general, and the American mathematician Marshall Stone, in particular, deliberately worked in the decades around mid-century to effect the transformation from a national community to one actively participating in an internationalizing mathematical world. The year 1938 was one of celebration and self-congratulation within the American mathematical research community. Fifty years earlier, four Columbia graduate students and two Columbia faculty members had met in New York City in order “to establish a mathematical society for the purpose of preserving, supplementing, and utilizing the results of their mathematical studies” [3, p. 4]. Within two decades, this extremely modest, local enterprise had grown into a national organization that supported two publications, its Bulletin started in 1891 and its Transactions first published in 1900, as well as regional sections in the midwest, on the west coast, and in the southwest [28, pp. 266–268 and 401–415]. By 1938, the leadership of that organization had recognized that the time was ripe both for chronicling its history and for showcasing the contributions to the store of mathematical knowledge made by the members of the vibrant and self-sustaining national community Received by the editors October 7, 2008. 2000 Mathematics Subject Classification. Primary 01A60, 01A70. c ©2009 American Mathematical Society Reverts to public domain 28 years from publication

Book Review: Introduction to circle packing: The theory of discrete analytic functions

Abstract: Introducing a new hobby for other people may inspire them to join with you. Reading, as one of mutual hobby, is considered as the very easy hobby to do. But, many people are not interested in this hobby. Why? Boring is the reason of why. However, this feel actually can deal with the book and time of you reading. Yeah, one that we will refer to break the boredom in reading is choosing introduction to circle packing the theory of discrete analytic functions as the reading material.

On the origin and development of subfactors and quantum topology

Abstract: We give an account of the beginning of subfactor theory and TQFT and some more recent developments.

The construction of solvable polynomials

Abstract: Although Leopold Kronecker’s 1853 paper “On equations that are algebraically solvable” is famous for containing his enunciation of the Kronecker-Weber theorem, its main theorem is an altogether different one, a theorem that reduces the problem of constructing solvable polynomials of prime degree μ to the problem of constructing cyclic polynomials of degree μ − 1. Kronecker’s statement of the theorem is sketchy, and he gives no proof at all. There seem to have been very few later treatments of the theorem, none of them very clear and none more recent than 1924. A corrected version and a full proof of the theorem are given. The main technique is a constructive version of Galois theory close to Galois’s own.

Reflections and prospectives

Abstract: Intellectual challenges and opportunities for mathematics are greater than ever. The role of mathematics in society continues to grow; with this growth comes new opportunities and some growing pains; each will be

Errata to “The construction of solvable polynomials”

Abstract: in order to include the case ν = 1. (When ν = 1, the argument of footnote 12 fails and no δ satisfies the requirements of the theorem. When the formula is stated as above, no δ is called for when ν = 1.) In the special case ν = 1, the definition of G(x) in (5.2) should be replaced by G(x) = x − sμ0 , where s0 is a nonzero Lagrange resolvent. (In this case, there is no m, but this G(x) has the μ needed roots αs0.) When ν = 1, the assertion to be proved in Section 7 reduces to a tautology. 2. Proposition 4.1 contains a serious error that does not affect the rest of the paper. The formula αsi → αsi+κ it gives for τ describes a permutation of the Lagrange resolvents, but does not describe an automorphism of Ω, so the proposition fails to provide the needed τ . (The τ constructed in the proof is an automorphism, but it does not combine with σ and η to generate the group.) Correction of the formula for τ implies corrections in the relations, but the main assertions remain:

On A. Weil

Abstract: When he was a college student, Taniyama studied the papers of A. Weil. He was apparently so impressed by this work that in 1953 he wrote a small note, “On Weil”, which was published in Sugaku no ayumi, volume 1, no. 1, the periodical of the New Mathematical Society, of which Taniyama was one of the original founders. It is reprinted in Taniyama’s collected works, in the original Japanese. It was not until 1955 that Taniyama met Weil in person, at the International Symposium on Number Theory, held in Tokyo. In his note, Taniyama expresses almost contradictory opinions, at one point praising Weil for his insight, creativity, and technical power, but at the same time criticizing Weil for not going far enough. In the penultimate paragraph Taniyama asks if there is any room left for revolutionary ideas in mathematics. The article ends with a devastating opinion concerning the state of Japanese mathematics of the 1950s.

About the cover: Christopher Clavius, Astronomer and Mathematician

Abstract: As early as 1232 John of Holy Wood, an English astronomer known usually by the Latin translation of his name, Sacro Bosco, observed that the Julian calendar was becoming increasingly inaccurate over time. It wasn’t until roughly 300 years later that Christopher Clavius, a German Jesuit, formulated the improved Gregorian calendar that was adopted by Pope Gregory XIII and instituted on Friday, October 15, 1582. This is the astronomical work for which Clavius is best known but he was a polymath who wrote on various branches of science and mathematics. Born in 1538, two years before St. Ignatius Loyola’s Society of Jesus was approved by the Catholic Church, Clavius became one of the earliest members of the Order that would have considerable influence on the history of astronomy and mathematics, particularly in the 16th and 17th centuries. The title page shown on the cover, with its armillary sphere, is from one of Clavius’s most important works, the In Sphaeram Ioannis de Sacro Bosco Commentarius (Rome, 1570), which not only presented the much earlier work of the 13th century Sacro Bosco but included evidence of Clavius’s awareness of the Copernican theory as described in the De Revolutionibus Orbium Cœlestium of 1543 (though Clavius remained loyal to the theory of Ptolemy). It also provides evidence of Clavius’s long time correspondence with Galileo, in which at one point he provided confirmation of some of Galileo’s calculations on Venus and Saturn. Galileo was sick in bed when he received Clavius’s confirming data and J. MacDonnell pointed out that “the letter brought him so much joy, it occasioned his immediate recovery” [1]. During that period Clavius was also corresponding with Viete and Kepler. As an author of scientific treatises as well as textbooks widely used in Jesuit schools, he was influential in introducing notation or promulgating that of others; e.g., the decimal point, the radical, applications of logarithms, and parentheses to group together algebraic symbols. His books were on topics in astronomy, spherical geometry and practical geometry, on Euclid (where he called attention to the problem involving the fifth postulate), as well as practical arithmetic. George Sarton referred to Clavius as “the most influential teacher of the Renaissance”. His reputation in science prompted the forming of the Clavius Mathematics Group in 1963, an organization of lay and religious mathematicians who organize four-week research meetings each summer in settings on campuses as widespread as Georgetown, the IAS at Princeton, McGill, Notre Dame, UC Berkeley, and the IHES outside Paris, among others.