Colloquia, AY 2005-2006
Colloquium, AY 2005-2006
Tea at 3:30 pm in 1425 Stevenson Center
| September 22, 2005 |
Paula Cohen, Texas A&M, College Station Subvarieties of Shimura varieties, special values of classical functions, and monodromy Abstract: Inspired by Hilbert’s 7th problem, Siegel (1932) and Schneider (1937) obtained the first significant results about the transcendence of periods of doubly periodic functions and special values of modular functions. In particular, they found a relation to the class fields occurring in algebraic number theory. Siegel formulated similar problems for G-functions, a special case of which is the classical hypergeometric function. The modern development of this circle of ideas was made possible by the work of Alan Baker and its outgrowth. In our lecture we focus on recent results, for example, the characterization of points at which hypergeometric functions take algebraic values. We describe the surprising role, first noticed by Wolfart, played by non-arithmetic monodromy groups acting on the complex ball. We also show how such problems are related to questions on subvarieties of Shimura varieties. The lecture will be self-contained and accessible to a general audience. |
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| October 6, 2005 |
Ian H. Sloan, University of New South Wales, Sidney Numerical integration in high dimensions – lifting the curse of dimension Abstract: Numerical integration in high dimensions confronts us with the curse of dimensionality — the number of function values needed to obtain an acceptable approximation can grow exponentially in the number of dimensions d. The exponential increase is clearly inevitable with any form of product integration rule, and for many theoretical settings is now known to be unavoidable no matter how the integration rule is chosen. It has been known since 1998 that the curse of dimensionality can in principle be overcome within the “weighted Sobolev space” setting introduced by Sloan and W\’ozniakowki, if the “weights” that describe the behaviour with respect to different variables satisfy a certain (necessary and sufficient) condition. In that work it was show that, under the appropriate condition on the weights, there exist integration rules for which the “worst-case error” is bounded independently of d. That 1998 result was non-constructive, giving no clue as to how we might construct “good” integration rules. More recently it has been shown that “good” rules can be found within the much smaller class of (shifted) lattice rules, and even more recently that good rules can be constructed one component at a time. This talk will review these developments, from early existence proofs and non-constructive methods to recent fast constructions of good integration rules in hundreds of dimensions, that may use hundreds of thousands of sample points. |
| October 13, 2005 |
William B. Johnson, Texas A&M, College Station A survey of non linear Banach space theory Abstract: There are three reasonable notions of geometric equivalence for metric spaces: Lipschitz equivalence, uniform equivalence, and Gromov’s notion of coarse equivalence (which has recently attracted interest from geometric analysts because of its relation to the Novikov and Baum-Connes conjectures). I’ll survey what is known about these types of equivalences when at least one of the metric spaces is a Banach space. When both spaces are Banach spaces, a fundamental question is: When does the existence of one of these non linear equivalences between the spaces imply the existence of a linear equivalance (i.e., an isomorphism)? If there is time I’ll also discuss the recently introduced concept of Lipschitz quotient maps, which are closely related to the non collapsing maps studied by David and Semmes. The talk is suitable for graduate students as well as faculty.
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| October 27, 2005 |
Shmuel Weinberger, University of Chicago Homology manifolds and their resolution Abstract: Homology manifolds originally were considered as an abstraction (the least you need for Poincare duality) and as a source of counterexamples. However, they are now important objects in their own right and fill in gaps in the theory of manifolds. This talk is mainly about the connections between manifold theory and homology manifolds, a conjectural picture and its heuristic evidence, and, finally some of the evidence that the hard-headed like: partial results. |
| November 3, 2005 |
Michael Lacey, Georgia Tech, Atlanta Convergence of Fourier Series: Past, Present, Future Abstract: Carleson’s celebrated Theorem on the pointwise convergence of Fourier series is the topic of this talk. We will review the statement, and explain why the proof is hard, as well as why one might want to know some elements of the proof. We conclude with a brief description of related results, and some difficult questions that remain unanswered. |
| November 10, 2005 |
Pawel Idziak, University of Cracow, Poland On-line algorithms for graphs and posets Abstract: Traditional design and analysis of algorithms assumes that the entire information of the input data is available for us before the algorithm is executed. This assumption is not met in many situations, where the complete knowledge of the entire input is not known in advance. Examples come from problems like: real-time systems, routing in communication networks, scheduling tasks on servers, paging in a virtual memory, dynamic storage allocation, etc. In these situations algorithms have to make their decision based only on an accessible piece of information. Data are delivered during the performance of the algorithm, while the algorithm cannot change its earlier decisions. Thus it often happens that a decision that seems to be correct at the moment leads to a solution that is far from being optimal in the future. On-line algorithms are supposed to deal with this kind of problems in a way that is as close to an optimal solution as possible. Thus, from a mathematical point of view they may be considered as two persons games: between Spoiler and an Algorithm. Spoiler presents data in the worst possible (from the Algorithm’s point of view) way, while Algorithms try to find a solution that is not far away from an optimal off-line solution. In this talk several on-line problems concerning graph coloring and poset chain covering are discussed. |
| November 24, 2005 |
Thanksgiving Break |
| December 1, 2005 |
Guido Weiss, Washington University, St. Louis A general description of Wavelets for those not in this field Abstract: I plan to to give a talk that describes the field of Wavelets as a pure mathematical subject and its properties that make it a good source for applications in many fields in science and engineering. I strongly believe that this subject is a very beautiful mathematical field. I hope to convince a rather wide audience of this. Students, Engineers, Scientists and Mathematicians having little background in Analysis should be able to understand the message I will try to give. |
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| January 12, 2006 |
Yanqin Fan, Department of Economics, Vanderbilt University Sharp Correlation Bounds and Their Applications Abstract: In this paper, we establish asymptotic properties, including the consistency and asymptotic normality, of nonparametric estimators of the sharp bounds on the correlation between two random variables. We demonstrate both theoretically and numerically that the sharp bounds may differ from the traditionally used bounds [-1,1] and the nonparametric estimators of the sharp bounds shed light on the strength of the type of dependence, linear or nonlinear, between two random variables. To facilitate inference on the true sharp bounds, we provide easy-to-compute estimators of the asymptotic variances of the nonparametric estimators of the sharp bounds. Using the sharp correlation bounds on the unobserved covariates, we derive sharp bounds on the correlation of durations in bivariate hazard rate models with unobserved heterogeneity and the correlation of dependent variables in bivariate log-linear regression models with unobserved covariates. These results provide insight on the selection of distributions of the unobserved heterogeneity in bivariate hazard rate models and unobserved covariates in log-linear regression models. |
| January 19, 2006 |
Frank Morgan, Williams College Double Bubbles and Gauss Spaces Abstract: The Ancient Greeks claimed and Schwarz (1884) proved that a round spherical “soap bubble” provides the least-perimeter way to enclose a given volume of air. Our Double Bubble Theorem (Annals of Math, 2002) says that the familiar double soap bubble provides the least-perimeter way to enclose and separate two given volumes of air. I’ll also discuss extensions to other spaces, many by students, and open questions. Featured today will be Gauss space, Euclidean space with Gaussian density. Despite the rotational symmetry of Gauss space, the best single bubble is not round, but another familiar shape. Gauss space is the model example of the important “new” category of manifolds with density. No special prerequisites: undergraduate math majors welcome. |
| January 26, 2006 |
George Metcalfe, Vienna University of Technology How to make your algebras dense: a proof-theoretic approach Abstract: Proof theory provides an algorithmic way of representing and reasoning about ordered algebraic structures. In this talk, I show how proof theory can be used to solve an algebraic problem important in the wider context of fuzzy logic; namely, that certain varieties are generated by their dense chains. The strategy consists of two parts. First, validity in all dense chains of the variety is shown to be equivalent to derivability in a calculus for the variety extended with a special density rule. It is then shown that applications of the density rule can be eliminated from proofs in the calculus. |
| February 2, 2006 |
Yehuda Shalom, Tel Aviv University (visiting Princeton) Some analytic methods in group theory Abstract: We shall try to explain how analytic tools, involving spectral analysis, ergodic theory and probability on groups, combine together to yield purely algebraic properties of some interesting classes of groups. We shall also use the main result, which is entirely elementary in its statement, as a motivation to discuss some aspects of the fundamental notions of amenability and property (T), assuming no prior familiarity. The talk should be accessible to every graduate student. Contact person: |
| February 9, 2006 |
Dean Billheimer, Department of Biostatistics, Vanderbilt University Compositional Data in Biomedical Research Abstract: Modern methods of compositional data analysis are not well known in biomedical research. Moreover, there appear to be few mathematical and statistical researchers working on compositional biomedical problems. Like other areas of science, biomedicine has many problems in which the relevant scientific information is encoded in the relative abundance of key species or categories. These vectors of relative amounts, or proportions, constitute compositional observations. I review standard approaches to compositional data analysis, and describe several recent theoretical advances. These newer methods exploit a vector space structure of the simplex that extends data modeling, and eases interpretation of results. In addition, I describe a problem in cancer research in which analysis of compositions plays an important role. The problem involves subcellular localization of the BRCA1 protein, and its role in breast cancer patient prognosis. This talk contains a tutorial component, and should be accessible to students with exposure to inner product spaces and multivariate statistics. |
| February 16, 2006 |
Leon Glass, Centre for Nonlinear Dynamics, Department of Physiology, McGill University Dynamics of Paroxysmal Tachycardia Abstract: Reentrant tachycardias are abnormal cardiac arrhythmias in which the period is set by the time it takes for the excitation to travel in a circuitous path. Simple conceptual models of reentrant tachycardia include waves circulating in rings or annuli, spiral waves in a plane, or scroll waves in three dimensions. In this talk I discuss various aspects of reentrant tachycardia including: the stability of the circulation as the path length of the reentrant circuit is decreased, the resetting of tachycardia by single pulses and predicting the effects of multiple pulses, control of instabilities during the reentrant rhythm by adjusting the timing of stimuli delivered during the course of the tachycardia, spontaneous breakup of rotating spiral waves and the spontaneous generation of bursting rhythms. Experimental models for these types of systems can be generated by growing embryonic chick heart cells in tissue culture and optically imaging the activity using voltage and calcium sensitive dyes. Mathematical models can be developed suitable for comparison with experimental and clinical data. |
| February 23, 2006 |
Benson Farb, University of Chicago Problems and progress in understanding the Torelli group Abstract: The Torelli group T(S) associated to a surface S is defined to be the group of homotopy classes of homeomorphisms of S acting trivially on H_1(S,Z). The study of T(S) connects to 3-manifold theory, symplectic representation theory, combinatorial group theory, and algebraic geometry. In this talk I will explain some of the main themes in this beautiful topic. I will then describe a new thread, very recently discovered by Chris Leininger and Dan Margalit and me, which might be summarized as “algebraic complexity implies dynamical complexity”. Much of this talk should be accessible to beginning graduate students.
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| March 2, 2006 |
From MV-algebras to AF-algebras via ordered groups: the first twenty years Abstract: Elliott’s theory yields a classification of AF-algebras in terms of countable Riesz groups. In particular, AF-algebras whose Murray von Neumann order of projections is a lattice correspond to (countable) lattice-ordered abelian groups with order-unit. These groups are categorically equivalent to (countable) MV-algebras, the algebras of Lukasiewicz infinite-valued logic. Propositions in this logic are states of knowledge in a R{‘e}nyi-Ulam game of Twenty Questions with lies, just as propositions in boolean logic are states of knowledge in the Twenty Questions game. We survey recent research on the interplay between these structures, including new representation theorems for lattice-ordered abelian groups, and decidability/undecidability results for the isomorphism problem of finitely presented AF C*-algebras. |
| March 9, 2006 |
Spring Break |
| March 16, 2006 |
Richard Kadison, University of Pennsylvania Re-examining the Pythagorean Theorem – A Functional Analyst’s View Abstract: In the colloquium talk we’ll study some surprising twists and connections with operator algebras, symmetric spaces, Schur’s work – on reviewing the basic theorem. We’ll start at the beginning. In the second lecture (on Friday), we’ll study the Schur-Horn extension and Pythagoras in II1 factors, as well as connections with the work of Kostant, Atiyah, and Guillemin-Sternberg. This may be a bit more technical.
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| March 23, 2006 |
Jonathan Rosenberg, University of Maryland An analogue of the Novikov Conjecture in complex algebraic geometry Abstract: We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain “higher Todd genera” are birational invariants, and implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). The conjecture is, in a certain sense, best possible, and unlike the usual Novikov Conjecture, it is already known to be true in all cases, though some variants are still open. An interesting biproduct of this work is a curious analogy between the homotopy category of smooth manifolds and the birational category of smooth projective varieties.
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| March 30, 2006 |
Noga Alon, Tel Aviv University (visiting Princeton) The structure of graphs and Grothendieck type inequalities Abstract: I will describe a connection between a classical inequality of Grothendieck, a constructive version of a powerful lemma of Szemeredi in Graph Theory, and integrality gaps of certain integer programming problems. The investigation of this connection suggests the definition of a new graph parameter, called the Grothendieck constant of a graph, whose study is motivated by algorithmic applications, and leads to several extensions of the inequality of Grothendieck, to an improvement of a recent result of Kashin and Szarek, and to a solution of a problem of Megretski and of Charikar and Wirth.
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| April 6, 2006 |
Yang Wang, Georgia Tech Hilbert’s 3rd Problem, Lattice Tiling, and Weyl-Heisenberg Bases Abstract: Steinhaus proposed in the 1950’s the following “infamous” problem: Is there a set T such that every rotation of T tiles the plane by the integer lattice? Jackson and Mauldin recently showed that such a T exists if the set is not required to be measurable. The question remains open for measurable sets. A similar question one may ask is that given a collection of lattices can one find a measurable set T that tiles Rn by each of the lattices in the collection. This question has a surprising link to time-frequency analysis, and to the classic Hilbert’s third problem.
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| April 13, 2006 |
Faculty Meeting |
April 20, 2006 |
Particles and Fluids Abstract: I will discuss systems in which particles interact with a fluid that carries them. The particles are agitated by thermal noise and constrained by inter-particle potentials. The system is modeled by a nonlinear Fokker-Planck equation, describing the particles, coupled with the Navier-Stokes equations, describing the fluid. In the absence of coupling to a fluid, the nonlinear Fokker-Planck equation has a gradient structure with multiple steady states. I will talk about some of the existing results and some of the open questions arising in the presence of coupling. |
Colloquium Chair: Gieri Simonett
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