Colloquia, AY 2006-2007
Colloquium, AY 2006-2007
Tea at 3:30 pm in 1425 Stevenson Center
| September 7, 2006 |
Faculty Meeting |
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| September 14, 2006 |
Quantum bluffing in entangled poker – An introduction to quantum game theory Abstract: Computers and networks that exploit the bizarre properties of quantum mechanics will have capabilities far exceeding those of the conventional computing environment. The encryption of data, the searching of databases, and even the play of simple games such as on-line poker will undergo profound changes when implemented in the quantum environment. This is because players who communicate their strategic choices via |
| September 21, 2006 |
Lattices in products of trees and simple groups Abstract: The study of groups of tree automorphisms viewed as analogs of semisimple Lie groups led to various interesting results. Studying lattices in products of trees one encounters both properties familiar from the case of Lie groups as well as new phenomena. The interplay between these led to a construction, jointly with Marc Burger, of a finitely presented torsion free simple group, this turned out also to be the first example of a simple group which is an amalgam of two free groups. More recently using a similar strategy (and many other ideas) P.-E. Caprace and B. Remy have shown the simplicity of a large family of Kac-Moody groups.
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| September 28, 2006 |
Compactifications of symmetric and locally symmetric spaces Abstract: Symmetric and locally symmetric spaces are special Riemannian manifolds closely related to Lie groups. They arise naturally in many different subjects, for example, as the moduli spaces of elliptic curves and abelian varieties, and the moduli spaces of positive definite quadratic forms and their equivalence classes. Many such natural spaces are non-compact, and an important problem is to compactify them. In fact, a number of different compactifications have been constructed, motivated by various applications. In this talk I will give a survey of compactifications of both symmetric and locally symmetric spaces, using simple examples such as the upper half-plane and its arithmetic quotients. Compactifications of symmetric spaces have often been studied by methods quite different from those those used for locally symmetric spaces. Here we will emphasize a uniform approach to both problems. |
| October 5, 2006 |
Richard Thompson’s group F: around the amenability problem Abstract: We well discuss the famous problem about amenability of Richard Thompson’s group F. This will include descriptions of the most popular criteria of amenability and some basic properties of the group F. Also we are going to give a survey of recent progress in this area and discuss some approaches to the problem. |
| October 12, 2006 |
Canceled Alexander Volberg, Michigan State University Scattering for Jacobi matrices and discrete Schroedinger operators, non-linear Fourier transforms and related questions of weighted Hilbert tansform Abstract: We introduce a certain notion of scattering, show its relations with non-linear Fourier transform on one side and with weighted estimates (with matrix weights) for the Hilbert transform on the other side. |
| October 19, 2006 |
Formal proofs in geometry Abstract: Traditional mathematical proofs are written in a way to make them easily understood by mathematicians. Routine logical steps are omitted. An enormous amount of context is assumed on the part of the reader. Proofs, especially in topology and geometry, rely on intuitive arguments in situations where a trained mathematician would be capable of translating those intuitive arguments into a more rigorous argument. In a formal proof, all the intermediate logical steps are supplied. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive, and yet less susceptible to logical errors. It is generally considered a major undertaking to transcribe a traditional proof into a formal proof. In recent years, a number of fundamental theorems in mathematics have been formally verified by computer, including the Prime Number Theorem, the Four Color Theorem, and the Jordan Curve Theorem. |
| October 26, 2006 |
Algebraic statistics for computational biology Abstract: This lecture gives an introduction to a book with this title which was published recently by Lior Pachter and myself. It concerns interactions between algebra and statistics and their emerging applications to computational biology. Statistical models of independence and sequence alignment will be illustrated by means of a fictional character, DiaNA, who rolls tetrahedral dice with face labels “A”, “C”, “G” and “T”. |
| November 2, 2006 |
4:10-5:00pm at SC 4327, Co-hosted by the Department of Physics and Astronomy Hidden noncommutative geometric structure of space-time Abstract: The geometry of space-time is reconstructed from the low-energy spectrum defined by the quarks and leptons. I show that there is a hidden noncommutative structure and that the dynamics of the unified geometrical theory is governed by the “Spectral Action Principle”. |
November 8, 2006 |
4:10-5:00pm at SC 1308, Special Lecture Zero-one laws in discrete mathematics Abstract: The fraction of n-vertex finite graphs that are connected grows to 1 as n grows to infinity. In that sense almost all finite graphs are connected. There are numerous results like that. Almost all graphs are Hamiltonian, not 3-colorable, rigid, etc. Each of these results required a separate proof. Is there a general phenomenon behind results of that sort? It turns out that much depends on the logical form of the property in question. In particular, every claim expressible in predicate logic is almost surely true or almost surely false on finite structures. This zero-one law was generalized in various directions. We will explain some of the results.
Contact person: Mark Sapir |
| November 9, 2006 |
Higher-dimensional analogs of stable curves Abstract: Stable curves were introduced in the 60s by Deligne-Mumford (and versions by Mayer, Knudsen, Grothendieck…). Stable maps is a more recent invention of Kontsevich. They have a myriad of applications: most notably to Gromov-Witten invariants and quantum cohomology, but also to such diverse topics as resolutions of singularities in positive characteristic and universal bounds for the number of solutions of diophantine equations. What happens if one replaces ‘curve’ in ‘stable curve or map’ by ‘higher-dimensional variety’? I will explain the current state of the art in the theory that results. |
| November 16, 2006 |
Homological Mirror Symmetry and Applications Abstract: In this talk we will give an elementary introduction to Homological Mirror Symmetry. We will discuss applications to classical problems in Algebraic Geometry. |
| November 23, 2006 |
Thanksgiving |
| November 30, 2006 |
On the superrigidity of malleable actions Please klick for the abstract. |
| December 7, 2006 |
A Taste of Compressed Sensing Abstract: The usual paradigm for encoding signals is based on the Shannon sampling theorem. If the signal is broad-banded then this requires a high sampling rate even though the information content in the signal may be small. Compressed Sensing is an attempt to get out of this dilemma and sample at close to the information rate. The fact that this may be possible is embedded in some old mathematical results in functional analysis, geometry and approximation. This talk will be an excursion into these topics which will focus on the relation between the number of samples we take of a signal and how well we can approximate the signal. It will take place in the discrete setting for vectors in Euclidean space. The talk should be understandable to graduate students and non specialists. |
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| January 11, 2007 |
Increasing and decreasing subsequences Abstract: We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,…,n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. We will also briefly discuss two variations of increasing and decreasing subsequences, viz., alternating subsequences and crossings and nestings of matchings. |
| January 18, 2007 |
Contact person: |
| January 25, 2007 |
On the Langlands functoriality for automorphic forms Abstract: The Langlands functorialty is a central conjecture in the modern theory of automorphic forms, the Langlands program. To motivate my lecture, I recall first some basic problems in number theory, including the famous Artin Conjecture for Galois representations. Then I will discuss the in some details the recent work on the Langlands functoriality and applications to number theory, including for example, the Inverse Galois Problem. |
Wednesday, February 28 |
Poincare’s variational problem in potential theory Abstract: In a famous Acta Mathematica article, Poincare has stated a variational principle as a heuristic basis for his work in potential theory. Later on, Carleman has developed in 2D some of Poincare’s ideas, but they were never put into the framework of modern mathematics. I will show in my talk how this can be done, and what perspectives Poincare’s programme opens today. In particular I will touch the theory of symmetrizable linear operators and some classical aspects of function theory related to the Beurling transform. Based on recent work with Dmitry Khavinson and Harold S. Shapiro. |
| March 8, 2007 |
Spring Break |
| March 15, 2007 |
Free probability and free entropy dimension Abstract: More than 20 years ago, Voiculescu found a noncommutative probabalistic notion called freeness that corresponds to the situation of words in a free group. The theory based on this definition is called free probability theory and in this theory free products play a role analogous to the usual (Cartesian) product of spaces in the classical probabalistic theory of independence. The parallels with classical probability theory are far-reaching and surprising. After introducing freeness, we will describe one of its fundamental examples, which is the asymptotic behavior of random matrices as the matrix size grows without bound. We’ll also describe the use of freeness to investigate von Neumann algebras, including applications of the related quantity, free entropy dimension. |
| March 22, 2007 |
Diffusion methods in image processing Abstract: One of the most famous approaches to the denoising and contrast enhancing of blurred images is based on the famous Perona – Malik equations. Unfortunately these equations are ill posed. Thus no mathematical justification for numerical algorithms based on them is possible. Since the Perona – Malik technique seems to produce excellent results, there have been proposed many modifications of the underlying equations with the aim to obtain, on the one hand side, a sound mathematical theory and, on the other hand side, to preserve the desirable features of the Perona – Malik equations. However, as shall be explained in this talk, these modifications result in an unavoidable smearing of sharp edges and, consequently, in an undesirable loss of information. In our talk we shall present a new approach to the Perona – Malik equations which does not have these shortcomings. In contrast to the widely used space regularization we propose a time regularization technique. It allows for a sound mathematical theory as well, but avoids blurring of sharp edges. We shall addresses a general mathematical audience without any prior knowledge of image processing, explain the diffusion theoretical approach in elementary terms, and illustrate the basic ideas by numerical experiments. |
| March 29, 2007 |
The Kadison-Singer problem in Mathematics and Engineering Abstract: We will see that the 1959 Kadison-Singer Problem in C*-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and engineering. This gives all these research areas common ground on which to interact as well as explaining why each of them has volumes of literature on their respective problems without a satisfactory resolution. We will look at some of the equivalences of KS in operator theory, Banach space theory, harmonic analysis, and applied math/engineering. |
| April 5, 2007 |
Faculty Assembly Contact person: |
| April 12, 2007 |
Discrete versus continuous in analysis Abstract: Analysis might be defined as the study of functions, and function spaces. Such a broad definition encompasses real-valued functions of a real variable, maps between Banach spaces of operators, and much, much more. While functions can be defined on discrete spaces (such as functions on the natural numbers, otherwise known as sequences), most analysts seem to regard the discrete aspects of the subject as being but stepping stones to the ultimate truth, which surely lies in the continuous realm. One might wonder, for instance, at the scant attention paid to difference equations in the undergraduate curriculum, compared to differential equations. In applied harmonic analysis, classical tools like the Fourier transform that apply to analog signals must be discretized in order to apply to digital signals. This has stimulated research on discrete versions of classical function spaces. For example I recently obtained new results on the “analysis” (continuous-to-discrete) and “synthesis” (discrete-to-continuous) operators on Lebesgue and Hardy spaces. I’ll describe this work near the end of the talk, after first exploring some continuous/discrete dichotomies drawn from differential equations, probability, mathematical physics and elementary geometry.
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| April 19, 2007 |
The Poincaré duality theorem and its converse Abstract: The Poincaré duality theorem established isomorphisms between the homology and cohomology of a topological manifold. The duality isomorphisms are the fundamental algebraic topology consequences of a space being locally Euclidean. The surgery theory developed over the last 40 years characterizes the homotopy types of manifolds of dimension >4 in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. The talk will describe a simplicial version of surgery theory, using a combinatorial version of sheaf theory to obtain a converse of the duality theorem in dimension >4: a simplicial complex is homotopy equivalent to a topological manifold if and only if it has sufficient Poincaré duality. |
Colloquium Chair (2006-2007): Gieri Simonett
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