# Mathematics Colloquia, Spring 2003

Thursdays 4:10 pm in 1431 Stevenson Center, unless otherwise noted
Tea at 3:30 pm in 1425 Stevenson Center

January 23, 2003  February 13, 2003 Glenn Webb, Vanderbilt University Mailborne Transmission of Anthrax: Modeling and Implications Abstract: A mathematical model is developed to analyze the transmission of inhalational anthrax through the postal system by cross-contamination of mail. The model consists of state vectors describing the numbers of cross-contaminated letters generated, the numbers of anthrax spores on these letters, the numbers of resulting infections in recipients, and matrices of transition probabilities acting on these vectors. The model simulates the recent outbreak in the US, and provides a general framework to investigate the potential impact of possible future outbreaks. Contact person: Dietmar Bisch February 5, 2003 Charles Chui, University of Missouri-St. Louis & Stanford University Note: Wednesday, 4:10 pm in SC 1431 Image Edge Analysis and Noise Removal Abstract: Inspired by the work of Mumford and Shah on problems in computer vision, the total variational norm is shown to be a natural choice for the Perona-Malik anisotropic diffusion approach to image edge enhancement, for which the bilateral filter provides a most effective computational scheme. We recently extended this to a trilateral filter for removing both white and impulse noises. Contact person: Larry Schumaker February 6, 2003 Stanley Chang, Wellesley College A New Invariant and the Surgery Exact Sequence Abstract: We will construct a higher" Hirzebruch-type invariant of compact manifolds based on the L2-signature and motivated by the work of Cheeger-Gromov. This invariant is useful in studying the structure set of manifolds whose fundamental group contains torsion. The talk will be geared towards a graduate student audience. Contact person: Guoliang Yu Matthias Hieber, University of Darmstadt (visiting UC Berkeley) Maximal Lp Regularity for Parabolic Evolution Equations Abstract: Lp properties of solutions for linear parabolic equations have far reaching consequences for many nonlinear problems, such as free boundary problems. In this talk we discuss the developments of the so-called maximal Lp regularity problem in the last year and show how it is related to heat-kernel bounds, imaginary powers, Fourier multipliers and the notion of R-boundedness. Contact person: Gieri Simonett Neil Robertson, Ohio State University The Strong Perfect Graph Theorem Abstract: This talk concerns the perfect graphs of Claude Berge. When for all (vertex) induced subgraphs H of a graph G the maximum clique size of H equals the chromatic number of H then G is said to be perfect. The simplest examples of perfect graphs are the 2-colorable (bipartite) graphs B and of nonperfect graphs are the simple circuits C(t) of length t>3. It is easy to see that the (edge-set) complements of such graphs B and C(t) are also perfect and not perfect, respectively. Berge, in a study of the Shannon capacity of graphs, further noted that the line-graphs of bipartite graphs and their complements are perfect, so that their capacity is easy to compute, while graphs with the above odd circuits or their complements as induced subgraphs (these are called the odd holes or antiholes, respectively, of G) are not perfect and the capacity is difficult to compute. From this evidence he made his famous conjecture in 1961 that a graph is perfect if and only it contains no odd hole or antihole. The direct corollary, that G is perfect if and only if its complement is perfect, was proved in 1972 by Lovasz. This established the Berge conjecture as a premier graph coloring problem. In 1976 Appel and Haken proved the 4-color conjecture for planar graphs, bringing Berge's conjecture to the forefront. Strong attacks on this problem were made by several graph theorists associated with Berge, Lovasz, Chvatal and Cornuejols over the years. In June of 2002 this conjecture was proved by another group (Chudnovsky, Robertson, Seymour and Thomas), using methods of structural graph theory. This talk will discuss the problem further and will describe the general methods and the line of proof in an intuitive way. A 160-page paper covering the proof details is available on the home page of Robin Thomas. More recent joint work of Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic has developed a polynomial-time algorithm to recognize a perfect graph, not depending on the graph structural decomposition theorems used to prove the Berge conjecture. Contact person: Mike Plummer Edward Swartz, Cornell University Representations of Matroids Note: Monday, 4:10 pm in SC 1431 Abstract: What is the nature of linear independence over fields of different characteristics? For a specific vector space, what are the possible geometric point configurations? Matroids, introduced by Whitney in 1935, are a framework for answering these and other questions involving notions of independence such as algebraic independence. In the 70's researchers of real hyperplane arrangements, the simplex algorithm and directed graphs were independently and simultaneously led to oriented matroids. This combinatorial abstraction of linear independence in an ordered field can always be realized by an arrangement of pseudospheres. We now know that if we allow homotopy spheres then all matroids have such a representation. Contact person: Paul Edelman Michael Burns, UC Berkeley Planar Operations on Subfactors Abstract: Jones' planar algebra formalism provides the most elegant and powerful description of the standard invariant of a finite index, extremal II1 subfactor, allowing the use of diagramatic techniques to prove results in the theory of operator algebras. After reviewing some of the theory of planar algebras, von Neumann algebras and subfactors, we will discuss a number of extensions of the planar algebra results. Contact person: Dietmar Bisch Nick Wright, Vanderbilt University Coarse Geometry and Scalar Curvature Note: Monday, 4:10 pm in SC 1431 Abstract: For manifolds, one of the most intuitive geometric properties is the curvature. The scalar curvature is dependent on the Riemannian metric however the topology also plays a role in determining whether there are metrics with positive curvature. Coarse geometry studies the large scale structure of a manifold and is a useful tool for analyzing curvature questions. I will describe the ideas and methods underlying coarse geometry. The relation with curvature is given by a geometric differential operator (the Dirac operator). The index theory for this operator gives various obstructions to positive scalar curvature. I will present some of these obstructions on open manifolds and draw conclusions for general closed manifolds. Contact person: Guoliang Yu Martin Kassabov, Yale University Kazhdan Property and Finite Graphs Note: Tuesday, 3:10 pm in SC 1424 Abstract: In this talk I survey several classical results about Kazhdan Property T and apply them to two combinatorial problems involving finite graphs --- construction of family of expanders and working time of product replacement algorithm in computational group theory. Kazhdan property T originated from the representation theory of Lie groups. Shortly after its introduction it was used by Margulis to construct an explicit example of a family of expanders. Unfortunately, the expanding constant of this family was unknown, because all proofs that a group has a property T were not quantitative, and the expanding constants of this family of expanders was unknown. In a resent paper, A. Lubotzky and I. Pak showed that Kazhdan property T of the group SLn(Z) implies that the working time of the product replacement algorithm on k-generated abelian groups is logarithmic in the size of the groups, but its dependence on k was unknown. A recent result by Y. Shalom gave an explicit bound of the Kazhdan constant for the group SLn(Z), which lead to quantitative bounds for the constants in these two combinatorial constructions. Contact person: Mark Sapir Sorin Popa, UCLA L2-Betti Numbers and the Fundamental Group of Finite von Neumann Factors Abstract: Click here to download the pdf file of the abstract. The fundamental group F(M) of a type II1 factor M was introduced by Murray and von Neumann in 1943 in connection with their notion of continuous dimension. It measures the extent to which amplifications'' of M are isomorphic to M (e.g., if the algebra of 2x2 matrices over M is isomorhic to M then 2 is in F(M)}. It is a puzzling and still poorly understood invariant. We will present results providing the first examples of factors M with trivial fundamental group. Thus, if G is the arithmetic group Z2 \rtimes SL(2, Z) and M=L(G) is the associated group von Neumann algebra then F(M)={1}. The proof uses in a crucial way the weak amenability'' of \Gamma=SL(2, Z) (i.e. Haagerup's approximation property or equivalently Gromov's a-T-menability) and the relative property (T) of Kazhdan-Margulis of the inclusion Z2 \subset Z2 \rtimes \Gamma. The combination of these two properties makes it possible to prove a unique decomposition of M as a cross-product M = L^\infty(T2) \rtimes \Gamma, thus allowing us to define l2-Betti number invariants bn(M) from the l2-Betti numbers bn(R), defined by Gaboriau in 2001, of the equivalence relation R induced by \Gamma on T2. Contact persons: Dietmar Bisch and Gennadi Kasparov Ralph McKenzie, Vanderbilt University Defining and Recognizing Structure in General Algebras; Congruence Lattices are the Key to Deep Results Abstract: In the last three decades of the twentieth century, universal algebra began to realize many of the lofty goals Garrett Birkhoff had envisioned for it in 1933. Especially notable is the ability to formulate and proof deep results about all finite and locally finite algebraic sysems. Tame congruence theory is an analysis of the possible ways a clone of operations on a set may be organized relative to a covering pair of congruences that it admits. Applied to all the covering pairs of congruences of a finite algebra A, and also those of finite algebras of functions derived from A, this theory reveals a wealth of previously unrecognized structural features in finite algebras, and provides natural and useful new ways of classifying them. The task of working out the implications and extending the insights of tame congruence theory has been the dominant theme of research in general algebra for the past twenty years. Many of the results discovered with its aid have since been extended by other means to all algebraic systems (without local finiteness assumptions). I originated this theory in 1981--84 (with the considerable help of my then graduate student David Hobby). In this talk, I will tell the story of how a long-running fascination with one little problem and several big problems, combined with stubbornness and luck, led to some big results. Contact person: Dietmar Bisch Zhong-Jin Ruan, University of Illinois at Urbana-Champaign Operator Spaces: A Natural Non-commutative Quantization of Functional Analysis Abstract: An operator space is a norm closed subspace of bounded operators on some Hilbert space together with a distinguished matrix norm''. Morphisms between operator spaces are completely bounded linear maps''. Operator space theory is a natural non-commutative quantization of functional analysis (Banach space theory). In this talk, I will first discuss some fundamental results in operator spaces, and then discuss some interesting applications to operator algebras and non-commutative harmonic analysis. Contact person: Guoliang Yu and Dechao Zheng Zhenghan Wang, Indiana University Topological Quantum Computation Abstract: An equivalent model of quantum computing based on topological quantum field theories has been proposed in the work of Freedman, Kitaev, Larsen and Wang. This new way of looking at quantum computation provides efficient quantum algorithms to approximately compute quantum invariants of links and 3-manifolds, and a possible way to realize a large scale quantum computer. We will start with a general introduction to quantum information science, and then discuss the connection to topology, computer science and condensed matter physics. Contact persons: Dietmar Bisch and Bruce Hughes Rostislav Grigorchuk, Texas A&M University The Ihara Zeta Function of Infinite Graphs, the KNS Spectral Measure and Integrable Maps Note: Special Colloquium, Monday, 4:10pm in SC 1431 (Part 1 of talk 4:10-5:00pm, 5 minutes break, Part 2 of talk 5:05-5:50pm) Abstract: We define the Ihara zeta function for Cayley graphs of infinite finitely generated groups. We extend the definition of the Ihara zeta function to infinite graphs which are limits of sequences Xn of finite k-regular graphs such that Xn+1 covers Xn. We associate to such a graph a measure mu with support in [-1,1] called the Kesten-von Neumann-Serre spectral measure. We present a few examples of computation of zeta function and a measure \mu for Schreier graphs of some fractal groups generated by finite automata. These computations are closely related to the integrability of some 2-dimensional mappings which are also in focus of our considerations. (joint work with A. Zuk (University of Chicago)) Contact person: Mark Sapir Yuri Bahturin, visiting Vanderbilt University Note: Tuesday, 4:10-5:00pm in SC 1431 Bicharacters on Hopf Algebras Abstract: The notion of skew-symmetric bicharacter on a Hopf algebra is dual to that of R-matrix widely used in mathematics and beyond. It appears in the theory of quantum groups, Yang - Baxter equations, etc. While R-matrices work in the case of finite-dimensional Hopf algebras, bicharacters are more universal and work fine in the case of infinite dimensions. One of the basic examples of bicharacters are the commutation factors on abelian groups used, in particular, to define so called color Lie superalgebras, a notion generated in physics few decades ago. In general, bicharacters on a commutative and cocommutative Hopf algebra H allow one to define Lie structures on the algebras with the coaction H. On the other hand, Hopf algebras whose structure includes a fixed skew-symmetric bicharacter form an important class of cotriangular Hopf algebras, dual to the triangular ones introduced by Drinfeld. If we want to classify such Lie structures or such Hopf algebras, we may apply so called Scheunert's trick, saying that any skew-symmetric bicharacter b(g,h) on a finitely generated abelian group G can be written in the form b(g,h)=a(g,h)[s(g,h)/s(h,g)], where a(g,h) is either trivial or the bicharacter defining ordinary Lie superalgebras and s(g,h) is a not necessarily skew-symmetric bicharacter. In particular, if we deform the product in a G-graded algebra using s(g,h), then the color commutator defined by b(g,h) becomes either an ordinary Lie bracket or an ordinary superbracket. The goal of this talk is to report on most recent result in this area, including the extension of Scheunert's trick to arbitrary cocommutative Hopf algebras of characteristic different from 2 and the classification of finite-dimensional algebras, which are commutative under a suitable generalized Lie bracket. Contact person: Mark Sapir

Colloquium Chair (Spring 2003): Dietmar Bisch