Georgia BenkartHow many walks of n steps are there from point A to point B on a graph? Often finding the answer involves clever combinatorics or tedious treading. But if the graph is the representation graph of a group, representation theory can facilitate the counting and provide much insight. The simply-laced affine Dynkin diagrams are representation graphs of the finite subgroups of the special unitary group SU(2) by the celebrated McKay correspondence. These subgroups are essentially the symmetry groups of the platonic solids, and the correspondence has been shown to have important connections with diverse subjects including mirror symmetry and the resolution of singularities. Inherent in McKay's correspondence is a rich combinatorics coming from the Dynkin diagrams. Some of the ideas involved in seeing this go back to Schur, who used them to establish a remarkable duality between the representation theories of the general linear and symmetric groups. There is a similar duality between the SU(2) subgroups and certain algebras that enable us to count walks and solve other combinatorial problems. In this case, the duality leads to connections with the Temperley-Lieb algebras of statistical mechanics, with partitions, with Catalan numbers, and much more.
Carolyn S. Gordon
Abstract. Inverse spectral problems ask how much information about an object is encoded in spectral data. For example, Mark Kac's question “Can you hear the shape of a drum?" asks whether a plane domain, viewed as a vibrating membrane, is determined by the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, by the characteristic frequencies of vibration. The lecture will focus on Kac's question and its generalization to Riemannian manifolds. We will consider methods for constructing manifolds with the same spectral data and compare examples of such “sound-alike” manifolds. We will also refer to related constructions on discrete and quantum graphs.
Abstract. Nowadays we are surrounded by numerous large information networks, such as the WWW graph, the telephone graph and various social networks. Many new questions arise. How are these graphs formed? What are basic structures of such large networks? How do they evolve? What are the underlying principles that dictate their behavior? How are subgraphs related to the large host graph? What are the main graph invariants that capture the myriad properties of such large sparse graphs and subgraphs.
In this talk, we discuss some recent developments in the study of large sparse graphs and speculate about future directions in graph theory.
Abstract. I will present an introduction to zeta functions of graphs along with some history and comparisons with other zetas from number theory and geometry such as Riemann’s and Selberg’s. Three kinds of graph zetas will be defined: vertex, edge and path. The basic properties will be discussed, including the Ihara formula saying that the zeta function is the reciprocal of a polynomial. I will then explore analogs of the Riemann hypothesis, zero (pole) spacings, and connections with expander graphs and quantum chaos. The graph theory version of the prime number theorem will be discussed. The graphs will be assumed to be finite undirected and possibly irregular. References include my joint papers with Harold Stark in Advances in Math.
Abstract. Outer space was introduced in the mid-1980s as a tool for studying the group Out(Fn) of outer automorphisms of a finitely-generated free group. The basic philosophy is that one should think of an automorphism of a free group as a topological object, either as a homotopy equivalence of a finite graph or as a diffeomorphism of a suitable three-manifold with free fundamental group. There are compelling analogies between the action of Out(Fn) on Outer space and the action of an arithmetic group on a homogeneous space or the action of the mapping class group of a surface on the associated Teichmuller space. In this talk I will first describe Outer Space and explain how it is used to obtain algebraic information about Out(Fn). I will then indicate how Outer Space is related to other areas, from infinite-dimensional Lie algebras to the mathematics of phylogenetic trees, and how ideas from Outer Space are currently expanding in new directions.
Abstract. One of the most widespread applications of learning theory is in ubiquitous search engines, which have to (and do!) classify enormous databases according to (almost) arbitrary criteria. Computer scientists have developed powerful algorithms for these very high-dimensional problems, which typically cannot be tackled by gradient-descent or similar optimization methods. These algorithms and the problems they attack provide very interesting mathematical challenges. The talk will discuss in particular the widely applied AdaBoost algorithm and its properties, as well as some variants. It will review joint work with Cynthia Rudin and Rob Schapire (co-inventor, with Freund, of AdaBoost, for which they were awarded the 2003 Gödel prize.)
Abstract. At the dawn of modern dynamics, in the 1920s, E. Artin and M. Morse discovered that geodesics on surfaces of constant negative curvature may be described by sequences of symbols via certain “coding” procedures. They found one of the first instances of what much later became widely known as “chaotic” behavior. Artin's code of geodesics on the modular surface is closely related to continued fractions. Morse's procedure is more geometric and more widely applicable. For 80 years these classical works provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. Major contributions were made by R. Bowen, C. Series, R. Adler and L. Flatto who interpreted and expanded the classical works in the modern lan guage of symbolic dynamics.
Quite surprisingly, there was room for new results in this well-developed area. Even more surprisingly, Gauss reduction theory leads to a variant of continued fractions that provides a particularly elegant coding of geodesics on the modular surface. This, in turn, brings about new connections with topological Markov chains that, mysteriously, are related to the five Platonic solids.
Abstract. Five ways in which crystals can grow (or shrink, or change their shapes) are discussed: motion by weighted mean curvature motion by surface diffusion motion by surface-attachment-limited kinetics with and without external driving forces, dendritic crystal growth, and motion of crystal aggregates in which individual crystals rotate. All are part of a family of motions which can be formulated and analyzed in the same general framework. Parts of this framework include a surface energy similar to that of soap bubbles (but varying with normal direction) and inner products determining the kinetics.