Noether Abstracts

Walking on Graphs the Representation Theory Way

posted Apr 5, 2014, 10:41 AM by AWM Web Editor   [ updated Apr 5, 2014, 10:41 AM ]

Georgia Benkart

AWM Emmy Noether Lecture
January 2014
Baltimore, MD

Abstract. How 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.

A Hasse principle for quadratic forms over function fields

posted Mar 2, 2013, 8:37 PM by AWM Web Editor   [ updated Mar 3, 2013, 5:38 AM ]

Raman Parimala

AWM Emmy Noether Lecture
January 2012
San Diego, CA

Abstract. The term `Hasse principle’ stands for the idea of finding integer solutions to equations by piecing together solutions modulo different prime powers. Solutions modulo prime powers lead to solutions in the completions of the field of rational numbers at p-adic valuations. A classical theorem of Hasse-Minkowski states that a quadratic form over rational numbers admits a nontrivial solution if it does over all p-adic completions and is indefinite over real numbers. There are more general theorems on the obstructions to the Hasse principle for the existence of rational points on homogeneous spaces under linear algebraic groups defined over number fields. We shall discuss formulations of the Hasse principle over function fields and some consequences for quadratic forms.

You Can’t Hear the Shape of a Manifold

posted Dec 6, 2010, 11:20 AM by AWM Web Editor

Carolyn S. Gordon

AWM Emmy Noether Lecture
January 2010
San Francisco, CA

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.

New Directions in Graph Theory

posted Jul 9, 2010, 10:39 AM by Glenna Buford

Fan Chung Graham

AWM Emmy Noether Lecture
January 2009
Washington, D.C.

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.

Fun with Zeta Functions of Graphs

posted Jul 9, 2010, 10:37 AM by Glenna Buford

Audrey A. Terras
University of California, San Diego

AWM Emmy Noether Lecture
January 2008
San Diego, CA

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.

Automorphisms of Free Groups, Outer Space and Beyond

posted Jul 9, 2010, 10:35 AM by Glenna Buford

Karen Vogtmann
Cornell University

AWM Emmy Noether Lecture
January 2007
New Orleans, LA

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.

Mathematical Results and Challenges in Learning Theory

posted Jul 9, 2010, 10:33 AM by Glenna Buford

Ingrid Daubechies
Princeton University

AWM Emmy Noether Lecture
January 2006
San Antonio, Texas

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.)

Symbolic dynamics for geodesic flows

posted Jul 9, 2010, 10:30 AM by Glenna Buford

Svetlana Katok 
The Pennsylvania State University

AWM Emmy Noether Lecture
January 2004
Phoenix, Arizona

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.

Five Little Crystals and How They Grew

posted Jul 9, 2010, 10:29 AM by Glenna Buford

Jean E. Taylor
Rutgers University and Courant Institute of Mathematical Sciences

AWM Emmy Noether Lecture
Thursday, January 16, 2003
Baltimore, Maryland

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.

Computing over the Reals: Where Turing Meets Newton

posted Jul 9, 2010, 10:26 AM by Glenna Buford

Lenore Blum
Distinguished Career Professor of Computer Science, Carnegie Mellon University

AWM Emmy Noether Lecture
Monday, January 7, 2002
9:00 a.m. - 9:50 a.m. 
Room 6 A/B San Diego Convention Center
San Diego, California

Abstract. The classical (Turing) theory of computation has been extraordinarily successful in providing the foundations and framework for theoretical computer science. Yet its dependence on 0's and 1's is fundamentally inadequate for providing such a foundation for modern scientific computation where most algorithms - with origins in Newton, Euler, Gauss, et al. - are real number algorithms.

In 1989, Mike Shub, Steve Smale and I introduced a theory of computation and complexity over an arbitrary ring or field R. If R is Z2 = {0,1}, the classical computer science theory is recovered. If R is the field of real numbers, Newton's algorithm, the paradigm algorithm of numerical analysis, fits naturally into our model of computation.

Complexity classes P, NP and the fundamental question "Does P= NP?" can be formulated naturally over an arbitrary ring R. The answer to the fundamental question depends on the complexity of deciding feasibility of polynomial systems over R. When R is Z2, this becomes the classical satisfiability problem of Cook-Karp-Levin. When R is the field of complex numbers, the answer depends on the complexity of Hilbert's Nullstellensatz.

The notion of reduction between problems (e.g. between traveling salesman and satisfiability ) has been a powerful tool in classical complexity theory. But now, in addition, the transfer of complexity results from one domain to another becomes a real possibility. For example, we can ask: Suppose we can show P = NP over the complex numbers (using all the mathematics that is natural here). Then can we conclude that P=NP over another field such as the algebraic numbers or even over Z2? (Answer: Yes and essentially yes.)

In this talk, I will discuss these results and indicate how basic notions from numerical analysis such as condition, round off and approximation are being introduced into complexity theory, bringing together ideas germinating from the real calculus of Newton and the discrete computation of computer science.

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