Abstracts of 2013-2014 UW-PIMS Mathematics Colloquia
June 6, 2014 at 2:30pm
Thomson Hall 101
Ravi Ramakrishna
Cornell University
Galois Representations

In the last 30 years representations of (infinite) Galois groups have played an increasingly important role in number theory. Indeed, arithmetic objects such as the Diophantine equation y2=x3+x2+1 or xn+yn=zn often have attached Galois representations that 'know' the solutions. This talk will survey of a small slice of this theory and will be accessible to mathematicians in all disciplines.

May 23, 2014 at 2:30pm
Loew Hall 101
Gene Abrams
University of Colorado at Colorado Springs
Leavitt path algebras - Something for everyone: algebra, analysis, dynamics, graph theory, number theory

The rings studied by students in most first-year algebra courses turn out to have what's known as the "Invariant Basis Number" property: for every pair of positive integers m and n, if the free left R-modules RRm and RRn are isomorphic, then m = n. For instance, the IBN property in the context of fields boils down to the statement that any two bases of a vector space must have the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the Fundamental Theorem for Finitely Generated Abelian Groups.

In seminal work completed in the early 1960's, Bill Leavitt produced a specifc, universal collection of algebras which fail to have IBN. While it's fair to say that these algebras were initially viewed as mere pathologies, it's just as fair to say that these now-so-called Leavitt algebras currently play a central, fundamental role in numerous lines of research in both algebra and analysis.

More generally, from any directed graph E and any field K one can build the Leavitt path algebra LK(E). In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined in 2004; over the ensuing decade, the subject has matured well into adolescence, currently enjoying a seemingly constant opening of new lines of investigation, and the significant advancement of existing lines. I'll give an overview of some of the work on Leavitt path algebras which has occurred in their first ten years of existence, as well as mention some of the future directions and open questions in the subject.

There should be something for everyone in this presentation, including and especially algebraists, analysts, flow dynamicists, and graph theorists. We'll also present an elementary number theoretic observation which provides the foundation for one of the recent main results in Leavitt path algebras, a result which has had a number of important applications, including one in the theory of simple groups. The talk will be aimed at a general audience; for most of the presentation, a basic course in rings and modules will provide more-than-adequate background.

May 19, 2014 at 2:30pm
Loew Hall 106
Shige Peng
Shandong University
Brownian motion under Knightian uncertainty and path-dependent risk

A typical measure of risk in finance must take into account of the uncertainty of probability model itself (called Knightian uncertainty). Nonlinear expectation and the corresponding non-linear distributions provides a deep and powerful tool: cumulated nonlinear i.i.d random variables of order 1/n tend to a maximal distribution, according a new law of large number, whereas, the accumulation of order 1/√n tends to a nonlinear normal distribution which is the corresponding central limit theorem. The continuous time uncertainty cumulation forms a nonlinear Brownian motion.

The related stochastic calculus under Knightian uncertainty provides us a powerful tool of valuation for path-dependent derivatives. The corresponding Feynman-Kac formula gives one to one correspondence between fully nonlinear parabolic partial differential equations and backward stochastic differential equations driven by the nonlinear Brownian motion.

May 16, 2014 at 2:30pm
Loew Hall 113
Victor Reiner
University of Minnesota
Factoring cycles

A classic result of Hurwitz, often credited to Dénes, says that in the symmetric group on n letters, there are nn-2 ways to factor an n-cycle into n-1 transpositions. Recent joint work with J. Lewis and D. Stanton (arXiv:1308.1468) uncovered a finite field q-analogue: in the general linear group GLn (Fq), there are (qn-1)n-1 ways to factor a Singer cycle into n reflections.

This talk will discuss what this means, and how to prove such things.

May 8, 2014 at 3:30pm (Joint Math/CSE Colloquim and CORE Seminar)
Electrical Engineering Building 105
Dan Spielman
Yale Institute for Network Science
Spectral Sparsification of Graphs

We introduce a notion of what it means for one graph to be a good spectral approximation of another, and prove that every graph can be well-approximated by a graph with few edges.

We ask how well a given graph can be approximated by a sparse graph. Expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor.

We also present an efficient randomized algorithm for constructing sparse approximations that only uses a logarithmic factor more edges than optimal.

Our algorithms follow from the solution of a problem in linear algebra. Given any expression of a rank-n symmetric matrix A as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.

This is joint work with Joshua Batson, Nikhil Srivastava and Shang-Hua Teng.

May 2, 2014 at 2:30pm
Loew Hall 101
Steffen Rohde
University of Washington
How to Draw Trees, and Why

Trees are ubiquitous in mathematics. They appear as basic objects in combinatorics and probability, as dendrites in topology and dynamics, and as real trees in geometry and analysis. In this talk, we will see that finite trees have a canonical embedding in the plane (via Shabat polynomials as Grothendieck dessin d'enfants), that conformal maps help to actually compute these embeddings, and that self-similar Julia sets from complex dynamics can be viewed as limits of these finite trees. We will also discuss motivation from probability theory and statistical physics, particularly stochastically self-similar trees (such as the Aldous' Continuum Random Tree), the Brownian map, and conjectured relations to Liouville Quantum Gravity.

April 25, 2014 at 2:30pm
Loew Hall 101
Roman Bezrukavnikov
Massachusetts Institute of Technology
Characters of finite Chevalley groups and categorification

Representation theory of finite groups seeks to understand functions on the group known as irreducible characters. A group like GL(n,F_q), the group of invertible square matrices with entries in a finite field F_q, originates in algebraic geometry, thus its irreducible characters should also be understood via algebraic geometry. A magnificent realization of that idea is provided by the theory of character sheaves created by Lusztig in 1980’s. I will explain how a systematic use of categorification allows to make some results of this theory including classification of character sheaves more transparent. The talk is based on a joint with with M. Finkelberg and V. Ostrik and a work in progress with D. Kazhdan and Y. Varshavsky.

April 18, 2014 at 2:30pm
Loew Hall 101
Dan Shumow
Senior SDE at Microsoft Research
From Algebraic Geometry to Politics: How Cryptography Connects Abstract Mathematics and Everyday Life

Cryptography is the mathematical science of keeping secrets and maintaining trust. Once an arcane study of primarily military significance, over the past several decades the advent of the internet has invigorated this science. There are now results in cryptography that draw from some of the most abstract branches of mathematics, such as Number Theory and Algebraic Geometry. While at the same time the applications of cryptography have proliferated widely. From the cellphones, to communications, to our personal data held in "the cloud" and the entertainment media we consume it is difficult to find a place where cryptography is not a part of modern life. Furthermore, government surveillance and electronic warfare have pulled it into the world of politics. This talk will focus on how the abstract mathematics used in cryptography have become of integral importance to the practical concerns of everyday life. To provide illustration I will draw on several examples of cryptography our daily lives and the news and discuss the underlying Mathematics.

March 7, 2014 at 2:30pm
Denny Hall 216
Neal Koblitz
University of Washington
A Mathematician in the World of Cybersecurity

I will describe some scary moments that have occurred in the history of elliptic curve cryptography almost from its inception; talk about some of the ironies of controversies in cryptography -- such as RSA's role as defender of strong encryption; and comment on the complicated role of the NSA in cryptography. At the end I will describe some weird recent conversations I have had relating to the NSA.

January 13, 2014 at 2:30pm
Thomson Hall 119
Bianca Viray
Brown University
The local to global principle for rational points

Let X be a connected smooth projective variety over Q. If X has a Q point, then X must have local points, i.e. points over the reals and over the p-adic completions Q_p. However, local solubility is often not sufficient. Manin showed that quadratic reciprocity together with higher reciprocity laws can obstruct the existence of a Q point (a global point) even when there exist local points. We will give an overview of this obstruction (in the case of quadratic reciprocity) and then show that for certain surfaces, this reciprocity obstruction can be viewed in a geometric manner. More precisely, we will show that for degree 4 del Pezzo surfaces, Manin's obstruction to the existence of a rational point is equivalent to the surface being fibered into genus 1 curves, each of which fail to be locally solvable. This talk will be suitable for a general audience.

January 10, 2014 at 2:30pm
Denny Hall 216
Wei Ho
Columbia University
Arithmetic invariant theory and applications

The origins of "arithmetic invariant theory" come from the work of Gauss, who used integer binary quadratic forms to study ideal class groups of quadratic fields. The underlying philosophy---parametrizing arithmetic and geometric objects by orbits of group representations---has now been used to study higher degree number fields, curves, and higher-dimensional varieties. We will discuss some of these constructions and highlight the applications to topics such as bounding ranks of elliptic curves and dynamics on K3 surfaces.

This talk is intended for a general mathematical audience.

November 22, 2013 at 2:30pm
Mary Gates Hall 231
Gil Kalai
Einstein Institute of Mathematics, Hebrew University of Jerusalem
Why quantum computers cannot work

Quantum computers are hypothetical devices based on quantum physics that can out-perform classical computers. A famous algorithm by Peter Shor shows that quantum computers can factor an n-digit integer in n³ steps, exponentially better than the number of steps required by the best known classical algorithms. The question of whether quantum computers are realistic is one of the most fascinating and clear-cut scientific problems of our time.

What makes it hard to believe that superior quantum computers *can* be built is that building them represents a completely new reality in terms of controlled and observed quantum evolutions, and also a new computational complexity reality. What makes it hard to believe that quantum computers *cannot* be built is that this may require profoundly new insights into the understanding of quantum mechanical systems.

My work is geared toward a negative answer, and I offer an explanation within the framework of quantum mechanics, for why quantum computers cannot be built.

I will also mention some highlights from a scientific debate on the matter between myself and Aram Harrow (started here).

October 25, 2013 at 2:30pm
Mary Gates Hall 231
Greg Blekherman
Georgia Institute of Technology
Nonnegative Polynomials, Moment Problems and Real Symmetric Tensor Decompositions

The study of nonnegative polynomials is a basic problem in real algebraic geometry. Truncated moment problem is a classical question in real analysis. Symmetric tensor decompositions are of interest in many areas of applied mathematics. The main goal of this talk is to explain the tight connections between these three topics. I will also present some recent results.

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Last modified: June 2, 2014, 11:55

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