Abstracts of 2013-2014 UW-PIMS Mathematics Colloquia
June 6, 2014 at 2:30pm
Loew Hall 101
Edward Witten
Institute for Advanced Study


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


May 16, 2014 at 2:30pm
Loew Hall 101
Victor Reiner
University of Minnesota


May 8, 2014 at 2:30pm
Sieg Hall 224
Dan Spielman
Yale Institute for Network Science


May 2, 2014 at 2:30pm
Loew Hall 101
Steffen Rohde
University of Washington


April 25, 2014 at 2:30pm
Loew Hall 101
Roman Bezrukavnikov
Massachusetts Institute of Technology


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: April 10, 2014, 09:35

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