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 y^{2}=x^{3}+x^{2}+1 or x^{n}+y^{n}=z^{n} 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 firstyear 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 Rmodules _{R}R^{m} and _{R}R^{n} 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 nowsocalled 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 L_{K}(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 morethanadequate background. 
May 19, 2014 at 2:30pm Loew Hall 106 

Shige Peng Shandong University 
Brownian motion under Knightian uncertainty and pathdependent 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 nonlinear 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. 
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 n^{n2} ways to factor an ncycle into n1 transpositions. Recent joint work with J. Lewis and D. Stanton (arXiv:1308.1468) uncovered a finite field qanalogue: in the general linear group GL_{n }(_{q}), there are (q^{n}1)^{n1} ways to factor a Singer cycle into n reflections. 
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 wellapproximated by a graph with few edges. 
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 selfsimilar 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 selfsimilar 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 padic 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 philosophyparametrizing arithmetic and geometric objects by orbits of group representationshas now been used to study higher degree number fields, curves, and higherdimensional 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. 
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 outperform classical computers. A famous algorithm by Peter Shor shows that quantum computers can factor an ndigit 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 clearcut scientific problems of our time. 
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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