| May 20, 2011 | |
| Eyal Lubetzky
Microsoft Research and UW |
The Static and Stochastic Ising Models |
|
The Ising model, one of the most studied models in mathematical physics, was
introduced in 1925 to model ferromagnetism. Over the last three decades,
significant effort has been dedicated to the analysis of stochastic
dynamical systems that both model the evolution of the Ising model and
provide efficient methods for sampling from it. In this talk I will survey
the rich interplay between the behaviors of the static and the dynamical
models as they both undergo a phase transition at the critical temperature.
In particular I will discuss a fundamental problem on the dynamical model
introduced by Glauber in 1963, which we recently settled in joint work with
Allan Sly. The talk will not assume any prior knowledge on the Ising model
and its critical phenomena. |
|
| April 29, 2011 | |
| Rekha Thomas
University of Washington |
From Hilbert's 17th Problem to Polynomial Optimization and Convex Algebraic Geometry |
|
Polynomial optimization concerns minimizing a polynomial subject to polynomial equations and inequalities. While this is a natural model for many applications, there are many difficulties (usually numerical and algorithmic) that have prevented their wide-spread use. However, in the last 10 years, several research streams in math and engineering have come together to breathe new life into this important class of problems. The story starts with Hilbert's work on nonnegative polynomials, but then goes on to use ideas from many branches of mathematics such as real algebraic geometry, convex analysis, functional analysis, optimization, probability and combinatorics. In particular, this is an area where algebra and analysis become naturally intertwined. I will attempt a (biased) survey of the main ideas that has helped in this development and defined a new field called "convex algebraic geometry." View
video of this lecture. |
|
| March 29, 2011 | |
| Hariharan Narayanan
Massachusetts Institute of Technology |
Testing the Manifold Hypothesis |
|
Increasingly, we are confronted with very high dimensional data sets. As a result, methods of avoiding the curse of dimensionality have come to the forefront of machine learning research. One approach, which relies on exploiting the geometry of the data, has evolved into a subfield called manifold learning. The underlying hypothesis of this field is that due to
constraints that limit the degrees of freedom of the generative process,
data tend to lie near a low dimensional submanifold. This has been
empirically observed to be the case, for example, in speech and video data.
Although there are many widely used algorithms motivated by this hypothesis,
the basic question of testing this hypothesis is poorly understood. We will
describe an approach to test this hypothesis from random data. |
|
| February 18, 2011 | |
| Béla Bollobás
University of Cambridge, University of Memphis, and Microsoft |
The Critical Probability of Percolation Percolation on Self-Dual Polygon Configurations |
|
In this talk I shall sketch some results Oliver Riordan of Oxford and I have obtained on critical probabilities in percolation.
Recently, Scullard and Ziff noticed that a broad class of planar percolation
models are self-dual under a simple condition which, in a parametrized
version of such a model, reduces to a single equation. They stated that the
solution of the resulting equation gave the critical point. However, just as
in the classical case of bond percolation on the square lattice, noticing
self-duality is simply the starting point: the mathematical difficulty is
precisely showing that self-duality implies criticality. Riordan and I have
managed to overcome this difficulty: we have shown that for a generalization
of the models considered by Scullard and Ziff self-duality indeed implies
criticality. View
video of this lecture. |
|
| January 14, 2011 | |
| Richard Stanley
Massachusetts Institute of Technology |
A Survey of Alternating Permutations |
|
An alternating permutation \(w=a_1\cdots
a_n\) of \(1,2,\dots,n\) is a permutation such that \(a_i>a_{i+1}\) if and only
if \(i\) is odd. If \(E_n\) (called an Euler number) denotes the number
of alternating permutations of \(1,2,\dots,n\), then \(\sum_{n\geq 0}E_n\frac{x^n}{n!}=\sec
x+\tan x\). We will discuss such topics as other occurrences of Euler numbers
in mathematics, umbral enumeration of classes of alternating permutations,
and longest alternating subsequences of permutations.
View
video of this lecture. |
|
| November 23, 2010 | |
| Christopher Hacon
University of Utah |
Birational Classification of Algebraic Varieties |
|
Complex algebraic varieties are defined by systems of polynomial equations over the field of
complex numbers. Their geometry has been extensively studied over the years. The 1 dimensional case corresponds to Riemann
surfaces. In dimension 2 we have the theory algebraic surfaces which was understood by the Italian school of Algebraic Geometry
at the beginning of the 20th century. The Minimal Model Program aims to generalize these results to higher dimensions. The 3
dimensional case was understood in the 1980s by celebrated work of Mori and others. In this talk I will discuss recent developments
on the classification of algebraic varieties in all dimensions.
(This talk is of an introductory nature and does not require
previous knowledge of the minimal model program.)
|
|
| October 29, 2010 | |
| Tatiana Toro
University of Washington |
Potential Theory Meets Geometric Measure Theory |
|
A central question in Potential Theory is the extent to which the geometry of a domain
influences the boundary regularity of solutions to divergence form elliptic operators.
To answer this question one studies the properties of the corresponding elliptic
measure. On the other hand one of the central questions in Geometric Measure Theory
(GMT) is the extent to which the regularity of a measure determines the geometry of its
support. The goal of this talk is to present a few instances in which techniques from
GMT and Harmonic Analysis come together to produce new results in both of these areas.
View
video of this lecture. |
|
| October 15, 2010 | |
| Sebastian Casalaina-Martin
University of Colorado, Boulder |
The Geometry of Riemann's Theta Functions |
|
Riemann's theta functions are solutions of the heat equation that carry a
tremendous amount of geometric information. Originally studied in connection
with elliptic integrals, it was later realized that the zero loci of these
functions, called theta divisors, carry geometric data of an associated Riemann
surface. These functions have been used extensively by algebraic geometers to
understand basic properties of complex projective manifolds, and more generally,
solution sets of algebraic equations over an arbitrary algebraically closed
field. In this talk, after reviewing some of the historical background on
addition formulas for elliptic integrals, I will discuss theta functions, and
the Riemann singularity theorem. Some recent results extending these classical
results will also be covered. Time permitting, I will present some applications
due to Clemens-Griffiths and Mumford. View
video
of this lecture. |
|
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