[CSDM] Fwd: Minerva Lectures at Princeton University

[Avi Wigderson]

-------- Original Message --------
Subject: 	Minerva Lectures at Princeton University
Date: 	Tue, 05 Feb 2013 14:41:45 -0500
From: 	Linda Brozyna <lbrozyna at math.princeton.edu>
To: 	undisclosed-recipients:;



We are pleased to present the Minerva Lectures featuring Professor
Terence Tao of UCLA:

PRINCETON UNIVERSITY MATHEMATICS DEPARTMENT:  MINERVA LECTURES

MINERVA LECTURE I:
Title: Sets with few ordinary lines
Monday, February 11, 2013, 4:30 p.m.
Abstract:  Given n points in the plane, an _ordinary line_ is a line
that contains exactly two of these points, and a _3-rich line_ is a line
that contains exactly three of these points.  An old problem of Dirac
and Motzkin seeks to determine the minimum number of ordinary lines
spanned by n noncollinear points, and an even older problem of Sylvester
(the "orchard planting problem") seeks to determine the maximum number
of 3-rich lines.  In recent work with Ben Green, both these problems
were solved for sufficiently large n, by combining tools from topology
(Euler's formula), algebraic geometry (the Cayley-Bacharach theorem, and
the classification of cubic curves), additive combinatorics (via the
group structure of said cubic curves), and even some Galois theory
(through the theorem of Poonen and Rubinstein that a non-central
interior point in the unit disk can pass through at most seven chords
connecting roots of unity).  We will discuss how these ingredients enter
into the solution to these problems in this talk.


MINERVA LECTURE II:
Title:  Polynomial expanders and an algebraic regularity lemma
Wednesday, February 13, 2013, 4:30 p.m.
Abstract:  The _sum-product phenomenon_ is the observation that given a
finite subset A in a ring, at least one of the sumset A+A or the product
set A.A is typically significantly larger than A itself, except when A
is "very close" to a field in some sense. Related to this is the fact
that for "typical" polynomials P of two variables, the set P(A,A) is
usually significantly larger than A itself.  Polynomials with this
property are known as expanding polynomials.  The class of expanding
polynomials has been fully classified in the context of the real or
complex fields by Elekes and Szabo, but the situation is not completely
resolved in the context of polynomials over finite fields (which is a
case of interest in theoretical computer science). Recently, however,
expansion in the setting of _large_ subsets of finite fields of large
characteristic has become better understood. The main new tool is an
_algebraic regularity lemma,_ which improves substantially over the
famous Szemeredi regularity lemma for graphs, in the setting of graphs
that come from the algebra of finite fields (or more precisely, are
definable in the language of that finite field).  The proof of this
lemma requires some nontrivial tools from model theory (including
nonstandard analysis), and modern algebraic geometry (in particular, the
theory of the etale fundamental group of an algebraic variety), thus
providing yet another example of algebraic geometry being applied in
modern combinatorics.  We discuss these connections in this talk.


MINERVA LECTURE III:
Title: Universality for Wigner random matrices
Thursday, February 14, 2013, 5:30 p.m.
Abstract:  Wigner random matrices are a basic example of a Hermitian
random matrix model, in which the upper-triangular entries are jointly
independent.  The most famous example of a Wigner random matrix is the
Gaussian Unitary Ensemble (GUE), which is particularly amenable to study
due to its rich algebraic structure.  In particular, the fine-scale
distribution of the eigenvalues is completely understood.  There has
been much recent progress on extending these distribution laws to more
general Wigner matrices, a phenomenon sometimes referred to as
_universality_.  In this talk we will discuss recent work of Van Vu and
myself on establishing several cases of this universality phenomenon, as
well as parallel work of Erdos, Schlein, and Yau.


**************

/All lectures will be in A02 McDonnell Hall. ///

*************

*Please refer to the attached PDF should you wish to post the lecture
information. ****
*

-- 
Linda Brozyna
Mathematics Dept.
Princeton University
609-258-3454



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