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Series: Research Horizons Seminar

Orthogonal Polynomials play a key role in analysis of random matrices. We discuss universality limits in the so-called unitary case, showing how the universality limit reduces to an asymptotic involving reproducing kernels associated with orthogonal polynomials. As a consequence, we show that universality holds in measure for any compactly supported measure.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

A starting point of geometric group theory is thinking of a group
as a geometric object, by giving it a metric induced from the
Cayley graph of the group. Gromov initiated a program of studying
groups up to quasi-isometries, which are ``bilipschitz maps up to bounded additive error". Quasi-isometries ignore local
structure and preserve asymptotic properties of a metric space. In the talk I will give a sample of results, examples, and open
questions in this area.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

Elliptic curves are solution sets of cubic polynomials in two
variables. I'll explain a bit of where they came from (computing the
arc length of an ellipse, hence the name), their remarkable group
structure, and some of the many roles they play in mathematics and
applications, from mechanics to algebraic geometry to cryptography.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

A useful parametrization of the one variable trigonometric moment problem
is given in terms of polynomials orthogonal on the unit circle. A
description of this parameterization will be given as well as some of its
uses. We will then describe a possible two variable extension.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

One of the basic problems arising in many pure and applied
areas of mathematics is to solve a system of polynomial equations.
Numerical Algebraic Geometry starts with addressing this fundamental
problem and develops machinery to describe higher-dimensional solution
sets (varieties) with approximate data. I will introduce numerical
polynomial homotopy continuation, a technique that is radically
different from the classical symbolic approaches as it is powered by
(inexact) numerical methods.

Series: Research Horizons Seminar

Hosted by: Huy and Yao

Research Horizons features Lunch Fun Break! The purpose is to create an opportunity for all graduate
students, new and experienced, domestic and international, to meet, eat and have fun.AGENDA: ***"Suggestion box" for graduate students will be displayed in Faculty Lounge Skiles 236.*** Propective students' visit on Friday, April 2. *** Game: "Can you comunicate in silience?" *** PIZZAs, soft DRINKs, relax and have fun. ***

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

We will have a chance to spend some time together to discuss issues
of relevance to the Graduate Program. Sort of like a "Town Hall
Meeting" of the graduate students and the graduate coordinator.
There are some things that I need to communicate to all of you,
but the format is otherwise unstructured, and I am seeking suggestions
on things which you would like to see addressed. So, please send me
comments on things which you would like to see discussed and I will
do my best to get ready for them.
Thanks, Luca Dieci.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

The Scenery Reconstruction Problem consists in trying to reconstruct
a coloring of the integers given only the observations made by
a random walk. For this we consider a random walk S and
a coloring of the integers X. At time $t$ we observe
the color $X(S(t))$. The coloring is i.i.d. and we show that
given only the sequence of colors
$$X(S(0)),X(S(1)),X(S(2)),...$$
it is possible to reconstruct $X$ up to translation
and reflection. The solution depends on the property of the
random walk and the distribution of the coloring.
Longest Common Subsequences (LCS) are widely used in genetics.
If we consider two sequences X and Y, then a common subsequence
of X and Y is a string which is a subsequence of X and of Y at the same
time. A Longest Common Subsequence of X and Y is a common
subsequence of X and Y of maximum length. The problem of the asymptotic
order of the flucutation for the LCS of independent random
strings has been open for decades. We have now been able to
make progress on this problem for several important cases.
We will also show the connection to the Scenery Reconstruction
Problem.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

The Hilbert transform is a foundational transform, with deep connections to
electrical charge, and analyticity. The `two weight inequality for the
Hilbert transform' concerns the most general setting in which the Hilbert
transform admits a (weighted) L^2 inequality. We will give a couple of
(surprising?) ways that this question arises. And we will indicate the
surprise that is behind the recent description of all setting in which the
two weight inequality holds.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

Sampling from and approximately counting the size of a large set
of combinatorial structures has contributed to a renaissance in research
in finite Markov chains in the last two decades.
Applications are wide-ranging from sophisticated card shuffles,
deciphering simple substitution ciphers (of
prison inmates in the California state prison), estimating the volume of
a high-dimensional convex body,
and to understanding the speed of Gibbs sampling heuristics in
statistical physics. More recent applications include rigorous estimates
on J.M. Pollard's (1979) classical Rho and Kangaroo algorithms for the
discrete logarithm problem in finite cyclic groups.
The lecture will be a brief (mostly self-contained) introduction to the
Markov Chain Monte Carlo (MCMC) methodology and applications, and will
include some open problems.