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Series: Research Horizons Seminar

Hosted by Yao Li and Ricardo Restrepo.

Deciding how to unknot a knotted piece of string (with its ends glued together) is not only a difficult problem in the real world, it is also a difficult and long studied problem in mathematics. (There are several notions of what one might mean by "unknotting" and I will leave the exact meaning a bit vague in this abstract.) In the past mathematicians have used a vast array of techniques --- from geometry to algebra, and even PDEs --- to study this question. I will discuss this question and (partially) recast it in terms of 4 dimensional topology. This new perspective will allow us to use a powerful new knot invariant called Khovanov Homology to study the problem. I will give an overview of Khovanov Homology and indicate how to study our unknotting question using it.

Series: Research Horizons Seminar

Hosted by: Yao Li and Ricardo Restrepo

Combinatorial mathematics exhibits a number of elegant, simply stated problems that turn out to be surprisingly challenging. In this talk, I report on a problem of this type on which I have been working with Noah Streib, Stephen Young and Ruidong Wang from Georgia Tech, as well as Piotr Micek, Bartek Walczak and Tomek Krawczyk, all computer scientists from Poland. Given positive integers $k$ and $w$, what is the largest integer $t = f(k,w)$ for which there exists a family $\mathcal{F}$ of $t$ vectors in $N^{w}$ so that: \begin{enumerate} \item Any two vectors in the family $\mathcal{F}$ are incomparable in the product ordering; and \item There do not exist two vectors $A$ and $B$ in the family for which there are distinct $i$ and $j$ so that $a_i\ge k +b_i$ and $b_j \ge k + a_j$. \end{enumerate} The Polish group posed the problem to us at the SIAM Discrete Mathematics held in Austin, Texas, this summer. They were able to establish the following bounds: \[ k^{w-1} \le t \le k^w \] We were able to show that the lower bound is essentially correct by showing that there is a constant $c_w$ so that $t \l c_w k^{w-1}$. But recent work suggests that the lower bound might actually be tight.

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.