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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.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

Orthogonal Polynomials and their generalizations have a great many
applications in areas ranging from signal processing to random matrices
to combinatorics. We outline a few of the connections, and present some
possible Ph. D Problems

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

Olof Sisask and myself have produced a new probabilistic
technique for finding `almost periods' of convolutions of subsets of
finite groups. In this talk I will explain how this has allowed us
to give (just recently) new bounds on the length of the longest
arithmetic progression in a sumset A+A.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

I will discuss some theorems and conjectures in the relatively new
field of arithmetic dynamics, focusing in particular on some methods
from number theory which can be used to study the orbits of points in
algebraic dynamical systems.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

In the preceeding talk, I outlined a framework for variational problems
and some of the basic tools and results. In this talk I will attempt
describe several problems of current interest.

Series: Research Horizons Seminar

I will describe several geometrical problems that arise from
the minimization of some sort of integral functional and the basic
relation between such minimization and partial differential equations.
Then I will make some further comments on my favorite kind of such
problems, namely those that have something to do with minimizing area of
surfaces under various side conditions.

Series: Research Horizons Seminar

The Research Horizons seminar this week will be a panel discussion on
the academic job market for mathematicians. The discussion will begin
with an overview by Doug Ulmer of the hiring process, with a focus on
the case of research-oriented universities. The panel will then take
questions from the audience. Professor Wick was hired last year at
Tech, so has recently been on the students' side of the process.
Professor Harrell has been involved with hiring at Tech for many years
and can provide a perspective on the university side of the process.

Series: Research Horizons Seminar

An assembly of atoms in a solid phase will be described through the notion
of Delone sets and related to tilings. The Hull and the tiling space wiill
be defined. It will be shown that the tiling space and the Hull can be
constructed through an inverse limit of CW-complexes built out of the
tiles and of the local patches. From then various cohomologies can be
defined and allow to distinguish between these atomic distributions. The
question of whether these topological invariant can be seen in experiments
will be addressed.

Series: Research Horizons Seminar

Dynamical systems theory is concerned with systems that change in time
(where time can be any semigroup). However, it is quite rare that one
can find the solutions for such systems or even a "sizable" subset of
such solutions. An approach motivated by this fact, that goes back to
Poincaré, is to study instead partitions of the (phase) space M of all
states of a dynamical system and consider the evolution of the elements
of this partition (instead of the evolution of points of M).
I'll explain how the objects in the title appear, some relations between
them, and formulate a few general as well as more specific open problems
suitable for a PhD thesis in dynamical systems, mathematical biology,
graph theory and applied and computational mathematics.
This talk will also serve to motivate and introduce to the topics to be
given in tomorrow's colloquium.

Series: Research Horizons Seminar

The Soul Theorem, proved by Cheeger and Gromoll forty
year ago, reveals a beautiful structure of noncompact complete
manifolds of nonnegative curvature. In the talk I will sketch
a proof of the Soul Theorem, and relate it to my current work
on moduli spaces of nonnegatively curved metrics.