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Series: Research Horizons Seminar

This is a survey talk on the knot concordance group and the homology cobordism group.

Series: Research Horizons Seminar

There has been much interest in the past couple of decades in identifying (extremal) regular graphs that maximize the number of independent sets, matchings, colorings etc. There have been many advances using techniques such as the fractional subaddtivity of entropy (a.k.a. Shearer's inequality), the occupancy method etc. I will review some of these and mention some open problems on hypergraphs.

Series: Research Horizons Seminar

We all know that the air in a room is made up by a huge number of atoms that zip around at high velocity colliding continuously. How is this consistent with our observation of air as a thin and calm fluid surrounding us? This is what Statistical Mechanics try to understand. I'll introduce the basic examples and ideas of equilibrium and non equilibrium Statistical Mechanics showing that they apply well beyond atoms and air.

Series: Research Horizons Seminar

In this chalk plus slides talk, I will give a few examples from my own experience to illustrate how one can use stochastic differential equations in various applications, and its theoretical connection to diffusion theory and optimal transport theory. The presentation is designed for first or second year graduate students.

Series: Research Horizons Seminar

A knot is a simple closed
curve in the 3-space. Knots appeared as one of the first objects of
study in topology. At first knot theory was rather isolated in
mathematics.
Lately due to newly discovered invariants and newly established
connections to other branches of mathematics, knot theory has become an
attractive and fertile area where many interesting, intriguing ideas
collide. In this talk we discuss a new class of knot
invariants coming out of the Jones polynomial and an algebra of
surfaces based on knots (skein algebra) which has connections to many
important objects including hyperbolic structures of surfaces and
quantum groups. The talk is elementary.

Series: Research Horizons Seminar

Tropical geometry provides a combinatorial approach for studying geometric objects by reducing them to graphs and polytopes. In recent years, tropical techniques have been applied in numerous areas such as optimization, number theory, phylogenetic trees in biology, and auction systems in economics. My talk will focus on geometric counting problems and their tropical counterpart. By considering these combinatorial gadgets, we gain newinsights into old problems, and tools for approaching new problems.

Series: Research Horizons Seminar

In 1665, Huygens
discovered that, when two pendulum clocks hanged
from a same wooden beam supported by two chairs, they synchronize in
anti-phase mode. Metronomes provides a second example of oscillators
that synchronize. As it can be seen in many YouTube videos,
metronomes synchronize in-phase when oscillating on top of the same
movable surface. In this talk, we will review these phenomena, introduce
a mathematical model, and analyze the the different physical effects.
We show that, in a certain parameter regime, the
increase of the amplitude of the oscillations leads to a bifurcation from the anti-phase synchronization being stable to the in-phase synchronization being stable. This may explain the experimental
observations.

Series: Research Horizons Seminar

After briefly describing my research interests, I’ll speak on two results that involve points moving around on surfaces. The first result shows how to “hear the shape of a billiard table.” A point bouncing around a polygon encodes a sequence of edges. We show how to recover geometric information about the table from the collection of all such bounce sequences. This is joint work with Calderon, Coles, Davis, and Oliveira. The second result answers the question, “Given n distinct points in a closed ball, when can a new point be added in a continuous fashion?” We answer this question for all values of n and for all dimensions. Our results generalize the Brouwer fixed point theorem, which gives a negative answer when n=1. This is joint work with Chen and Gadish.

Series: Research Horizons Seminar

Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will try to answer these questions
by presenting many examples, results and open problems.
Note: This is an introductory general audience talk unrelated to the colloquium.

Series: Research Horizons Seminar

In this talk, we describe transforming a theoretical algorithm from
structural graph theory into open-source software being actively used for
real-world data analysis in computational biology. Specifically, we apply
the r-dominating set algorithm for graph classes of bounded expansion in
the setting of metagenome analysis. We discuss algorithmic improvements
required for a practical implementation, alongside exciting preliminary
biological results (on real data!). Finally, we include a brief
retrospective on the collaboration process. No prior knowledge in
metagenomics or structural graph theory is assumed.
Based on joint work with T. Brown, D. Moritz, M. O’Brien, F. Reidl and T.
Reiter.