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Series: Research Horizons Seminar

Series: Research Horizons Seminar

Series: Research Horizons Seminar

Series: Research Horizons Seminar

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.