I will briefly present our Stochastics Group and its main interests, and will continue with some of the problems I have worked on in recent years.
Mathematical billiards naturally arise in mechanics, optics, acoustics, etc. They also form the most visual class of dynamical systems with evolution covering all the possible spectrum of behaviours from integrable (extremely regular) to strongly chaotic. Billiard is a (deterministic) dynamical system generated by an uniform (by inertia) motion of a point particle within a domain with piecewise smooth walls ("a billiard table"). I will introduce all needed notions on simple examples and outline some open problems. This talk is also a preparatory talk to a Mathematical Physics seminar (on Monday April 8) where a new direction of research will be discussed which consider physical billiards where instead of a point (mathematical) particle a real physical hard sphere moves. To a complete surprise of mathematicians and PHYSICISTS evolution of a billiard may completely change (and in different ways) in transition from mathematical to physical billiards. It a rare example when mathematicians surprise physicists. Some striking results with physicists are also already obtained. I will (again visually) explain at the end of RH why it is surprising that there could be difference between Math and Phys billiards.
An element of the braid group can be visualized as a collection of n strings that are braided (like a hair braid). Braid groups are ubiquitous in mathematics in science, as they record the motions of a number of points in the plane. These points can be autonomous vehicles, particles in a 2-dimensional medium, or roots of a polynomial. We will give an introduction to and a survey of braid groups, and discuss what is known about homomorphisms between braid groups.
The Mapper algorithm constructs compressed representations of the underlying structure of data but involves a large number of parameters. To make the Mapper algorithm accessible to domain experts, automation of the parameter selection becomes critical. This talk will be accessible to graduate students.
For a polytope P, the h-vector is a vector of integers which can be calculated easily from the number of faces of P of each dimension. For simplicial polytopes, it is well known that the h-vector is symmetric (palindromic) and unimodal. However in general the h-numbers may even be negative. In this talk I will introduce the tropical h-vector of a polytope, which coincides with the usual h-vector of the dual polytope, if the polytope is simple. We will discuss how they are related to toric varieties, tropical geometry, and polytope algebra. I will also discuss some open problems.
In this talk, we will discuss various ways to describe three-manifolds by decomposing them into pieces that are (maybe) easier to understand. We will use these descriptions as a way to measure the complexity of a three-manifold.
This is a survey talk on the knot concordance group and the homology cobordism group.
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.
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.