Mathematical Research Seminar - Archive

A linear map between matrix spaces is called completely positive (cp) if it preserves positive semidefiniteness, even when ampliated to higher-dimensional spaces. In this talk we demonstrate how cp maps are closely related to a particular class of convex sets: the solution sets of linear matrix inequalities (LMIs), known as spectrahedra.
Spectrahedra generalize polyhedra and have gained prominence since the 1990s due to their central role in semidefinite programming (SDP). The main result will explain how a cp map gives rise to a linear Positivstellensatz certificate and vice-versa.
The talk is based on joint works with Bill Helton and Scott McCullough.

In this talk, we will describe a construction of some combinatorial structures, especially regular graphs and digraphs, using finite groups. We will point out some interesting results obtained by using some particular finite groups. Further, we will show how some of the obtained combinatorial structures can be described geometrically.

These are exciting times for cryptography!

For the first time in a long time new asymmetric cryptography algorithms were standardised by NIST (National Institute of Standards and Technology). In the talk we will explain: Why was this necessary? How do these algorithms behave? Will the future of cryptography even involve solutions based on quantum mechanics?

In a nucleation process, formation of small nuclei initiates displacement of one equilibrium by another. Typically, nucleation is local: diameter of the nuclei is much smaller than the time-scale of convergence to the new state. We will discuss a few simple models in which this is not true; instead, the nuclei generate lower-dimensional structures that grow and interact significantly before most of the space is affected. Analysis of such models includes a variety of combinatorial and probabilistic methods.

The talk will be aimed at the general audience.