Raziskovalni matematični seminar - Arhiv

A prototypical problem in extremal graph theory is determining which graphs are minimizers or maximizers of the density of a fixed graph H, possibly with some additional constraints. For example, considered among most important conjectures in extremal combinatorics, the famous conjecture by Sidorenko and Erdős-Simonovits claims that the density of every bipartite graph H is asymptotically minimized by quasirandom graphs among all graphs with the same edge density.

In this talk we will focus on directed graphs with the property that their homomorphism density is maximized by transitive tournaments. We prove that for any bipartite graph H whose edges are oriented in the same direction between both parts (that is, a directed graph that admits a homomorphism to a directed edge ), the n-vertex transitive tournament maximizes the number of homomorphisms from H among all oriented n-vertex graphs.

Joint work with Igor Balla, Bartlomiej Kielak, Daniel Král’, and Filip Kučerák.

ZOOM link: https://upr-si.zoom.us/j/94947596338?pwd=Ala2JolIlOnXb1jINtebXmk7ZlHjb9.1

Arising from applications in machine learning such as the problem of approximate sampling from a given probability distribution or optimization, special types of stochastic processes such as McKean-Vlasov SDEs became highly attractive in recent years. The theory about them is quite interesting as it shows that in many ways they exhibit different characteristics than classical diffusion processes. In this talk, I will discuss the behaviour of interacting particle systems and their gradient flows. In particular, the focus will be on WassersteinFisher-Rao and Fisher-Rao gradient flows and deriving conditions which result in the convergence to the target measures. I will consider various approaches to the problem as well as numerical methods that can be used for simulations.