Serdecznie zapraszamy na warsztaty przygotowujące do wykładu im. prof. S. Łojasiewicza. Odbędą się one w piątek, 7 czerwca, w sali 1016 budynku Wydziału Matematyki i Informatyki UJ.

Program warsztatów:

10:00 - 10:45 Jakub Kozik, Rudiments of the probabilistic method
11:00 - 11:45 Bartosz Walczak, Szemerédi regularity lemma
11:45 - 12:15 poczęstunek
12:15 - 13:00 Andrzej Grzesik, Dense graph limits
13:15 - 14:00 Mikołaj Frączyk, Sparse graph limits

 

Abstrakty:

Jakub Kozik, Rudiments of the probabilistic method

We will take a quick tour on several basic applications of the probabilistic method.
Our main subject of study will be the problems of hypergaph coloring.
Special attention will be paid to Lovász Local Lemma.

Bartosz Walczak, Szemerédi regularity lemma

Szemerédi's regularity lemma, informally speaking, asserts that every graph can be partitioned into a bounded number of parts so that the edges between parts behave almost randomly. We will present typical applications of the lemma for embedding or counting copies of a fixed subgraph in a given dense graph.

Andrzej Grzesik, Dense graph limits

The theory of graph limits, developed in the last 20 years in a series of papers by Lovász et al., provides an analytic description of large graphs, which are often used to model interesting structures and phenomena of the world, for example internet, social networks, neurons in a brain, transistors in a chip or interactions between particles. The theory occurred to be extremely useful and allow solving many longstanding open problems in graph theory. In the talk we will provide an intuitive introduction to limits of dense graph sequences and the notion of a graphon, as well as present some applications in the extremal graph theory.

Mikołaj Frączyk, Sparse graph limits

We’ll discuss the notions of local convergence of graphs. The topology of local convergence, often called the Benjamini-Schramm topology, is well suited to study the statistics of bounded degree finite graphs whose size goes to infinity. The local limits of such sequences of graphs are random, typically infinite, bounded degree graphs enjoying additional property called unimodularity. Unimodular random graphs are widely useful in percolation theory, measured group theory and inspired several recent developments in differential geometry and algebraic topology. In my talk we will give several examples of locally convergent sequences and explain the mass transport formula, which is used to both define and study the unimodular random graphs.