Přednášky v podzimním semestru 2016

Přednášky se konají na Ústavu matematiky a statistiky,budova č.8 v areálu Přírodovědecké fakulty, Kotlářská 2, Brno.

### 9. listopadu 2016, v 16:00 v posluchárně M1

#### Wieslaw Kubiś (Institute of Mathematics, Czech Academy of Sciences, Prague)

Generic objects and infinite games

#### Abstract:

Let F be a fixed class of small' mathematical structures (e.g. finite graphs, finite-dimensional normed spaces, etc.) and assume that a notion of embedding' has been defined so that we can say that one structure is an extension of another. We say that a structure is `big' if it can be build as the union (or, more formally, colimit) of a chain of embeddings in F. Fix a big structure U. We consider the following infinite game for two players: Player I chooses a structure S_0 from F. Player II responds by its extension S_1, again in F. Player I responds by an extension S_2 of S_1. And so on. We say that Player II wins if the union of the infinite chain of S_ns is isomorphic to U, otherwise Player I wins. We say that U is generic, if Player II has a winning strategy.

In the talk I will present examples of generic objects in several areas of mathematics. Further, I will show some of their basic properties and relations to the theory of universal homogeneous models.

Záznam přednášky ZDE

### 12. října 2016, v 16:00 v posluchárně M1

#### Laszlo Erdös (IST, Rakousko)

Universality of random matrices and log-gases

#### Abstract:

The Wigner-Dyson-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a  sense that  they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution.
We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and  PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution.
We also show related results for log-gases that represent a  universal model for strongly correlated systems.

Záznam přednášky ZDE

Aktualizováno Pátek, 11 Listopad 2016 13:12