UNCE

Seminář projektu UNCE/SCI/022

Methods of Algebra and Logic

Matematicko-fyzikální fakulta UK

Sokolovská 83, 186 00 Praha 8 - Karlín

Středa 27. 06. 2018 od 13:00; Katedra Algebry - seminární místnost

Program

Lectures will be cca 30 min., with a generous break before or after the third lecture.

Vita Kala: Universal quadratic forms over number fields

A universal form is a positive definite quadratic form with integral coefficients which represents all positive numbers - a classical example over the integers is the sum of four squares x^2 + y^2 + z^2 + w^2. I shall discuss some recent results concerning the number of variables required by a universal form over a real quadratic field, and the (non-)existence of universal forms with Z-coefficients.

Alexandr Kazda: Constraint Satisfaction Problem and friends

The Constraint Satisfaction Problem (CSP) is a very general computational problem. In this talk, we will explain why universal algebra is suitable for describing the complexity of CSP and how we can transfer our tools from CSP to various similar problems: valued CSP (VCSP), counting CSP (#CSP), and especially promise CSP (PCSP).

Andrew Moorhead (Vanderbilt University): Supernilpotence in Universal Algebra

The commutator for groups can be seen as a special case of a commutator theory for general algebraic systems. With such a commutator it is possible to define analogues of abelianness, solvability and nilpotence. This commutator theory has recently been extended to something called higher commutator theory, from which one can define a stronger type of nilpotence which is called supernilpotence. In this talk we will discuss some of the reasons that supernilpotence is a useful concept.

Benjamin Vejnar: The complexity of the homeomorphism relation between compact spaces

We study the complexity of the homeomorphism relation of compact metric spaces when restricted to some subclasses such as continua, regular continua or regular compacta. The complexity of an equivalence relation on a Polish space is compared with some others using the notion of Borel reducibility.

Jonathan Verner: Complexity of Ultrafilters

As is well known, constructing a non-trivial ultrafilter requires (some amount of) the axiom of choice so these objects are inherently complicated. It is not clear, however, whether they are all created equal or whether some are more complex than others. In this talk I will introduce several different "measures" of complexity of ultrafilters, give an overview of the known results, and show that even after more than half a century of intensive research, basic questions still remain unanswered.