Past Kinderseminar

22 October 2014
12:30
Francesca Balestrieri
Abstract

Given any family of varieties over a number field, if we have that the existence of local points everywhere is equivalent to the existence of a global point (for each member of the family), then we say that the family satisfies the Hasse principle. Of more interest, in this talk, is the case when the Hasse principle fails: we will give an overview of the "geography" of the currently known obstructions.

18 June 2014
11:00
Constantin Gresens
Abstract
A word w has finite width n in a group G if each element in the subgroup generated by the w-values in G can be written as the product of at most n w-values. A group G is called verbally elliptic if every word has finite width in G. In this talk I will present a proof for the fact that every finitely generated virtually nilpotent group is verbally elliptic.
11 June 2014
10:30
Robert Leek
Abstract
"Show that there is a function $f$ such that for any sequence $(x_1, x_2, \dots)$ we have $x_n = f(x_{n + 1}, x_{n + 2}, \dots)$ for all but finitely many $n$." Fred Galvin. Problem 5348. The American Mathematical Monthly, 72(10):p. 1135, 1965.\\ This quote is one of the earliest examples of an infinite hat problem, although it's not phrased this way. A hat problem is a non-empty set of colours together with a directed graph, where the nodes correspond to "agents" or "players" and the edges determine what the players "see". The goal is to find a collective strategy for the players which ensures that no matter what "hats" (= colours) are placed on their heads, they will ensure that a "sufficient" amount guess correctly.\\ In this talk I will discuss hat problems on countable sets and show that in a non-transitive setting, the relationship between existence of infinitely-correct strategies and Ramsey properties of the graph breakdown, in the particular case of the parity game. I will then introduce some small cardinals (uncountable cardinals no larger than continuum) that will be useful in analysing the parity game. Finally, I will present some new results on the sigma-ideal of meagre sets of reals that arise from this analysis.
4 June 2014
10:30
Kristen Pueschel
Abstract
Riley and Dison's hydra groups are a family of group and subgroup pairs $(G_k, H_k)$ for which the subgroup $H_k$ has distortion like the $k$-th Ackermann function. One wants to know if finite quotients can distinguish elements that are not in $H_k$, as a positive answer would allow you to construct a hands-on family of finitely presented, residually finite groups with arbitrarily large Dehn functions. I'll explain why we get a negative answer.
28 May 2014
10:30
Ilya Kazachkov
Abstract
In the late 70s -- early 80s Makanin came up with a very simple, but very powerful idea to approach solving equations in free groups. This simplicity makes Makanin-like procedures ubiquitous in mathematics: in dynamical systems, geometric group theory, 3-dimensional topology etc. In this talk I will explain loosely how Makanin's algorithm works.
21 May 2014
10:30
Dennis Dreesen
Abstract
The Haagerup property is a group theoretic property which is a strong converse of Kazhdan's property (T). It implies the Baum-Connes conjecture and has connections with amenability, C*-algebras, representation theory and so on. It is thus not surprising that quite some effort was made to investigate how the Haagerup property behaves under the formation of free products, direct products, direct limits,... In joint work with Y.Antolin, we investigated the behaviour of the Haagerup property under graph products. In this talk we introduce the concept of a graph product, we give a gentle introduction to the Haagerup property and we discuss its behaviour under graph products.
14 May 2014
10:30
Emily Cliff
Abstract
In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\ This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.
7 May 2014
10:30
Henry Bradford
Abstract

An important moral truth about the mapping class group of a closed orientable surface is the following: a generic mapping class has no power fixing a finite family of simple closed curves on the surface. Such "generic" elements are called pseudo-Anosov. In this talk I will discuss one instantiation of this principle, namely that the probability of a simple random walk on the mapping class group returning a non-pseudo Anosov element decays exponentially quickly.

30 April 2014
10:30
Alejandra Garrido Angulo
Abstract
For any infinite group with a distinguished family of normal subgroups of finite index -- congruence subgroups-- one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for $SL(n,\mathbb{Z})$ where $n\geq 3$, by Mennicke and Bass, Lazard and Serre. \\ Groups acting on infinite rooted trees are a natural setting in which to ask this question. In particular, branch groups have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.

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