Past Kinderseminar

This talk will introduce various aspects of modern cryptography. After introducing RSA and some factoring algorithms, I will move on to how elliptic curves can be used to produce a more complex form of Diffie--Hellman key exchange.

This talk is about the ordinary representation theory of finite groups of Lie type. I will begin by carefully reviewing algebraic groups and finite groups of Lie type and the construction and properties of (ordinary) Gelfand--Graev characters. I will then introduce generalized Gelfand--Graev characters, which are constructed using the Lie algebra of the ambient algebraic group. Towards the end I hope to give an idea of how generalized Gelfand--Graev characters can and have been used to attack Lusztig's conjecture and the role this plays in the determination of the character tables of finite groups of Lie type.

The representation theory of groups is surrounded by deep and difficult conjectures. In this talk we will take a tour of (some of) these problems, including Alperin's weight conjecture, Broué's conjecture, and Puig's finiteness conjecture.

We will present different ideas leading to and evolving around geometry over the field with one element. After a brief summary of the so-called numbers-functions correspondence we will discuss some aspects of Weil's proof of the Riemann hypothesis for function fields. We will see then how lambda geometry can be thought of as a model for geometry over $\mathbb{F}_\mathrm{un}$ and what some familiar objects should look like there. If time permits, we will

explain a link with stable homotopy theory.

Much of group theory is concerned with whether one property entails another. When such a question is answered in the negative it is often via a pathological example. We will examine the Rips construction, an important tool for producing such pathologies, and touch upon a recent refinement of the construction and some applications. In the course of this we will introduce and consider the profinite topology on a group, various separability conditions, and decidability questions in groups.

The semisimplicity problem is the long-standing conjecture that the group algebra $KG$ of a $p'$-group $G$ over a field $K$ of characteristic $p\geqslant 0$ has zero Jacobson radical. We will discuss recent advances in connection with this problem.

I will describe in detail the first construction of infinite, finitely generated torsion groups due to Golod in early 60s --

these groups are special cases of the so-called Golod-Shafarevich groups. If time allows, I will discuss some related constructions and open problems.

Using the theory of formal groups, Landweber´s exactness theorem provides means to construct interesting invariants of topological spaces out of geometric objects. I will illustrate the resulting connection between algebraic geometry and stable homotopy theory in the special case of elliptic curves.

We begin by proving the abc theorem for polynomial rings and looking at a couple of its consequences. We then move on to the abc conjecture and its equivalence with the generalized Szpiro conjecture, via Frey polynomials. We look at a couple of consequences of the abc conjecture, and finally consider function fields, where we introduce the abc theorem in that case.

Presheaves on categories crop up everywhere! In this talk, I'll give a

gentle introduction to 2-categories, and discuss the notion of a

presheaf on a 2-category. In particular, we'll consider which

2-categories such a presheaf might take values in. Only a little

familiarity with the notion of a category will be assumed!

The subgroup growth of finitely generated groups was seen last term, in a lecture of Dan Segal. This time, we see representation growth, and how it is similar to, and different from, subgroup growth.