# Düsseldorf Research Seminar in Pure Mathematics

This is a seminar with two purposes:

1. We have invited guests giving talks about their research.

2. We give talks among ourselves introducing each other to our own research. These talks should explain basic notions; interruptions, discussion and deviations from the original plan are very welcome.

The standard time and place is

**Mondays, 16:30-17:30, Room 25.22.00.72**. Occasional deviations from this rule (e.g. when a speaker is not here on Monday) will be marked next to the talk.

If you are interested in speaking or inviting a guest for this seminar, please contact one of the organizers:

Alejandra Garrido

Andrea Fanelli

David Bradley-Williams

Peter Arndt

# List of scheduled talks:

- Monday, 25. 9. 2017: Paula Lins (Bielefeld), Bivariate zeta functions of some finitely generated nilpotent groups --- Abstract
- Some Monday during the semester: You? Get in touch!

# List of past talks:

- Friday, 21. 7. 2017: Ingo Blechschmidt (Uni Augsburg), First steps in synthetic algebraic geometry --- Abstract
- Monday, 17. 7. 2017: Benedikt Schilson (HHU D'dorf), Kummer varieties in arbitrary characteristics --- Abstract
- Monday, 3. 7. 2017: Benedict Meinke (HHU D'dorf), Complex symplectic structures on nilmanifolds --- Abstract
- Monday, 26. 6. 2017: Marlis Balkenhol (HHU D'dorf), Wave equation methods for Selberg’s 3/16 theorem --- Abstract
- Monday, 19. 6. 2017: Ronan Terpereau (Université de Bourgogne, Dijon), A symplectic version of the Chevalley restriction theorem --- Abstract
- Monday, 12. 6. 2017: Sasa Novakovic , Rationality of varieties and categorical representability --- Abstract
- Monday, 22. 5. 2017: Kevin Langlois, Some results on singularities of algebraic varieties with torus action --- Abstract
- Monday, 8. 5. 2017: Oliver Bräunling (Uni Freiburg), Homology torsion growth of knot exteriors --- Abstract
- Monday, 13. 02. 2017: Oihana Garaialde, Our game: counting cohomology algebras --- Abstract
- Friday, 10. 02. 2017: Hiromu Tanaka (Imperial College London, UK), Minimal model programme in positive characteristic --- Abstract
- Monday, 06. 02. 2017: Jesus Martinez Garcia (MPIM, Bonn), Geometric Invariant Theory and Moduli of log Pairs --- Abstract
- Tuesday, 31. 01. 2017: Jone Uria Albizuri (University of the Basque Country), Various types of fractal groups --- Abstract
- Monday, 16. 01. 2017: Anitha Thillaisundaram (Lincoln, UK), Automorphisms of finite p-groups --- Abstract
- Monday, 19. 12. 2016: Matteo Vannacci, Wreath products in Group Theory --- Abstract
- Tuesday, 13. 12. 2016: David González Álvaro (Madrid), invited by Marcus Zibrowius, What is sectional curvature? --- Abstract
- Monday, 28. 11. 2016: Florian Severin, An Introduction to Model Theory --- Abstract
- Monday, 21. 11. 2016: Andrea Fanelli, (Blow-)up and downs --- Abstract
- Monday, 14. 11. 2016: Peter Arndt, The field with one element --- Abstract

# Abstracts:

- Monday, 14. 11. 2016: Peter Arndt, The field with one element

There is a mathematical phantom called the field with one element. This is not (yet?) a mathematical object but rather a collection of phenomena occurring in combinatorics, group theory, algebraic geometry and topology (so every Düsseldorf pure mathematician could run into it at some point). These occurrences can be seen as limit cases for mathematics involving finite fields F_p, for p-->1. I will explain some of the sightings of this phantom, skirting all the above areas (plus a bit of logic). Finally, I will list a few of the approaches for making the field with one element into an actual mathematical object. - Monday 21. 11. 2016: Andrea Fanelli, (Blow-)up and downs

After recalling some basic notions in classical algebraic geometry, I will introduce the birational classification problem. The fundamental construction (and motivation) is --guess what?-- the blow-up of a point in a smooth surface. I will start recalling the notation (local coordinates, parameters, "around what?"...) and then give an idea of the techniques by Castelnuovo and friends. I will finish up with an idea of what happens in higher dimension. - Monday 28. 11. 2016: Florian Severin, An Introduction to Model Theory

Once the essential definitions are established, I will state the Compactness theorem for first-order logic (FOL). A standard application is the introduction of non-standard models, e.g. of the natural numbers or the reals. I will further mention some combinatorial consequences. A generalized version of another corollary oft Compactness, the Löwenheim-Skolem theorem, leads to seemingly paradoxical observations in ZF(C) set theory. As an example, I will present Skolem's paradox (and try to convince you, why it does not yield a proper contradiction). - Tuesday 13. 12. 2016: David González Álvaro (Madrid), What is sectional curvature?

For a smooth manifold endowed with a Riemannian metric different notions of curvature can be defined locally. In this talk we will introduce the definition of sectional curvature, which can be seen as a generalization of the Gaussian curvature in surfaces. We will review classical results on manifolds admitting metrics with certain bounds for the sectional curvature, and we will conclude discussing related open problems. - Monday 19. 12. 2016: Matteo Vannacci, Wreath products in Group Theory

The wreath product construction is a way to combine two given finite groups into a new finite group. It turns out that this is an interesting combinatorial and group theoretical object. In this seminar, I will introduce this construction and describe how to use it to manufacture new interesting infinite groups. - Monday 16. 01. 2017: Anitha Thillaisundaram, Automorphisms of finite p-groups

We discuss two conjectures, one disproved and one still unsolved, regarding automorphisms of finite p-groups. - Tuesday 31. 01. 2017: Jone Uria Albizuri, Various types of fractal groups

The aim of this talk is to introduce some basic notions about groups acting on regular rooted trees in order to discuss and clarify the notion of being fractal for this kind of groups. For this purpose we will define three types of fractality and we will show that they are not equivalent, by giving explicit examples. - Monday 06. 02. 2017: Jesus Martinez Garcia, Geometric Invariant Theory and Moduli of log Pairs

Geometric Invariant Theory (GIT) is a tool to construct moduli spaces of objects. It has been used to construct moduli of vector bundles, curves, and more recently Fano varieties. I will describe how we can use GIT to construct moduli of log pairs (X,D) formed by a projective hypersurface X and a hyperplane section D over algebraically closed fields. Log pairs are natural elements of birational geometry and compactifying their moduli is an important approach to their classification. We will use computational algebraic technology tools to carry out our analysis and we will illustrate the setting for cubic surfaces. Time permitting, I will speak of some applications to complex differential geometry. - Friday 10. 02. 2017: Hiromu Tanaka, Minimal model programme in positive characteristic

This is a survey talk on the minimal model programme. The minimal model theory is a classification theory in algebraic geometry. Its origin is the theory of Riemann surfaces or the classification theory of algebraic surfaces established by the Italian school of algebraic geometry in the early 20th century. In this talk, I explain about the current status and the goal of that theory. If we have time, we also discuss the main difficulty in positive characteristic. - Monday 13. 02. 2017: Oihana Garaialde, Our game: counting cohomology algebras

In mathematics, there are plenty of well-known counting or classifying problems. In this talk, we introduce a counting problem in cohomology algebras.

Let p be a prime number. We shall begin by presenting some explicit cohomology algebras of finite p-groups so as to understand the complexity of such algebras. Our aim is to count isomorphism types of cohomology algebras of p-groups that have a certain common group property without explicit computations. - Monday, 8.5. 2017: Oliver Bräunling (Uni Freiburg), Homology torsion growth of knot exteriors

One can assemble the torsion homology cardinalities of the Z-tower of a knot exterior in a generating function. While the asymptotic growth of these cardinalities is well-understood, and thus suggests a little bit about the shape of the generating function, I will discuss to what extent the generating function possesses an analytic continuation and the things "that happen" in this continuation. - Monday, 22. 5. 2017: Kevin Langlois , Some results on singularities of algebraic varieties with torus action

In this talk, we will recall the combinatorial description of Altmann-Hausen obtained in 2006 for describing actions of algebraic tori on normal affine varieties. The idea is that for any such a variety with an action of an algebraic torus T, one can construct naturally a T-equivariant proper modification f:X'-->X such that X' resolves the indeterminacy locus of the rational quotient Y of X, making X' a toric fibration over Y. More precisely, this approach will tell us how one can encode the geometry of the map f and how describe the fibers of the fibration X'-->Y. In particular, if Y is of dimension 0, we recover the classical construction of an affine toric variety by its polyhedral cone. It is desirable to have a dictionnary for determining the 'type' of a singularity of X in term of its combinatorial datum, especially for singularities appearing in the minimal model program. If the time permits, we will present a recent work where we give a criterion for X to be terminal, canonical, or log canonical in the case where Y is of dimension 1, extending some results of Liendo-Suess obtained in 2013. - Monday, 12. 6. 2017: Sasa Novakovic, Rationality of varieties and categorical representability

It is a very old and prominent problem in algebraic/arithmetic geometry to determine whether a variety admits a rational point or is birational to the projective space. I will explain how the derived category of complexes of coherent sheaves can detect the existence of rational points, being necessary for the variety to be rational. I do not want to tell you more. Just be surprised! - Monday, 19. 6. 2017: Ronan Terpereau (Université de
Bourgogne, Dijon), A symplectic version of the Chevalley restriction
theorem

(joint work with Michael Bulois, Christian Lehn, Manfred Lehn) Let G be a connected linear algebraic group and let V be a polar representation of G with Cartan subspace c and Weyl group W (those terms will be explained during the talk!). It is expected that the symplectic reduction (V \oplus V^*)///G identifies with the quotient (c \oplus c^*)/W as a symplectic variety. In this talk I will give some examples and explain what is known about this conjecture. - Monday, 26. 6. 2017: Marlis Balkenhol (HHU D'dorf), Wave equation methods for Selberg’s 3/16 theorem

Let H be the upper half plane of the complex plane equipped with the hyperbolic metric. For a discrete subgroup Γ of PSL_{2}(**R**), the quotient Γ\H is a Riemannian surface and thus has a Laplace-Beltrami operator Δ. Selberg conjectured that for a Hecke congruence subgroup Γ_{0}(q) the smallest non-trivial eigenvalue of Δ is at least 1/4, and proved that it is at least 3/16. In this talk, I will present a new approach to the problem considering a wave equation on a different quotient Γ'\H. - Monday, 3. 7. 2017: Benedict Meinke (HHU D'dorf), Complex symplectic structures on nilmanifolds

- Monday, 17. 7. 2017: Benedikt Schilson (HHU D'dorf), Kummer varieties in arbitrary characteristics

The Kummer variety associated to an abelian variety A is the quotient of A by the sign involution. Classically, mathematicians studied such varieties coming from Jacobians of genus-2-curves over the complex numbers. In the case that the characteristics of the ground field is different from 2 many results can be generalized to higher dimension. In the "wild" case, i.e. char(k)=2, there are only few results, nearly all of them concerning surfaces. After a short introduction I will explain how to get open affine subschemes of wild Kummer varieties and have a look at its singular points. - Friday, 21. 7. 2017: Ingo Blechschmidt (Uni Augsburg), First steps in synthetic algebraic geometry

We describe how the internal language of certain toposes, the associated little and big Zariski toposes of a scheme, can be used to give simpler definitions and more conceptual proofs of the basic notions and observations in algebraic geometry. The starting point is that, from the "internal point of view" of the little Zariski topos, sheaves of rings and sheaves of modules look just like plain rings and plain modules. In this way, some concepts and statements of scheme theory can be reduced to concepts and statements of intuitionistic linear algebra. This simplifies working with sheaves and brings conceptual clarity. The internal language of the big Zariski topos goes even further. It incorporates Grothendieck's functor-of-points philosophy in order to cast modern algebraic geometry, relative to an arbitrary base scheme, in a naive language reminiscient of the classical Italian school. The base scheme looks like the one-element set from this point of view. The talk gives an introduction to this topos-theoretic point of view of algebraic geometry. No prior knowledge about toposes is supposed. - Monday, 25. 9. 2017: Paula Lins (Bielefeld), Bivariate zeta functions of some finitely generated nilpotent groups

Over the last few decades, zeta functions have been used as tools in various areas of asymptotic group theory. In this talk, I will define two bivariate zeta functions of groups for a large class of finitely generated nilpotent groups and discuss some arithmetic and analytic properties of their local factors. These zeta functions encode, respectively, the numbers of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. We will also discuss the fact that our bivariate zeta functions specialise to the (univariate) class number zeta function. Moreover, in case of nilpotency class 2, our bivariate representation zeta function also specialises to the (univariate) twist representation zeta function. As a consequence, our results yield analytic and arithmetic properties of these univariate zeta functions.