# Düsseldorf Doctoral 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 25.22.03.73**. Occasional exceptions 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:

Peter Arndt

David Bradley-Williams

Kevin Langlois

This seminar is part of the activities of the Research Training Group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology.

# List of scheduled talks:

- Monday, 17. 12. 2018: Aleksandra Kwiatkowska (Münster), Kechris-Pestov-Todorcevic correspondence and its application to homeomorphism groups of dense trees --- Abstract
- Monday, 14. 01. 2019: Sebastian Posur (Siegen), TBA --- Abstract
- Wanna give a talk? Get in touch!

# Abstracts:

- Monday, 17. 12. 2018: Aleksandra Kwiatkowska (Münster), Kechris-Pestov-Todorcevic correspondence and its application to homeomorphism groups of dense trees

I will discuss the Kechris-Pestov-Todorcevic correspondence relating the Ramsey property for a family of finite structures and dynamical properties, like the extreme amenability, of the automorphism group of the corresponding countable limit structure. I will then apply this correspondence to compute universal minimal flows of homeomorphism groups of dense trees called Ważewski dendrites. - Monday, 14. 01. 2019: Sebastian Posur (Siegen), TBA

# List of past talks:

- Monday, 10. 12. 2018: Emily Norton (Bonn), Do finite groups of Lie type and Cherednik algebras speak to each other? --- Abstract
- Monday, 03. 12. 2018: J. M. Moreno Fernández (Bonn), Infinity structures and higher products --- Abstract
- Monday, 19. 11. 2018: Sean Tilson (Wuppertal), Algebraic topology and hints at brave new algebra
- Monday, 5. 11. 2018: Hsueh-Yung Lin (Bonn), On the fundamental groups of compact Kähler threefolds --- Abstract
- Monday, 22. 10. 2019: Stéphanie Cupit-Foutou (Bochum), An overview on affine versus projective spherical varieties --- Abstract
- Monday, 15. 10. 2018: Urban Jezernik (University of the Basque Country, Leioa), Fibres of word maps --- Abstract
- Monday, 18. 6. 2018: Xiaolei Wu (Bonn), Conjugacy growth and finiteness properties for families of subgroups --- Abstract
- Wednesday, 13. 6. 2018: Gabriel Lehéricy (Paris/Konstanz), Using quasi-orders to describe C-groups. --- Abstract
- Monday, 11. 6. 2018: Gabriel Lehéricy (Paris/Konstanz), Quasi-orders as a uniform approach to orders and valuations. --- Abstract
- Wednesday, 6. 6. 2018: Andriy Regeta (Köln), Automorphism groups of affine toric varieties --- Abstract
- Monday, 04. 6. 2018: Saba Aliyari (HHU), An introduction to tropical geometry --- Abstract
- Tuesday, 30. 5. 2018: Fabio Bernasconi (Imperial College London), An introduction to the Minimal Model Program and singularities --- Abstract
- Tuesday, 29. 5. 2018: Fabio Bernasconi (Imperial College London), Pathologies for Fano varieties and singularities in positive characteristic --- Abstract
- Tuesday, 22. 5. 2018: Arthur Forey (Paris), Local densities --- Abstract
- Tuesday, 15. 5. 2018: Jeanine Van Order (Bielefeld), Bounds for Mordell-Weil ranks via bounds for Fourier-Whittaker coefficients. --- Abstract
- Monday, 7. 5. 2018: Boulos El Hilany (MPIM, Bonn), Constructing polynomial systems with many positive solutions using tropical geometry --- Abstract
- Monday, 23. 4. 2018: Panagiotis Konstantis (Marburg), The existence of almost complex structures on special manifolds --- Abstract
- Monday, 16. 4. 2018: Claudio Quadrelli (Milan), The Bloch-Kato conjecture and maximal pro-p Galois groups of fields --- Abstract
- Monday, 9. 4. 2018: Coffee, Cake and Daylight --- Celebrating the completion of the (Un)Common room
- Monday, 5. 2. 2018: Carsten Feldkamp (HHU D'dorf), Results on the Magnus property --- Abstract
- Monday, 29. 01. 2018: Claudia Schoemann (Mainz), Unitary representations of p-adic U(5) --- Abstract
- Monday, 22. 01. 2018: Davide Veniani (Mainz), Recent advances about lines on quartic surfaces --- Abstract
- Monday, 15. 01. 2018: Julian Brough (Wuppertal), Using characters to determine solubility criterion for finite groups --- Abstract
- Monday, 8. 01. 2018: Kevin Langlois (HHU D'dorf), Algebraic cycles and log homogeneous resolutions of singularities --- Abstract
- Tuesday, 19. 12. 2017: Cinzia Casagrande (Turin), On the Fano variety of linear spaces contained in two odd-dimensional quadrics --- Abstract
- Monday, 18. 12. 2017: Sylvy Anscombe (UCLan, Preston), Valued Fields: Discrete versus Tame --- Abstract
- Monday, 11. 12. 2017: Matteo Vannacci (HHU D'dorf), On self-similar finite p-groups --- Abstract
- Monday, 27. 11. 2017: Luca Tasin (Bonn), On some results about projective structures --- Abstract
- Monday, 20. 11. 2017: Zaniar Ghadernezhad (Freiburg), Non-amenablility of automorphism groups of generic structures --- Abstract
- Tuesday, 14. 11. 2017: Tobias Hemmert (HHU D'dorf), KO-theory of compact homogenous spaces --- Abstract
- Monday, 13. 11. 2017: Mima Stanojkovski (Bielefeld), Intense automorphisms of finite groups --- Abstract
- Tuesday, 07. 11. 2017: David Bradley-Williams (HHU D'dorf), Classes of metrically homogeneous graphs, Ramsey properties and automorphism groups --- Abstract
- Monday, 23. 10. 2017: Alastair Litterick (Bielefeld/Bochum), Complete Reducibility and Subgroup Structure of Reductive Groups Abstract
- Monday, 16. 10. 2017: Alejandra Garrido (HHU D'dorf), What is a topological full group? --- Abstract
- Monday, 25. 9. 2017: Paula Lins (Bielefeld), Bivariate zeta functions of some finitely generated nilpotent groups --- Abstract
- 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 archive:

- 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 Selbergs 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

Hyperkähler manifolds play an important role in mathematics and physics since they are Ricci-flat and thus satisfy the Einstein equation. Just as Kähler manifolds naturally have a symplectic structre, Hyperkähler manifolds have a complex symplectic structre, but the converse is not true in general. In my talk I will explain an approach to find examples of complex symplectic manifolds that are not Hyperkähler manifolds. - 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. - Monday, 16. 10. 2017: Alejandra Garrido (HHU D'dorf), What is a topological full group?

I will attempt to answer the question in the title. - Monday, 23. 10. 2017: Alastair Litterick (Bielefeld/Bochum), Complete Reducibility and Subgroup Structure of Reductive Groups

Reductive algebraic groups have been studied intensively since at least the mid-1900s, for instance due to their close connections with the finite simple groups. Over the complex numbers, reductive groups such as the general linear group are well-behaved, for the same reason that complex representations of finite groups are nice: Modules are completely reducible. In the late 1990s, Serre generalised this concept from representation theory to all reductive groups. We'll look at this definition, and see how it combines algebraic geometry, representation theory and geometric invariant theory in understanding the subgroup structure of reductive groups over other fields. - Tuesday, 07. 11. 2017: David Bradley-Williams (HHU D'dorf), Classes of metrically homogeneous graphs, Ramsey properties and automorphism groups

The random graph is the unique homogeneous and universal countably infinite graph. Indeed that is a consequence of the fact that it can be constructed as the Fraïssé limit of the class of finite graphs. A wider class of graphs are homogeneous when you add predicates for distances (in the graph metric). There is a catalogue of such graphs by Cherlin, which is a conjectural classification. This will be a light tour of topics and I'll attempt to visit some of these interesting related topics in a suitably introductory fashion. - Monday, 13. 11. 2017: Mima Stanojkovski (Bielefeld), Intense automorphisms of finite groups

Let G be a finite group and let Int(G) be the subgroup of Aut(G) consisting of those automorphisms (called 'intense') that send each subgroup of G to a conjugate. Intense automorphisms arise naturally as solutions to a problem coming from Galois cohomology, still they give rise to a greatly entertaining theory on its own. We will discuss the case of groups of prime power order and we will see that, if G has prime power order but Int(G) does not, then the structure of G is (surprisingly!) almost completely determined by its nilpotency class. The results I will present are part of my PhD thesis. - Tuesday, 14. 11. 2017: Tobias Hemmert (HHU D'dorf), KO-theory of compact homogenous spaces

The aim of this talk is to explain some computations (old and new) of real topological K-theory of homogeneous spaces G/H where G is a compact Lie group and H is a closed subgroup. I will start by giving a brief introduction to KO-theory and will then outline the computations, which are essentially representation-theoretic and go via the computation of the so-called Witt ring of G/H. - Monday, 20. 11. 2017: Zaniar Ghadernezhad (Freiburg), Non-amenablility of automorphism groups of generic structures

The study of amenable groups is originated in the works of von Neumann in his analysis of Banach-Tarski paradox. Since then amenability, non-amenability and paradoxicality has been studied for various groups appearing in different parts of mathematics. A topological group $G$ is amenable if every $G$-flow has an invariant Borel probability measure. Well-known examples of amenable groups are finite groups, solvable groups and locally compact abelian groups. The study of amenability of topological groups benefit from various viewpoints that ranges from analytic approach to combinatorial. Kechris, Pestov, and Todorcevic established a very general correspondence which equates a stronger form of amenability, called extreme amenability, of the automorphism group of an ordered Fra\"iss\'e structure with the Ramsey property of its finite substructures. In the same spirit Moore showed a correspondence between the automorphism groups of countable structures and a a structural Ramsey property, which englobes F\o lner's existing treatment. Hrushovski's generic constructions generalizes the Fra\"iss\'e's-limit structures by considering a notion of strong-substructure rather than substructure. In this talk we will consider automorphism groups of certain Hrushovski's generic structures. We will show that they are not amenable by exhibiting a combinatorial/geometric criterion which forbids amenability. - Monday, 27. 11. 2017: Luca Tasin (Bonn), On some results about projective structures.
- Monday, 11. 12. 2017: Matteo Vannacci (HHU D'dorf), On self-similar finite p-groups

Groups acting on rooted trees provide counterexamples to a number of problems in group theory and they have been extensively studied. A particularly interesting subclass is given by self-similar groups. We show that this class is in some sense very rigid, as we prove that there are only finitely many finite p-groups of a given rank acting faithfully and self-similarly on the rooted p-regular tree. - Monday, 18. 12. 2017: Sylvy Anscombe (UCLan, Preston), Valued Fields: Discrete versus Tame
- Tuesday, 19. 12. 2017: Cinzia Casagrande (Turin), On the Fano variety of linear spaces contained in two odd-dimensional quadrics

We will talk about the geometry of the Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space of dimension 2m+2. The variety G is isomorphic in codimension one to the blow-up X of P^{2m} at 2m + 3 points. I will explain how to show the existence of a birational map between G and X, and how this map allows to determine the cones of nef, movable and effective divisors of G, and the automorphism group of G. This generalizes to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m = 1). Some of these results are a joint work with Carolina Araujo (IMPA). - Monday, 8. 01. 2018: Kevin Langlois (HHU D'dorf), Algebraic cycles and log homogeneous resolutions of singularities

The algebraic Betti numbers of a complex projective manifold X are usually defined as the dimensions of the spaces generated by the algebraic cycle classes in the even cohomology H^2i(X, C). Algebraic stringy invariants are extensions of this construction to the more general case of algebraic varieties with log terminal singularities. Recently, Batyrev and Gagliardi have adressed many conjectures concerning the behavior of the algebraic stringy Euler number with respect to the minimal model program. In this note we show that the Batyrev-Gagliardi conjectures hold true for projective varieties admitting a log homogeneous resolution. This is a joint work with Clelia Pech and Michel Raibaut. - Monday, 15. 01. 2018: Julian Brough (Wuppertal), Using characters to determine solubility criterion for finite groups

Recently many authors have used a variety of graphs to encode information about groups and then studied these graphs to determine group structure. In this talk I will consider two graphs call the vanishing graph and the block graph. In particular, I will discuss the implications of certain edges and use these to provide solubility criterion for a group. - Monday, 22. 01. 2018: Davide Veniani (Mainz), Recent advances about lines on quartic surfaces

The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines. - Monday, 29. 01. 2018: Claudia Schoemann (Mainz), Unitary representations of p-adic U(5)

We study the parabolically induced complex representations of the unitary group in 5 variables, U(5), defined over a p-adic field. We determine the points and lines of reducibility and the irreducible subquotients of these representations. Further we describe - except several particular cases - the unitary dual in terms of Langlands quotients. - Monday, 5. 2. 2018: Carsten Feldkamp (HHU D'dorf), Results on the Magnus property

It is known that the fundamental groups of compact, non-orientable surfaces possess the Magnus property. The aim of this talk is to present a generalisation of that theorem for some combinations of amalgamated and direct products. A group G possesses the Magnus property if for every two elements u,v in G with the same normal closure, the element u is conjugate in G to v or v^{-1}. The Magnus property was named after Wilhelm Magnus who proved it for free groups in 1930. - Monday, 16. 4. 2018: Claudio Quadrelli (Milan), The Bloch-Kato conjecture and maximal pro-p Galois groups of fields

Abstract: The absolute Galois group of a field is the most interesting group for a number theorist. It is also a very mysterious group, as in general very little is known about its structure. The recent proof of the Bloch-Kato conjecture by V. Voevodsky provides new possibilities to investigate the structure of such groups, via Galois cohomology. After introducing gently the cohomology of a (profinite) group, I will present some new results on the structure of pro-p groups whose Galois cohomology behaves like the Galois cohomology of absolute Galois groups. In particular, such results provide new obstructions for the realization of a pro-p group as the maximal pro-p Galois group (and thus also as the absolute Galois group) of a field. - Monday, 23. 4. 2018: Panagiotis Konstantis (Marburg), The existence of almost complex structures on special manifolds.

We will show in this introductory talk some topological techniques to determine the existence of almost complex structures on special manifolds. These techniques involve characteristic classes, homotopy theory and topological K-theory. - Monday, 7. 5. 2018: Boulos El Hilany (MPIM, Bonn), Constructing polynomial systems with many positive solutions using tropical geometry.

The number of positive solutions to a system of two polynomials in two variables defined over the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. In this talk, I am going to present a construction of a system as above having 7 positive solutions. This is achieved using tools developed in tropical geometry. When the corresponding tropical hypersurfaces intersect transversally, one can easily estimate the positive solutions to the system using the classical combinatorial patchworking for complete intersections. This generalization is applied to construct a system as above having 6 positive solutions. It turns out that this bound is sharp. The main result is proved using non-transversal intersections of tropical curves. - Tuesday, 15. 5. 2018: Jeanine Van Order, Bounds for Mordell-Weil ranks via bounds for Fourier-Whittaker coefficients.

Let E be an elliptic curve defined over the rationals, p a prime, and K an imaginary quadratic field. I will explain how to show that the Mordell-Weil group of E over the Z_p^2-extension of K is finitely generated up to higher Heegner points, ultimately by using bounds for Fourier-Whittaker coefficients of certain automorphic forms. - Tuesday, 22. 5. 2018: Arthur Forey (Paris), Local density

The local density of a complex or real analytic set X at a point x is the limit of the normalized volume of X intersected with balls centered at x of radii going to zero. One can give a geometric interpretation of this limit in terms of tangent cones. A similar theory has been developed for p-adic sets. I will present all those notions of densities and an extension to more general valued fields. - Monday, 28. 5. 2018: Fabio Bernasconi (Imperial College London), An introduction to the Minimal Model Program and singularities

The Minimal Model Program has become a central tool in the birational classification of higher-dimensional algebraic varieties. Despite its technicalities, the main ideas can be explained in geometric terms. I will describe it for surfaces, introducing the notions of Kodaira dimension and minimal models and showing with some examples that singularities already lurk around in dimension two. I will then explain what are the difficulties appearing starting from dimension three and state the full MMP conjecture. - Tuesday, 29. 5. 2018: Fabio Bernasconi (Imperial College London), Pathologies for Fano varieties and singularities in positive characteristic

After the seminal work of Hacon and Xu, a large part of the MMP has been established for threefolds over fields of characteristic p > 5. However very little is known in small characteristic, especially in the study of threefold singularities. In this talk I will review the cone construction for Fano-type varieties and I will explain how this allows to construct pathological examples of singularities in positive characteristic starting from Fano varieties violating vanishing theorems . As an application, I will show how to construct log del Pezzo surfaces in characteristic two and three violating Kodaira vanishing, thus deducing the existence of klt not Cohen-Macaulay threefold singularities and I will use some recent examples of Totaro to construct non-normal plt centres. - Monday, 4. 6. 2018: Saba Aliyari (HHU), An introduction to tropical geometry

This talk will be an introductory talk about tropical geometry. First I will define the basic notions of tropical semi-rings and tropical curves and then by focusing on R^2 I will present the tropical version of Bezout's theorem. - Wednesday, 6. 6. 2018: Andriy Regeta (Köln), Automorphism groups of affine toric varieties

The automorphism group of an affine algebraic variety $X$ does not has the structure of an algebraic group, but it has the structure of a so-called infinite-dimensional algebraic group. It is well-known that two reductive algebraic groups are isomorphic if and only if they the same rank and the same weights of root subgroups. We will prove an analog result in infinite-dimensional case. Let $Y$ be an affine toric variety. Root subgroups of $Aut(X)$ and $Aut(Y)$ have the same weights if and only if $Y \cong X$. Moreover, using dynamics of binational transformations of surfaces we prove that if $dim X = dim Y = 2$, then $Aut(Y) \cong Aut(X)$ as an abstract group if and only if $Y \cong X$. - Monday, 11. 6. 2018: Gabriel Lehéricy (Paris/Konstanz), Quasi-orders as a uniform approach to orders and valuations.

Orders and valuations play a central role in real algebra. The theory of ordered fields present strong similarities with the theory of valued fields, which is why several attempts have been made to find a common generalization of these two theories. In one of his papers, Fakhruddin realized that quasi-orders ( i.e. reflexive, transitive and total binary relations) can be used as a generalization of orders and valuations. After recalling a few basic facts of valuation theory, I will present Fakhruddin's theory of quasi-ordered fields and show that it is a very practical generalization of ordered and valued fields. I will then introduce quasi-ordered groups and explain how they relate to ordered and valued groups. - Wednesday, 13. 6. 2018: Gabriel Lehéricy (Paris/Konstanz), Using quasi-orders to describe C-groups. [Model Theory research seminar]

A C-relation is a ternary relation interpretable in the set of branches of a tree. A C-group is a group endowed with a C-relation compatible with the group operation; examples of such structures are totally ordered groups and valued groups. C-minimality was introduced as an analog of o-minimality where the order is replaced by a C-relation. Macpherson and Steinhorn gave a partial description of C-minimal groups. Simonetta and Delon completely classified abelian valued C-minimal groups. However, there is still no complete classification of C-groups. In my talk, I will show how we can use quasi-orders to study C-groups. I will show that any C-group is basically a “mix” of ordered and valued groups, so that ordered and valued groups are the "building blocks" of C-groups. I will then use this result to describe abelian C-minimal groups. - Monday, 18. 6. 2018: Xiaolei Wu (Bonn), Conjugacy growth and finiteness properties for families of subgroups

I will start the talk with definitions of word growth and conjugacy growth. Then I will relate conjugacy growth to some interesting problems related to finiteness properties of groups. In particular, I will discuss an interesting conjecture due to Juan-Pineda and Leary. The conjecture says a group admits a finite model for the classifying space for the family of virtually cyclic subgroups if and only if it is virtually cyclic. I will discuss some interesting cases which can be solved using conjugacy growth. This is a joint work with Timm von Puttkamer. - Monday, 15. 10. 2018: Urban Jezernik (University of the Basque Country, Leioa), Fibres of word maps

A word is an element of a free group. Given any other group, one can evaluate this word at any tuple of its elements. This evaluation is called a word map. We will survey some recent developments in inspecting fibres of such maps. - Monday, 22. 10. 2018: Stéphanie Cupit-Foutou (Bochum), An overview on affine versus projective spherical varieties

- Monday, 5. 11. 2018: Hsueh-Yung Lin (Bonn), On the fundamental groups of compact Kähler threefolds

From a Hodge-theoretic and a deformation-theoretic point of view, compact Kähler manifolds are natural generalizations of projective manifolds. Whether the fundamental group of a compact Kähler manifold can always be realized as the fundamental group of a projective manifold is still an open problem. In this talk, we will explain our solution to this problem in dimension 3. (Joint work with B. Claudon and A. Höring.)

- Monday, 19. 11. 2018: Sean Tilson (Wuppertal), Algebraic topology and hints at brave new algebra.

I will introduce early successes of algebraic topology at answering geometric and topological questions. By investigating and enhancing classical invariants with more structure, natuarlly, more information is retained. We will see how these enhancements naturally lead to analogs of classical algebra, so dubbed Brave New Algebra. - Monday, 03. 12. 2018: Jose M. Moreno Fernández (Bonn), Infinity structures and higher products

Abstract: Given a topological space X, the higher Whitehead products constitute a (partially defined, subtle) operation on the homotopy groups of X. These higher products are invariants of the homotopy type of X. We will show how, from Daniel Quillen's point of view for rational homotopy theory, L-infinity algebras can be used to effectively detect and compute these products rationally. The talk will be elementary. It will be based on examples and computations, and no advanced knowledge of homotopy theory, graded Lie algebras or homological algebra will be assumed. - Monday, 10. 12. 2018: Emily Norton (Bonn), Do finite groups of Lie type and Cherednik algebras speak to each other?

This talk is about unexplained coincidences of decomposition numbers between seemingly unrelated objects. The decomposition matrix of a unipotent block of a finite group of Lie type in cross characteristic has a square submatrix indexed by the unipotent characters. Many low-rank examples of these decomposition matrices were computed in recent years by Dudas and Malle. In many cases, the matrices obtained are identical on the principal series characters, which are indexed by the irreducible characters of the Weyl group, to decomposition matrices I computed for the rational Cherednik algebra at a corresponding parameter. I will explain structural parallels and differences between the two theories and summarize the numerical data, and I will provide examples that show that we cannot in general expect the decomposition matrix of the Cherednik algebra to appear as a submatrix of the decomposition matrix of the finite group.

Given a smooth real manifold M of even dimension, it is a basic problem to understand whether M admits a complex projective structure and then study the algebraic invariants of such structures. In this talk I will report on some recent results on this topic, in particular concerning Chern numbers and Spin six manifolds. This is based on joint works with S. Schreieder.