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, 16:3017:30, Room 25.22.02.81 . 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 BradleyWilliams
Andrea Fanelli
Alejandra Garrido
Kevin Langlois
This seminar is part of the activities of the Research Training Group GRK 2240: Algebrogeometric Methods in Algebra, Arithmetic and Topology.
List of scheduled talks:

Wanna give a talk? Get in touch!
 Monday, 15. 01. 2018: Julian Brough (Wuppertal), Using characters to determine solubility criterion for finite groups  Abstract
 Monday, 22. 01. 2018: Davide Veniani (Mainz), Recent advances about lines on quartic surfaces  Abstract
 Monday, 29. 01. 2018: Claudia Schoemann (Göttingen), TBA  Abstract
 Monday, 5. 2. 2018: Carsten Feldkamp (HHU D'dorf), TBA  Abstract
List of past talks:
 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 odddimensional 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 selfsimilar finite pgroups  Abstract
 Monday, 27. 11. 2017: Luca Tasin (Bonn), On some results about projective structures  Abstract
 Monday, 20. 11. 2017: Zaniar Ghadernezhad (Freiburg), Nonamenablility of automorphism groups of generic structures  Abstract
 Tuesday, 14. 11. 2017, 14:00  15:00: Tobias Hemmert (HHU D'dorf), KOtheory of compact homogenous spaces  Abstract
 Monday, 13. 11. 2017: Mima Stanojkovski (Bielefeld), Intense automorphisms of finite groups  Abstract
 Tuesday, 07. 11. 2017, 16:3017:30: David BradleyWilliams (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 pgroups  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 blowup 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 firstorder logic (FOL). A standard application is the introduction of nonstandard 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 LöwenheimSkolem 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 pgroups
We discuss two conjectures, one disproved and one still unsolved, regarding automorphisms of finite pgroups.  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 wellknown 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 pgroups so as to understand the complexity of such algebras. Our aim is to count isomorphism types of cohomology algebras of pgroups 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 Ztower of a knot exterior in a generating function. While the asymptotic growth of these cardinalities is wellunderstood, 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 AltmannHausen 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 Tequivariant 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 LiendoSuess 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 LaplaceBeltrami operator Δ. Selberg conjectured that for a Hecke congruence subgroup Γ_{0}(q) the smallest nontrivial 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 Ricciflat 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 genus2curves 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 functorofpoints 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 oneelement set from this point of view. The talk gives an introduction to this topostheoretic 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 mid1900s, 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 wellbehaved, 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 BradleyWilliams (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), KOtheory of compact homogenous spaces
The aim of this talk is to explain some computations (old and new) of real topological Ktheory 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 KOtheory and will then outline the computations, which are essentially representationtheoretic and go via the computation of the socalled Witt ring of G/H.  Monday, 20. 11. 2017: Zaniar Ghadernezhad (Freiburg), Nonamenablility of automorphism groups of generic structures
The study of amenable groups is originated in the works of von Neumann in his analysis of BanachTarski paradox. Since then amenability, nonamenability 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. Wellknown 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'slimit structures by considering a notion of strongsubstructure 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 selfsimilar finite pgroups
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 selfsimilar groups. We show that this class is in some sense very rigid, as we prove that there are only finitely many finite pgroups of a given rank acting faithfully and selfsimilarly on the rooted pregular 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 odddimensional quadrics
We will talk about the geometry of the Fano manifold G parametrizing (m1)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 blowup 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 BatyrevGagliardi 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 (Göttingen), TBA
 Monday, 5. 2. 2018: Carsten Feldkamp (HHU D'dorf), TBA
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.