Seminar of Algebra and Number Theory

The talks during the summer semester 2020 take place in zoom. Please contact Gabriela Weitze-Schmithüsen, if you would like to participate.

Next seminars:

Past seminars:

  • 14.7.2020 - Sven Caspart (KIT, Karlsruhe)
    Title: 
    Translation Manifolds - beyond two dimensions
    Abstract:
    The concept of a (finite) translation surface is a well studied with much progress in recent years. However, all this is for surfaces, i.e. two dimensional. In this talk we will tackle the question: What happens in higher dimension? To this end we will introduce translation manifolds, a higher dimensional generalisation of a translation
    manifold. We will see how some concepts, in particular singularities, generalise to the higher dimensional case and what can and cannot happen as well as some phenomena unknown to the two-dimensional world.
    Towards the end we take a closer look at removable singularities and how to identify them.

  • 7.7.2020 - Mirko Stappert (Universität des Saarlandes)
    Titel:
    (Höhere) Algebraische K-Theorie
    Abstract:
    Algebraische \(K\)-Theorie ist eine Folge von Invarianten von Ringen. Sie hat tiefgründige Beziehungen zu unterschiedlichsten Gebieten wie Zahlentheorie, Geometrie/Topologie und Operatoralgebren. Zum Beispiel ist sie sowohl in der Klassifikation von hochdimensionalen Mannigfaltigkeiten als auch von \(C^*\)-Algebren unerlässlich.
    In diesem Vortrag werden wir historisch starten und zunächst die unteren \(K\)-Gruppen (\(K_0, K_1\)) definieren und ihre Bedeutsamkeit an vielen zahlentheoretischen und geometrischen Beispielen beleuchten. Danach werden wir Quillens höhere \(K\)-Theorie motivieren und definieren. Im letzten Teil soll es dann noch eine kurze Einführung in die modernen Berechnungverfahren durch Spurmethoden und Topologische-Hochschild-Homologie geben.

  • 30.6.2020 - Eduard Duryev (Institut de mathématiques de Jussieu, Paris)
    Title: SL(2,Z) orbits of origami and spaces of isoperiodic deformations.
    Abstract: An origami, or a square-tiled surface, is a collection of unit squares with pairs of sides identified by parallel translations, so that they form a closed surface. The group SL(2,Z) acts on the set of square-tiled surfaces by applying affine transformation followed by a suitable cut-and-paste operation. Understanding orbits of this action is a vastly open question. I will address its progress in genus 2.
    We will see that square-tiled surfaces naturally sit on closed leaves of isoperiodic foliation, or rel leaves, of a bigger moduli space of translation surfaces. Such leaf in turn admits a square-tiling of their own and square-tiled surfaces in question are nothing more than points of the squares with rational coordinates. I will show how this helps to better understand the main question.

 

  • 19.5.2020 - Manuel Kany (Universität des Saarlandes)
    Title: Generische Tori und ein Dichtheitssatz von Prasad und Rapinchuk
    Abstract: 
    G. Prasad und A. Rapinchuk haben 2013 ein Paper herausgebracht, in dem sie sich unter anderem generischen Tori widmen. Der Vortrag soll dazu dienen Grundlagen im Bereich der algebraischen Gruppen zu legen und die Argumente im Beweis zu Therorem 9.10 in "Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces" nachvollziehbar zu machen.

    Es sei K ein Körper mit unendlich vielen Elementen und \(K\) der algebraische Abschluss von \(k\). Eine algebraische Gruppe über \(K\) ist ein lineare algebraische Varietät, sodass die Gruppenoperationen Morphismen von Varietäten sind. Ein Torus ist eine algebraische Gruppe, die isomorph zu einer zusammenhängenden, abgeschlossenen Untergruppe der Diagonalmatrizen \(D_n(K)\) ist für eine natürliche Zahl \(n\).

    Sowohl die Weyl-Gruppe \(N_G(T)/C_G(T)\) (Normalisator/Kommutator) von \(T\), als auch die Galois-Gruppe \(Gal(K/k)\) operieren auf den Charakteren \(X(T)\) von \(T\). Man erhält zwei Untergruppen in \(Aut(X(T))\). Stimmt die Untergruppe, die von der Weyl Gruppe kommt, mit der von der Galois-Gruppe kommenden überein und ist der Torus maximal bezüglich Inklusion, so nennen wir ihn generisch.

    Das Theorem liefert nun: Ist die Kommutatorgruppe eines Elements \(g\) aus \(G\) ein generischer Torus und ist ein weiteres Element \(x\) in \(G\) von unendlicher Ordnung gegeben, das nicht mit \(g\) kommutiert, so erzeugen \(x\) und \(g\) eine Gruppe, die im Falle einer fast-einfachen Gruppe, deren Wurzeln in \(\Phi(G,T)\) alle die gleiche Länge haben, Zariski-dicht in \(G\) ist.

  • 5.5.2020 - Alan McLeay (Université du Luxembourg)
    Title: Homeomorphic subsurfaces and the omnipresent arcs
    Abstract: No surface \(S\) of finite-type admits a subsurface homeomorphic to \(S\), unless the inclusion map is homotopic to the identity.  To prove this fact, one only needs to count the genus or number of punctures on the surface.  For surfaces of infinite-type, we will show that more "interesting" homeomorphic subsurfaces can occur.  In exploring this, we are led naturally to a subclass of arcs and a new graph on which many big mapping class groups act.
  • 3.2.2020 - Anja Randecker (NYU Shanghai)
    Title: Random walks on hyperbolic manifolds
    Abstract: For any group with a measure, we can define random walks on this group. If the group acts on a space (such as the Poincaré disk), we also obtain random walks on this space. And if a random walk is not recurrent, then it defines a hitting measure on the boundary of the space (such as the circle).
    In general, it is an open question whether the hitting measure is in the same class as the Lebesgue measure. In a joint work with Giulio Tiozzo, we have answered this question negatively for a certain class of groups.
    Previous knowledge on random walks is not required. I will start by introducing random walks in general, before explaining properties of random walks on hyperbolic spaces and the main tool for our result: cusp excursion.

  • 27.1.2020 - Stefano Francaviglia (University of Bologna) 
    Title:
    Axes of automorphism in Outer Space and its bordification
    Abstract: In complete analogy with Teichmuller space, the so called Outer Space of the free group \(F_n\) is the moduli space of marked graphs whose fundamental group is \(F_n\). The group \(\mathrm{Out}(F_n)\) naturally act on this space by change of marking. Given an automorphism \(\phi\), the set \(\mathrm{Min}(\phi)\) is the set of points of Outer Space that are minimally displaced by \(\phi\). In this talk we will show that \(\mathrm{Min}(\phi)\) consists exactly of the set of train track for \(\phi\) and we will discuss some finiteness property of \(\mathrm{Min}(\phi)\).

  • 20.1.2020 - Julian Meyer (Universität des Saarlandes)
    Title: 
    Fundamentalbereiche und Modulformen zu Fricke-Gruppen
    Abstract:
    Bei der Beschäftigung mit Modulformen kommen immer wieder Kongruenzuntergruppen von \(SL_2(\mathbb{Z})\) vor, insbesondere Gruppen der Form

    \[
    \Gamma_0(n) := \{ \gamma \in SL_2(\mathbb{Z}) \ \big| \ \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \ \text{mit} \ n|c \} \subset SL_2(\mathbb{Z})
    \]

    für ein \(n \in \mathbb{N}\). Hierzu hat man die Atkin-Lehner-Involutionen \(\gamma_Q \in SL_2(\mathbb{R)} \forall \ Q\|n\). Diese Normalisieren die Gruppen \(\Gamma_0(n)\) und man betrachtet dann die Gruppen

    \[
    \Gamma_0(n)^* := \Gamma_0(n) \cup \bigcup_{Q\|n} \gamma_Q \Gamma_0(n) \ \subset SL_2(\mathbb{R)}.
    \]

    Diese Gruppen bilden dann den gesamten Normalisator von \(\Gamma_0(n)\) und sind maximale, diskrete Untergruppen von \(SL_2(\mathbb{R})\). \\
    Für \(n=p\) eine Primzahl erhält man nur die Fricke-Involution \(\gamma_p := \begin{pmatrix} 0 & -\frac{1}{\sqrt{n}} \\ \sqrt{n} & 0 \end{pmatrix}\). In meinem Vortrag wird es um die Fundamentalbereiche der so erhaltenen Fricke-Gruppen \(\Gamma_0(p)^*\) gehen. Ich werde darstellen, wie man sich zusammenhängende, unbeschränkte Fundamentalbereiche für die Fricke-Gruppen beschaffen kann, zum Einen geometrisch und zum Anderen durch nachrechnen. Zum Abschluss werde ich noch schildern, wie die Geometrie dieser Fundamentalbereiche die Abschätzung von Fourier-Koeffizienten von Modulformen für die Fricke-Gruppen beeinflusst.

  • 31.7.2019 - Manuel Kany (Universität des Saarlandes)
    Title: 
    About trace fields of translation surfaces
    Abstract: Es sei \((X,w)\) eine Translationslfläche, mit \(w\) holomorphe 1-Form. Für eine Richtung \(v\) in \( \mathbb{R})^2\) bilden die Geodätischen in diese Richtung eine Blätterung für \((X,w)\). Wir konstruieren Zykel in der singulären Homologie \(H_1(X, \mathbb{R}))\), die für lange Zeit den Geodätischen folgen und poincaré dual zu den 1-Formen \(Re(w),~ Im(w)\) in \(H^1(X, \mathbb{R})\) sind.

  • 31.7.2019 - Pascal Kattler (Universität des Saarlandes)
    Title: 
    On covers of the pillowcase

  • 24.7.2019 - Luca Junk (Universität des Saarlandes)
    Title:
     Quantum Automorphism Groups of Graphs and Coherent Algebras
    Abstract:
    The symmetry of a graph is encoded in its automorphism group. In this talk we will consider a
    generalization of this, called quantum automorphism group, which captures more information about the
    graph. In particular, I will present a recent result by Lupini, Mančinska and Roberson about the asymptotic
    behaviour of quantum automorphism groups when the size of the graphs goes to infinity, which is in analogy
    with a classical result of Erdős and Rényi about automorphism groups.

  • 15.7.2019 - Bharathwaj Palvannan (University of Pennsylvania)
    Title: Codimension two cycles in Iwasawa theory
    Abstract: Classical Iwasawa theory studies a relationship, called the Iwasawa main conjecture, between a
    \(p\)-adic \(L\)-function and a Selmer group.  This relationship involves codimension one cycles of an
    Iwasawa algebra. This talk will discuss results on the topic of higher codimension Iwasawa theory. We will
    consider the restriction to an imaginary quadratic field of an elliptic curve defined over the rational numbers
    with good supersingular reduction at an odd prime. We shall also consider the tensor product of Hida
    families. This is joint work with Antonio Lei.

  • 8.7.2019 - Martin Ulirsch (Goethe Universität Frankfurt)
    Title: What is the fundamental group of a tropical curve?
    Abstract: An (abstract) tropical curve is a finite metric graph (possibly with further decorations such as
    integer vertex weights or sheaves of harmonic functions). The topological fundamental group of the
    underlying graph is a finitely generated free group that classifies all topological covers. One might suspect
    that this is all there is. In this talk I intend to convince you that there are at least three other different
    candidates that answer the question in the title: one that classifies tropical admissible covers,  another that
    classifies realizable tropical admissible covers, and a third that seems particularly suitable to compactify
    Culler-Vogtmann outer space. This gives a new perspective on the classical correspondence theorem for
    algebraic and tropical Hurwitz numbers and allows us to (re-)construct tropical/logarithmic compactifications
    of the moduli space of curves with level structures and of profinite Teichmüller space.

  • 20.5.2019 - Vincent Delecroix (Université Bordeaux 1, MPI Bonn)
    Title:
    The dynamics of translation flows in some infinite origamis
    Abstract:
    Origamis are coverings of the square torus ramified over one point. The linear flows on the torus
    pull-back as translation flows on the covers. In the case the covering is finite, we know many (but not all)
    dynamical properties: periodic orbits, ergodicity, mixing properties, etc thanks to the fact that it is a so-called
    Veech surface. I will discuss several techniques that applies in the infinite setting and various examples
    where the dynamics is still mysterious.

  • 24.4.2019 - Manuel Kany (Universität des Saarlandes)
    Title:
    Spurkörper von Translationsflächen
    Abstract: Man bezeichnet mit Spurkörper einer Translationsfläche \((X,w)\) die Körpererweiterung, die man erhält, wenn man zu \(\mathbb{Q}\) die Elemente \(Spur(A)\) adjungiert, wobei die Matrizen \(A\) Elemente der Veechgruppe \(SL(X,w)\) zu \((X,w)\) sind.
    Diese Körpererweiterung ist algebraisch über \(\mathbb{Q}\). Wir lernen erste Ergebnisse von McMullen, Kenyon und Smilie kennen und zeigen damit eine Aussage von Hubert und Lanneau:
    Enthält die Veech Gruppe \(SL(X,w)\) ein Pseudo-Anosov Element, so ist der Spurköper total reel.

  • 15.4.2019 - Markus Kirschmer (RWTH Aachen)
    Title:
    Quaternäre quadratische Formen
    Abstract:
    Nach einem klassischen Ergebnis von Gauß entsprechen die quadratischen Formen in zwei
    Variablen über Z bekanntlich den Idealen quadratischer Erweiterungen von Z. 
    Analog dazu korrespondieren
    auch die quadratischen Formen in vier Variablen über Z bestimmten Idealen in Quaternionenordnungen. 
    In dem Vortrag möchte ich diese Korrespondenz auf beliebige algebraische Zahlkörper ausdehnen.
    Weiter
    möchte ich zeigen, wie die Arithmetik in Quaternionenordnungen
    ausgenutzt werden kann, um die Isometrie-
    klassen im Geschlecht einer quaternären quadratischen Form explizit zu bestimmen.

  • 28.1.2019 - Tobias Columbus (Universität des Saarlandes)
    Title:
    Derivators and Cocartesian Squares
    Abstract:
    Focussing mostly on the ideas, I will give a gentle introduction to derivators and -- if time permits -- discuss some work in progress on a certain class of cocartesian squares and their relation to (pre-)derivators.

  • 21.1.2019 - Ferrán Valdez (UNAM, Morelia, Mexiko)
    Title: Around Big Mapping Class Groups
    Abstract: We introduce Big Mapping Class Groups, discuss their general properties, some recent results regarding their simplicial actions and space of non-trivial quasimorphisms.

  • 27.11.2018 - David Torres-Teigell (Goethe-Universität Frankfurt)
    Title: Calculation of Masur-Veech volumes by counting of square-tiled surfaces
    Abstract: The Hodge bundle H of holomorphic one-forms over the moduli space of genus g compact Riemann surfaces is naturally stratified according to the multiplicities a=(a_1,...,a_n) of the zeroes of the differential. Each stratum H(a) can be parametrised around a point (X,ω) by the so-called period coordinates in the relative cohomology group H^1(X,Z(ω);C), which is a complex vector space of dimension 2g+n-1. These coordinates allow us to define the Masur-Veech measure on the stratum by pulling back the Lebesgue measure, normalised so that the unit cube in the lattice H^1(X,Z(ω); Z + i·Z) has volume one. Masur and Veech proved that this measure is finite (more precisely, its restriction by disintegration to the "hyperboloid" formed by differentials of area one is). We will first explain the approach of Eskin-Masur, Eskin-Okounkov and Kontsevich-Zorich to the computation of these volumes via enumeration of points with integral coordinates, and then apply results on the classification of squared-tiled surfaces and Euler characteristics of arithmetic Teichmüller curves to specific cases.

  • 12.11.2018 - Ingrid Irmer (MPI Bonn)
    Title: Simple curves in covering spaces of surfaces
    Abstract: Let S' --> S be a finite cover, where S is a surface. When does pi_1(S') contain a curve that is simple in S? This talk surveys variants of this question, and related conjectures about mapping class groups and 3-manifolds.

  • 31.08.2018 - Kai Krämer (Universität des Saarlandes)
    Title:
    Liftungen einfach geschlossener Kurven auf endlichen regulären Überlagerungen von geschlossenen Fläche
    Abstract: Es wird die Einfache-Kurven-Homologie sc_p(H_1(S',Z)) zu einer endlich-blättrigem Überlagerung p:S'-->S vorgestellt. Dabei ist S eine geschlossene orientierbare Fläche. Außerdem wird mittels einer expliziten Familie von Beispielen gezeigt, dass H_1(S',Z) nicht immer gleich sc_p(H_1(S',Z)) ist.

  • 19.07.2018 - Valentijn Karemaker (University of Pennsylvania)
    Title:
    Dynamics of Belyi maps
    Abstract: A (genus 0) Belyi map is a finite map from the projective line to itself, branched exactly at 0, 1, and infinity. Such maps can be described combinatorially by their generating systems.
    Assuming further that 0, 1, and infinity are both fixed points and the unique ramification points above 0, 1, and infinity respectively yields dynamical Belyi maps, since the resulting maps can be iterated and will therefore exhibit dynamical behaviour.
    In this talk, we will discuss several results on the dynamics, reductions, and monodromy of dynamical Belyi maps, and the interplay between these.
    (This is joint work with J. Anderson, I. Bouw, Ö. Ejder, N. Girgin, and M. Manes.)

  • 12.07.2018 - Markus Baumeister (RWTH Aachen)
    Title: Classification of geodesic self-dual surfaces
    Abstract: In this talk, I give a glimpse into the classification of geodesic self-dual surfaces.
    Geodesic duality is an interesting kind of duality between discrete (combinatorial) surfaces. In the first part of the talk, I develop the definitions of these discrete surfaces with a special focus on the computational aspect.
    In the second part of the talk, I present a group theoretic approach to the classification of self-dual surfaces that satisfy certain regularity conditions. Finally, I close with a partial classification of
    regular self-dual surfaces.

  • 05.07.2018 - Andrea Thevis (Universität des Saarlandes)
    Title: Strata of \(p\)-Origamis
    Abstract: We begin by recalling some basic definitions from topology and the theory of translation surfaces. Using these we define \(p\)-origamis which arise as normal covers of the torus. They are the objects of main interest in the talk. The goal is to classify the types of singularities of \(p\)-origamis. If time permits, I describe connections to Teichmüller theory and geometric group theory.
  • 07.06.2018 - Mang Zhao (Universität des Saarlandes)
    Title:
    Combinatorial Model of Frieze Patterns
    Abstract:
    Frieze Patterns, a special two-dimensional array, which satisfies unimodular rule for every 2x2 adjacent entries over positive integers, has been invented by H.S.M. Coxeter in 1971. Afterwards, many interesting properties, like periodicity, linear recurrence relations and quiddity cycles, have been successfully investigated for frieze patterns. The combinatorial model of frieze patterns over positive integers has been systematically introduced by Sophie Morier-Genoud in 2015. In the last year, Michael Cuntz invented another important approach to the extension of frieze patterns over any subset of a commutative ring and gave a method to combine 2 frieze patterns.
    In our Bachelor thesis, we look at the operational rules for combination of 2 frieze patterns and give a general “decomposition” form for all frieze patterns. Moreover, we focus on the research on a specific domain of frieze patterns, which is called "simple".

  • 07.05.2018 - Pascal Kattler (Universität des Saarlandes)
    Title: The relative cohomology of abelian covers of the flat pillowcase
    Abstract:
    This talk discusses the paper "The Relative Cohomology Of Abelian Covers Of The Flat
    Pillowcase" from Chenxi Wu. We first introduce finite, abelian covers of a surface, called the flat
    pillowcase. Then we calculate its relative cohomology. We have a natural action from the deck group
    on the cohomology. So we can regard the cohomology as a linear representation of the deck group.
    Finally we decompose this representation in irreducible representations.

  • 25.01.2018 - Oliver Lorscheid (IMPA)
    Title: What is a tropical scheme?
    Abstract: The tropicalization of a classical variety can be seen as a combinatorial gadget that allows us to study certain properties of the classical variety. This technique has been successfully applied to various problems in algebraic geometry. Recently, Jeffrey and Noah Giansiracusa have endowed tropicalizations with the structure of a semiring scheme, a notion that emerged from \(\mathbb{F}_{1}\)-geometry. It is expected that this aproach will provide a more sophisticated foundation for tropical geometry and that it will allow for a deeper understanding of various combinatorial techniques in the area. In this talk, we will first review the concepts of a tropical variety and of tropicalizations, and then turn to an explanation of the role of tropical scheme theory for tropical geometry.

  • 26.10.2017 - Andrea Thevis (Universität des Saarlandes)
    Title: Invariant differential forms in arbitrary characteristic
    Abstract: In this talk I first give a short introduction in invariant theory of finite groups and explain the difficulties when computing invariant rings of finite groups. In the second half of the talk we present an algorithm computing rings of invariant differential forms. We then consider rings of invariant differential forms for permutation groups and pseudo-reflection groups. Using our algorithm, we compute the rings of invariant differential forms for certain examples and study their structure.

  • 07.12.2017 - Christian Steinhart (Universität des Saarlandes)
    Title: Little strolls in Outer Space
    Abstract: The (Culler-Vogtmann) Outer Space is the moduli space of weighted graphs with a marking. In this talk, I will first give a short recap on the definition of Outer Space and the Thurston metric on it. Afterwards I will introduce an isometric embedding between Outer Spaces of different degree and show that you can "wiggle" these embeddings in an isometric manner. We want to know whether these are all the isometric embeddings between Outer Spaces. Our approach is based on the existence and properties of geodesics in these spaces.

  • 24.07.2017 - David Torres-Teigell (Universität des Saarlandes)
    Title: Translation surfaces and Teichmüller curves
    Abstract: Translation surfaces are compact surfaces whose transition maps outside a finite set of points are translations. They can also be defined as polygons with parallel sides identified by translations, or as pairs of a compact Riemann surface of genus \(g\) together with an abelian differential. In particular, there is a natural projection \(\mathcal{H}_{g} \to \mathcal{M}_{g}\) from the space \(\mathcal{H}_{g}\) of translation surfaces of genus \(g\) to the moduli space \(\mathcal{M}_{g}\) which parametrises compact Riemann surfaces of genus \(g\). The group \(\mathrm{SL}(2,\mathbb{R})\) acts naturally on \(\mathcal{H}_{g}\) and we are interested in studying the orbits \(\mathrm{SL}(2,\mathbb{R})\cdot (X,\omega)\subset \mathcal{H}_{g}\) of this action and their projections to \(\mathcal{M}_{g}\). Whenever the projection of such an orbit is an algebraic curve \(C\), we call \(C\to\mathcal{M}_{g}\) a Teichmüller curve. In this talk we will give an overview of the theory and introduce all these concepts. At the end of the talk, we will present the striking ``magic wand'' theorem of Eskin, Mirzakhani and Mohammadi, that claims that orbit closures \(\overline{\mathrm{SL}(2,\mathbb{R})\cdot (X,\omega)}\) are particularly nice objects in \(\mathcal{H}_{g}\).

  • 17.07.2017 - Christian Steinhart (Universität des Saarlandes)
    Title: Thurston metric on Outer Space
    Abstract: The talk will give an introduction to (Culler-Vogtmann) Outer Spaces and the Thurston metric on it. First we will learn how to compute a distance of two points in the Outer Space, which are essentially weighted graphs with a marking. Afterwards we will talk about some properties of the metric.

  • 26.06.2017 - Samuel Lelièvre (Université Paris-Sud 11)
    Title: Ring arithmetic in number fields and flat surfaces
    Abstract: This talk highlights links between flat surfaces, Fuchsian groups, gcd algorithms. Enumeration problems for orders in number fields, sub-monoids in Fuchsian groups, orbits for the linear action of a Fuchsian group on the plane, Farey trees are considered. Beyond the well-known integers, golden integers are discussed.

  • 19.06.2017 - Benjamin Peters (Karlsruher Institut für Technologie)
    Title: Hurwitzräume von Translationsflächen
    Abstract: Hurwitzräume klassifizieren (verzweigte) Überlagerungen mit vorgegeben topologischen Eigenschaften wie das Geschlecht der überlagerten Fläche, der Grad der Überlagerung und die Anzahl sowie Ordnung der Verzweigungspunkte. Wir betrachten den Hurwitzraum aller Überlagerung des Torus sind von Grad zwei mit vier Verzweigungspunkten. Alle Fläche in diesem Raum sind auf natürliche Weise Translationsflächen und wir untersuchen, welche Bahnabschlüsse es in diesem Hurwitzraum gibt. Genauer: Wir beschreiben alle affin-invarianten Untermannigfaltigkeiten dieses Raums, die durch Automorphismen gegeben sind. Insbesondere bestätigt dies die allgemeinere Vermutung, dass es in jedem Hurwitzraum affin-invariante Untermannigfaltigkeiten jeder Dimension gibt.

  • 12.06.2017 - Quentin Gendron (Leibniz Universität Hannover)
    Title: Über die Geometrie der Schichten Abelschen Differentiale
    Abstract: Sei \(\mathcal{M}_{g,n}\) der Modulraum der projektiven glatten algebraischen Kurven \(X\) von Geschlecht \(g\geq 2\) mit \(n\) paarweise verschidenen Punkte \(z_{i}\in X\). Für jede Partition \(k=(k_{1},\ldots,k_{n})\) von \(2g-2\) definieren wir die "Schicht" \[\mathcal{M}_{g}(k)=\Big\{(X,z_{1},\cdots,z_{n}); \sum_{i=1}^{n}k_{i}z_{i}\sim K_{X}\Big\}\subset\mathcal{M}_{g,n},\] wo \(K_{X}\) der kanonische Divisor von \(X\) ist. In diesem Vortrag werde ich Geometrie der Vergißabbildung nach \(\mathcal{M}_{g}\) der Schichten diskutieren. Dafür werde ich den Abschluss der Schichten in der Deligne-Mumford'sche Kompaktifiezierung von \(\mathcal{M}_{g,n}\) betrachten. Teils dieser Vortrag ist eine Zusammenarbeit mit M. Bainbridge, D. Chen, S. Grushevsky und M. Möller.

  • 12.06.2017 - Anja Randecker (University of Toronto)
    Title: The Veech group of the Chamanara surface
    Abstract: Finite translation surfaces can be obtained by gluing the edges of finitely many polygons via translations. The group \(\mathrm{SL}(2,\mathbb{R})\) acts on all these translation surfaces by affine transformations and the stabilizer of a translation surface under this action is called Veech group. In the last years, interest arose also in infinite translation surfaces, i.e. those glued from infinitely many polygons. One of the first results was proven by Chamanara who studied the Veech group of an infinite translation surface that is related to the dynamical system of the baker's map. In this talk, I will introduce the example of Chamanara and explain some of the tools that are used to determine the Veech group. The talk is based on joint notes with Frank Herrlich that are available at https://arxiv.org/abs/1612.06877.

  • 29.05.2017 - Ariyan Javanpeykar (Johannes Gutenberg-Universität Mainz)
    Title: Finiteness results for Fano varieties with good reduction
    Abstract: Shafarevich proved that the set of isomorphism classes of elliptic curves over Q with good reduction outside a fixed set of prime numbers is finite. Analogous finiteness statements were proven by Faltings in 1983 for abelian varieties of fixed dimension, and curves of fixed genus >1. Furthermore, Scholl subsequently proved similar finiteness results in 1985 for two-dimensional Fano varieties. We show that Scholl's result can be extended to certain classes of Fano threefolds, and present explicit examples to show that similar finiteness results do not hold for all Fano threefolds. This is joint work with Daniel Loughran.

  • 15.05.2017 - Christoph Karg (KIT/Universität Heidelberg)
    Title: Fenchel-Nielsen-Koordinaten auf dem Teichmüllerraum von Flächen unendlichen Typs
    Abstract: Für geschlossene Flächen von Geschlecht g ≥ 2 kann man den zugehörigen Teichmüllerraum anschaulich durch 6g-6 Parameter beschreiben, den sogenannten Fenchel-Nielsen-Koordinaten. Ein wichtiges Hilfmittel hierzu ist die Zerlegung der Flächen in endlich viele topologische Hosen. Wir interessieren uns in diesem Vortrag für Flächen, deren Fundamentalgruppe unendlich viele Erzeuger hat. In diesem Fall bestehen die Flächen aus unendlich vielen Hosen und müssen dementsprechend durch unendlich viele Parameter beschrieben werden. Wir werden in diesem Vortrag notwendige Bedingungen formulieren, sodass der Teichmüllerraum gutartige Eigenschaften aufweist. Dieser Vortrag basiert auf einem Artikel von Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su and Zongliang Sun.

  • 30.01.2017 - Alejandro Soto (Goethe-Universität Frankfurt)
    Title: Toric geometry over valuation rings
    Abstract: Toric geometry has been a very important subject of algebraic geometry, mainly due to its combinatorial description. This has allowed many concrete constructions and examples, which were absent formerly. Dealing with families of toric varieties leads to the study of toric schemes over discrete valuation rings (DVR). In the 70's Mumford showed that these also admit a combinatorial description which generalize in a canonical way the theory over a field. Lately in the setting of tropical geometry, there has been a necessity of dealing with toric schemes over arbitrary rank one valuation rings. Those were introduced by Gubler in 2011. In this talk I will show that many properties of toric schemes over DVR extend to the setting of arbitrary rank one valuation rings. In particular, the existence of an equivariant completion.

  • 23.01.2017 - Robert Kucharczyk (ETH Zürich)
    Title: Klassifizierende Räume für absolute Galoisgruppen
    Abstract: Die absolute Galoisgruppe des Körpers Q der rationalen Zahlen ist ein wichtiges Objekt der modernen Mathematik. Während sich ihr maximaler abelscher Quotient über die Operation auf Einheitswurzeln einfach beschreiben lässt, gibt die Kommutatoruntergruppe, die gleich der absoluten Galoisgruppe der maximalen zyklotomischen Erweiterung von Q ist, noch einige Rätsel auf. In diesem Vortrag werde ich die Konstruktion eines topologischen Raumes präsentieren, dessen pro-endliche Fundamentalgruppe kanonisch zu letztgenannter Gruppe isomorph ist; ferner kann der Raum als klassifizierender Raum für diese Gruppe in einem geeigneten pro-endlichen Formalismus aufgefasst werden. Die klassische Fundamentalgruppe des Raumes (über Homotopieklassen von Schleifen definiert) ist dann eine bemerkenswerte Untergruppe der absoluten Galoisgruppe. Dieser Vortrag basiert auf einer gemeinsamen Arbeit mit Peter Scholze (Bonn).

  • 28.11.2016 - David Torres-Teigell (Universität des Saarlandes)
    Title: Cutting out Teichmüller curves with modular forms
    Abstract: Teichmüller curves are totally geodesic curves C inside the moduli space of Riemann surfaces. By results of Möller, the Jacobian of a point in C always contains a subvariety that admits real multiplication. In particular, there exists certain "Prym-Torelli" map that allows us to see the curve C inside a Hilbert modular variety parametrising abelian varieties with real multiplication. In this talk we will introduce the Gothic Teichmüller curves, discovered earlier this year by McMullen-Mukamel-Wright, and describe their Prym-Torelli images inside a Hilbert modular surface. Our main objective is to cut this image out as the vanishing locus of some Hilbert modular form and use this description to calculate their Euler characteristics.

  • 21.11.2016 - Daniele Alessandrini (Universität Heidelberg)
    Title: The PSL(2,C) geometry of the Lagrangian Grassmannian
    Abstract: Quasi-Fuchsian representations of surface groups in PSL(2,C) are very important in Teichmüller theory. Their limit set in CP^1 is a circle, and the complement is a cocompact domain of discontinuity whose quotient is the union of two copies of the surface.
    We want to understand how these properties generalize to higher rank lie groups. Quasi-Hitchin representations in Sp(4,C) are considered the analog of Quasi-Fuchsian representations. I will describe the action of these representations on the Lagrangian Grassmannian of C^4, where Guichard and Wienhard proved they have a cocompact domain of discontinuity. The quotient of this domain by the action of the representation is a 6-manifold, and I will describe its topology. This is joint work with Sara Maloni and Anna Wienhard.

 


 

AG Schulze-Pillot
AG Weitze-Schmithüsen
Fachrichtung Mathematik UdS