Lehre im Sommersemester 2022

Lehre im Sommersemester 2021

Lehre im Wintersemester 2020/21

Lehre im Sommersemester 2020

Lehre im Wintersemester 2019/20

Lehre im Sommersemester 2019

Lehre im Wintersemester 2018/19

Lehre im Sommersemester 2018

Lehre im Wintersemester 2017/18

Lehre im Sommersemester 2017

Lehre im Wintersemester 2016/17

Lehre im Sommersemester 2016

Lehre im Wintersemester 2015/16

Lehre im Sommersemester 2015

Lehre im Wintersemester 2014/15

Lehre im Sommersemester 2014

Wintersemester 2023/24

Mi, 08. Nov. 2023, Frankfurter Seminar – Kolloquium des Instituts für Mathematik

• Annette Huber-Klawitter (Universität Freiburg): Linear Relations of 1-Periods
  1-Periods are complex numbers obtained by integrating an algebraic $1$-form defined over $\mathbf{Q}$ (e.g. $dx/x$) over a chain with algebraic end points. The set contains many interesting numbers (e.g., the values of $\log$ in algebraic numbers). Their transcendence and the relations between them are a classical question of transcendence theory. We now have complete picture, explaining the relations qualitatively in terms of obvious relations and also quantitatively, by which we mean dimension formulas. In the talk we are going to explain some of these general results and then discuss the application to the values of the hypergeometric function–recovering results of Wolfart. (joint work with G. Wüstholz)

Mi, 06. Dez. 2023, Oberseminar Algebra und Geometrie

• Helge Ruddat (University of Stavanger): Global Smoothings of Toroidal Crossing Varieties
  As a natural generalization of normal crossing singularities, I am going to define toroidal crossing singularities and toroidal crossing varieties and explain how to produce them in large quantities by subdividing lattice polytopes. I will then explain the statement of a global smoothing theorem proved jointly with Felten and Filip. The theorem follows the tradition of well-known theorems by Friedman, Kawamata-Namikawa and Gross-Siebert. In order to apply a variant of the theorem to construct (conjecturally all) projective Fano manifolds with non-empty anticanonical divisor, Corti and Petracci discovered the necessity to allow for particular singular log structures that are known by the inspiring name “admissible"'. I will explain the beautiful classical geometric curve-in-surface geometry that underlies this notion and hint at why we believe that we can feed these singular log structures into the smoothing theorem in order to produce all 98 Fano threefolds with very ample anticanonical class by a single method.

Sommersemester 2023

Wintersemester 2022/23

Sommersemester 2022

• Patrick Kennedy-Hunt (University of Cambridge): The Logarithmic Hilbert Scheme and its Tropicalisation
  Let X be a scheme equipped with a SNC divisor D. I will sketch the construction of a proper moduli space of subschemes Z in expansions of X such that Z satisfies an appropriate transversality condition to D. The central insight is a tropical understanding for when Z is flat over the Artin fan of X, and how this behaves in families. I will present the example of the logarithmic linear system, a toric modification of projective space. The associated fan is closely related to the geometry of tropical curves and secondary polytopes. Joint work with Dhruv Ranganathan.

Mi, 18. Mai 2022, Oberseminar Algebra und Geometrie

• Andreas Mihatsch (Universität Bonn): δ-Forms and Intersection Numbers
  δ-Forms were invented by Gubler–Künnemann, they are certain hybrid objects on non-archimedean spaces that combine the smooth differential forms of Chambert-Loir–Ducros with tropical intersection theory. In this talk, I will first present a purely local definition of δ-forms that is based on the combinatorics of skeletons in Berkovich spaces. I will then explain an application to intersection numbers on integral models: The lengths of artinian complete intersections on the model agree with certain integrals of δ-forms in the generic fiber.

• Daniel Greb (Univ. Duisburg-Essen): Invariant rings of reductive representations and singularities of the Minimal Model Program
  I will discuss my recent work with Braun, Langlois, and Moraga showing that invariant rings of finite-dimensional representations of (linearly) reductive groups over an algebraically closed field of characteristic zero have Kawamata log-terminal (klt) singularities. I will spend most of the time on explaining what klt singularities are, why the klt condition is a very natural and geometric condition, and why smaller classes of singularities are not sufficient in order to understand arbitrary reductive quotient singularities. Then, I will discuss some applications of the main result, e.g. to varieties important in geometric representation theory and to certain moduli spaces. If time permits, I will discuss some ideas of the proof.

Mi, 15. Juni 2022, Oberseminar Algebra und Geometrie

• Andreas Bäuerle (Univ. Tübingen): Gorenstein 3-Fanos von Picardzahl 1 mit einer 2-Torus Wirkung; Gorenstein Fano 3-folds of Picard number 1 with a 2-torus action
  Wir klassifizieren die dreidimensionalen, nichttorischen, Q-faktoriellen, log terminalen, Gorenstein Fano Varietäten von Picardzahl eins, die eine effektive Wirkung eines zweidimensionalen Torus besitzen.
  We classify the non-toric, Q-factorial, log terminal, Gorenstein Fano threefolds of Picard number one that admit an effective action of a two-dimensional torus.

• Petra Schwer (Otto-von-Guericke-Universität Magdeburg): What is a building and why should one care?
  Groups like GL_n, SL_n or SP_n  play an important role in many areas of mathematics. It has been known for a long time that some of their properties (when studied over the reals or complex numbers) are best understood via the associated symmetric spaces. Jaques Tits later introduced buildings as a tool to study the respective groups over other field and developed, together with Bruhat, a theory that also captures reductive groups evaluated over non-archimedian local fields with discrete valuations, like the p-adic numbers.
  In this talk I will explain how some of the subgroup structures of such a reductive group over a non-Archimedian local field can be explained via Coxeter combinatorics and the geometry of an (affine) Bruhat-Tits building, its apartments and retractions. The building for example simultaneously encodes the (affine) flag variety and (affine) Grassmannian associated to the group. But it also permits to explain more complicated structures such as representation theoretic data or other associated varieties in purely combinatorial terms.

Wintersemester 2021/22

Sommersemester 2021

Mi, 21. April 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Noémie Combe (MPI Leipzig): The realm of Frobenius manifolds
  This talk will focus on different facets of so-called Frobenius manifolds, a mathematical object that arose in the process of axiomatisation of Topological Field Theory (TFT). Until 2019 there were three main classes: 1. Quantum cohomology, in relation to Gromov--Witten invariants. 2. Saito manifold (unfolding spaces of singularities), in relation to Landau--Ginzburg models. 3. The moduli space of solutions to Maurer--Cartan equations appearing in the Barannikov--Kontsevich theory, related to Gerstenhaber--Batalanin--Vilkoviskiy algebras. In a result 2020, in a joint work with Yu. Manin, we have proved that there exists a very unexpected bridge between algebraic geometry (involving moduli spaces of curves, Gromov--Witten invariants, unfolding spaces of isolated singularities, known as the Saito manifold) and statistical manifolds, central objects for machine learning, information geometry and decision theory. In this talk we will discuss different aspects of Frobenius manifolds in particular the Saito manifold (unfoldings of isolated singularities) and consider relations to Grothendieck—Teichmuller theory.

Mi, 05. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Marcin Lara (IMPAN/Warsaw): Fundamental groups of rigid spaces, geometric arcs and specialization morphism
  We introduce a new category of coverings in rigid geometry, called geometric coverings, and show it is classified by a certain topological fundamental group. Geometric coverings generalize the class of étale coverings, introduced by de Jong, and its various natural modifications, and have certain desirable properties that were missing from those older notions: they are étale local and closed under taking infinite disjoint unions. The definition is based on the property of unique lifting of “geometric arcs". On the way, we answer some questions from the foundational paper of de Jong. In a separate project, for a formal scheme over a complete rank one valuation ring, we prove existence of a specialization morphism from the de Jong fundamental group of the rigid-analytic generic fiber to the pro-étale fundamental group of the special fiber.
  This is joint work with Piotr Achinger and Alex Youcis.

Mi, 19. Mai 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Andrea Petracci (FU Berlin): On deformations of toric varieties and applications to moduli of Fano varieties
  Toric varieties are algebraic varieties whose geometry is encoded in certain discrete combinatorial objects, such as cones and polyhedra. After recalling the deformation theory of toric affine varieties (due to Klaus Altmann), in this talk I will show some applications to the local study of the recently constructed moduli space of K-polystable Fano varieties (i.e. Fano varieties admitting a Kähler-Einstein metric). In particular, I will explain that this moduli space is singular - this is joint work with Anne-Sophie Kaloghiros.

Mi, 02. Juni 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Michael Groechenig (Toronto): Hypertoric Hitchin systems and p-adic integration
  The first half of this talk will be devoted to a panoramic overview of p-adic integration for Hitchin systems. In particular, I will explain the main ideas that were used in joint work with Dimitri Wyss and Paul Ziegler to resolve the Hausel--Thaddeus conjecture.
  In the second half we will turn to concrete examples. A construction due to Hausel and Proudfoot associates to a graph a complex-analytic integrable system. In joint work with Michael McBreen we introduce a formal-algebraic analogue of their spaces and compute the p-adic volumes of the fibres in graph-theoretic terms.

Mi, 23. Juni 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Mattia Talpo (Pisa): Derived categories of parabolic sheaves with rational weights
  I will talk about some of my past work (joint with N. Sibilla and S. Scherotzke) describing semi-orthogonal decompositions for the derived category of parabolic sheaves with rational weights on certain log schemes. I will start by recalling the notion(s) of parabolic bundles and sheaves on a pair, their relationship with bundles and coherent sheaves on (finite and infinite) root stacks, and then I will explain how to apply known results about semi-orthogonal decompositions on root stacks.

Mi, 30. Juni 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Victoria Hoskins (Radboud University Nijmegen): The geometry and cohomology of moduli spaces of vector bundles and Higgs bundles
  Moduli spaces are geometric solutions to classification problems in algebraic geometry. One of the most classical examples is moduli of vector bundles and Higgs bundles on a Riemann surface, which has very rich geometry and has connections with representation theory and mathematical physics. I will describe the geometry of these moduli spaces and survey some results on their various cohomological invariants. Finally I will present some joint work with Simon Pepin Lehalleur on the motives of these moduli spaces, which unify different cohomological invariants and also encode Chow groups describing subvarieties of these moduli spaces.

Mi, 14. Juli 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Fabio Bernasconi (University of Utah): Log liftability for del Pezzo surfaces and applications to singularities in positive characteristic
  In a recent work with Arvidsson and Lacini, we proved a liftability result to characteristic zero for singular del Pezzo surfaces over perfect fields of characteristic $p>5$. I will explain exactly what notion of liftability we use for singular surfaces (and pairs), and I will sketch some parts of the proof. From this, I will explain the importance of this result for positive characteristic birational geometry: we prove a Kawamata-Viehweg vanishing theorem for such surfaces that I successively used, in a work with Kollár, to deduce properties of singularities of klt threefolds and new liftability results for threefolds Mori fibre spaces in positive characteristic.

Mi, 04. August 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Xuesen Na (University of Maryland): Limiting configuration of SU(1,2) Higgs bundles
  The moduli space of Higgs bundles, or the space of solutions of Hitchin equations has been a focus of intensive studies in algebraic geometry, symplectic geometry and topology. Recently the asymptotics near the ends of the moduli space has been investigated by studying behavior of solutions for (E,t\Phi) as $t\to\infty$ by Mazzeo et al (2014), Mochizuki (2016) and Fredrickson (2018) for some cases of SL(n,C) Higgs bundles.
  In this talk I will present a new result of the limiting behavior of solutions SU(1,2) Hitchin equation, as a first step of extending the study to the G-Higgs bundle with G a real rank-one Lie group. The proof relies on construction of approximate solutions by gluing local models on disks to decoupled solutions which converge to limiting configuration after appropriate scaling. A by-product of the study is an explicit description of spectral data of generic SU(1,2) Higgs bundle by Hecke transformations.

Wintersemester 2020/21

Sommersemester 2020

Mi, 13. Mai 2020, Oberseminar Algebra und Geometrie (on Zoom)

Sommersemester 2019

Mi, 03. April 2019, Oberseminar Algebra und Geometrie

• Gabino González-Diaz (Universidad Autónoma de Madrid): Galois action on Riemann surfaces and their associated solenoids
  Let S be a compact Riemann surface uniformised by a Fuchsian group Γ. For any element σ ∈ G := Gal(C/Q) the natural Galois action of G on the coefficients of the algebraic equation corresponding to S yields a new Riemann surface S^σ with uniformising group Γ^σ.
  Little seems to be known about the relationship between Γ and Γ^σ as subgroups of PSL_2(R). In this talk I will attempt to show that some invariants of this Galois action can be found by studying the action of G on the solenoid associated to S. I will apply these results to present explicit non-Galois-conjugate (arithmetic) Fuchsian groups.

Mi, 22. Mai 2019, Oberseminar Algebra und Geometrie

• Alexander Betts (MPIM Bonn): Non-abelian Kummer maps for curves
  The Q_ℓ -pro-unipotent non-abelian Kummer map associated to a curve X is a certain function controlling the existence of Galois-invariant paths between points of X, and plays an important role in the non-abelian Chabauty method for finding rational points. In this talk, I will report on a project with Netan Dogra, in which we compute these maps explicitly when the base field is p-adic - by relating the problem to complex-analytic curve families over the punctured disc, we give a description of this map in terms of harmonic analysis on the reduction graph of X. As a result, we are able to prove injectivity results for these maps.

Do, 06. Juni 2019, Oberseminar Algebra und Geometrie

• Lucia Mocz (Universität Bonn): A New Northcott Property for Faltings Height
  The Faltings height is a useful invariant in arithmetic geometry. In particular, it plays a key role in Faltings' proof of the Tate conjecture for abelian varieties, in which it is crucially used that the Faltings height satisfies a specific Northcott property. Here we demonstrate a different Northcott property: namely, assuming the Colmez Conjecture and the Artin Conjecture, we show that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension which have bounded Faltings height, along with an unconditional statement within isogeny classes.

• Dr. Hanieh Keneshlou (Max-Planck-Institut Leipzig): Moduli of 6-gonal genus 11 curves with several pencils
  Considering a smooth d-gonal curve C of genus g, one may naturally ask about the existing possible number of pencils of degree d on C. Motivated by some questions of Michael Kemeny, in this talk we will focus on this question for hexagonal curve of genus 11. We describe a unirational irreducible component of the schemes of 6-gonal genus 11 curves possessing certain numbers of pencils.

Mi, 12. Juni 2019, Oberseminar Algebra und Geometrie

• Dr. Samouil Molcho (Hebrew University of Jerusalem): The Logarithmic Picard Group and its Tropicalization
  The Jacobian of a family C ---> S of smooth curves is an Abelian variety, that is, a proper smooth group scheme over S. On the other hand, when the family of smooth curves is allowed to degenerate to a nodal curve, there is in general no way to extend the Jacobian to a proper, smooth group scheme over the limiting nodal curve as well. However, in the setting of logarithmic geometry, such a degeneration of the Jacobian does exist: it is the logarithmic Picard scheme. In this talk I will define the logarithmic Picard group, discuss its main properties, and describe its structure. I will focus in particular on one of the essential pieces of this structure, which is the tropicalization of the logarithmic Picard scheme. This tropicalization, the tropical Picard scheme, can be understood as a moduli space on an associated tropical curve, and is closely related to the tropical Jacobian of a tropical curve. I will describe the tropical Picard scheme in detail and provide the necessary background in log geometry. 

Mo, 24. Juni 2019, Oberseminar Algebra und Geometrie

• Prof. Dr. Katherine E. Stange (University of Colorado, Boulder): An illustration in number theory: abelian sandpiles and Schmidt arrangements
  This talk ranges over some number theoretical topics I've come across recently which have interesting visual aspects.  I'll discuss a question in abelian sandpiles and relate it to Schmidt arrangements, which are pictures, from the world of hyperbolic geometry, that illustrate properties of certain rings of integers.  The talk is largely colloquium-style and accessible to a general mathematical audience.

Mi, 26. Juni 2019, Oberseminar Algebra und Geometrie

• Prof. Dr. Jonathan Wise (University of Colorado, Boulder): Complete moduli of line bundles and divisors
  The Jacobian of a smooth curve is an abelian variety, but if the curve is allowed to degenerate to acquire nodes, it may not be possible to degenerate the Jacobian simultaneously without sacrificing its smoothness, properness, or group structure. Similarly, the Abel-Jacobi morphism from a curve to its Jacobian fails to extend to extend to nodal curves. I will discuss how logarithmic geometry can correct these problems.

Do, 18. Juli 2019, Oberseminar Algebra und Geometrie

• Dmitry Zakharov (Central Michigan University): Covers of algebraic curves and graphs with a finite group action
  A classical subject of algebraic geometry is the study of curves with a group action. For example, a hyperelliptic curve is a curve with a Z/2Z-action such that the quotient is the projective line. Covers of a given curve X with structure group G are classified by monodromy representations from the fundamental group of X to G.
  Tropical geometry studies degenerations of algebraic objects by means of certain polyhedral counterparts, which record the combinatorial structure of the degenerations. The tropical counterpart of an algebraic curve is a metric graph with vertex weights and is called a tropical curve. In my talk, I will describe a theory of G-covers of tropical curves, where G is a finite group. A G-cover of a tropical curve is a map of graphs together with an action of G on the source graph commuting with the cover. The principal complication is that the action is not required to be free, so the cover may have non-trivial stabilizers that vary along the graph.
  I will focus mostly on the case when the structure group is abelian, which corresponds, in the number-theoretic setting, to class field theory. I will show that, when G is abelian, G-covers of a tropical curve T with given stabilizers are classified by a cohomology group that generalizes the first simplicial cohomology group of T with coefficients in G. I will also describe the relationship between cyclic covers of the curve T and the torsion in its Jacobian.
  Joint work with Yoav Len and Martin Ulirsch.

Wintersemester 2018/19

Sommersemester 2018

Wintersemester 2017/18

Do., 16. November 2017, Oberseminar Algebra und Geometrie

• Yuichiro Hoshi (RIMS, Kyoto University): On torsion points on a curve with good reduction over an absolutely unramified base
  Robert Coleman made a conjecture concerning ramification of torsion points on curves over absolutely unramified complete discrete valuation rings. In this talk, after reviewing the conjecture, we discuss some results related to it.
  

Do, 30. November 2017, Oberseminar Algebra und Geometrie

• Katharina Hübner (Universität Heidelberg): The tame site
  Let X be a scheme over a field of characteristic p> 0. The étale cohomology groups of X with p-torsion coefficients are not very well behaved. For instance, the group H¹(A¹_k,et,Z/pZ) vanishes if k has characteristic different from p, but it is not even finitely generated if the characteristic of k is p. We propose a definition of a "tame site" which does not have these problems and whose fundamental group is the tame fundamental group which is already known.
  

Do, 01. Februar 2018, Oberseminar Algebra und Geometrie

• Veronika Wanner (Universität Regensburg): Subharmonic functions on non-archimedean curve
  A. Thuillier developed a theory of subharmonic functions on smooth non-archimedean curves.
  There is also a notion of subharmonic functions by A. Chambert-Loir and A. Ducros using their bigraded real-valued differential forms on Berkovich analytic spaces. I will give an introduction to both theories and I will explain why a continuous function on the analytification of a smooth algebraic curve over a non-archimedean field is subharmonic in the sense of A. Thuillier if and only if it is subharmonic in the sense of A. Chambert-Loir and A. Ducros.

Mi, 07. Februar 2018, Oberseminar Algebra und Geometrie

• Magnus Carlson (KTH Stockholm), Cohomology rings of number fields and applications
  In this talk I will explain how to, given a number field K and O_K its ring of integers, compute the étale cohomology ring H^*(Spec O_K,Z/nZ). I will explicitly describe this ring structure in terms of arithmetical data coming from K. To show that this structure is of arithmetic interest, I will discuss two situations where this ring structure gives non-trivial applications. The first application gives an infinite family of totally imaginary quadratic number fields {K_i} such that Aut(PSL(2,q²)), for q an odd prime power, cannot be realized as an unramified Galois group over K_i, but its maximal solvable quotient can. For the second application, I will discuss recent work of Maire, where he studies the cohomological dimension of quotients of the maximal unramified pro-2 extension of a totally imaginary number field K. This talk is based on joint work with Tomer Schlank.

Do, 08. Februar 2018, Oberseminar Algebra und Geometrie

• Marco Maculan (Paris): Stein spaces in non-archimedean geometry
  A celebrated theorem of Serre states that an algebraic variety X is affine if and only if, for every coherent sheaf F on X and every positive integer q, the cohomology group H^q(X,F) vanishes.
  In complex geometry, where Cartan proved this result first, the equivalent of affine varieties are called Stein spaces.
  In this talk I will present some results, old and new, concerning the situation over a non-archimedean field. As local compactness will have a key role, Berkovich spaces will come into play. Joint work with J. Poineau.

• Madeline Brandt (University of California, Berkeley), Tropical Superelliptic Curves
  Given a smooth curve defined over a valued field, it is a difficult problem to compute the Berkovich skeleton of the curve. In theory, one can find a semistable model for the curve and then find the dual graph of the special fiber, and this will give the skeleton. In practice, these procedures are not algorithmic and finding the model can become difficult. It is known how to find the Berkovich skeleton of genus one and genus two curves; more recently, the hyperelliptic case has also been solved. In this talk, we present the solution for superelliptic curves y^n=f(x). This involves studying the covering from the curve to P^1, and recovering data about the Berkovich skeleton from the tropicalization of P^1 together with the marked ramification points. Throughout the talk we will study many examples in order to get a feel for the difficulties of this problem and how the procedure is carried out.

Sommersemester 2017

Wintersemester 2016/17

19. Oktober 2016, Oberseminar

• Ariyan Javanpeykar (Universität Mainz): The Lang-Vojta conjecture and integral points on the moduli of smooth hypersurfaces
  Siegel proved the finiteness of the number of solutions to the unit equation in a number ring, i.e., for a number field K with ring of integers O, the equation x+y = 1 has only finitely many solutions in O*. That is, reformulated in more algebro-geometric terms, the hyperbolic curve P^1-{0,1,infty} has only finitely many "integral points". In 1983, Faltings proved the Mordell conjecture generalizing Siegel's theorem: a hyperbolic complex algebraic curve has only finitely many integral points. Inspired by Faltings's and Siegel's finiteness results, Lang and Vojta formulated a general finiteness conjecture for "integral points" on complex algebraic varieties: a hyperbolic complex algebraic variety has only finitely many "integral points".
  In this talk we will explain the Lang-Vojta conjecture and we will explain some of its consequences for the arithmetic of homogeneous polynomials over number fields. This is joint work with Daniel Loughran.

09. November 2016, Oberseminar

• Daniel Greb (Universität Duisburg-Essen): The Miyaoka-Yau Inequality and uniformisation of canonical models
  After an introduction to the basic goals and notions of higher-dimensional birational geometry and the minimal model program, I will concentrate on the case of varieties of general type. By the seminal work of Birkar-Cascini-Hacon-McKernan (~2006) the minimal model program is known to work for these, so that every smooth projective variety of general type admits a minimal as well as a canonical model. Motivated by Riemann's Uniformisation Theorem in one complex variable, I will then describe approaches to higher-dimensional uniformisation theorems. Time permitting, at the end of my talk I will explain the proof of a recent result (with Kebekus, Peternell, and Taji) that establishes the Miyaoka-Yau Inequality (MYI) for minimal varieties of general type and characterises those varieties for which the MYI becomes an equality as quotients of the unit ball by a cocompact discrete subgroup of PSU(1, n).

16. November 2016, Oberseminar

• Arne Smeets (University of Leuven): Pseudo-split varieties and arithmetic surjectivity
  Let X → Y be a dominant morphism of smooth, proper, geometrically integral varieties over a number field k, with geometrically integral generic fibre. One can ask the following question: for which places v of k is the induced map X(k_v) → Y(k_v) surjective? I will address this question using techniques coming from toroidal/logarithmic geometry and analytic number theory. In particular, I will explain why this set of places is a so-called frobenian set, and I will present a necessary and sufficient geometric criterion for X(k_v) → Y(k_v) to be surjective for all but finitely many places v of k; this can be seen as an "optimal" version of the celebrated Ax-Kochen theorem, and generalizes a result of Denef previously conjectured by Colliot-Thélène. (Joint work with D. Loughran and A. Skorobogatov.)

11. Januar 2017, Oberseminar

• Stefan Wewers (Universität Ulm): Semistabile Reduktion und der Führerexponent einer Picard-Kurve
  Der Führer einer glatten projektiven Kurve Y über einem Zahlkörper ist eine wichtige arithmetische Invariante. Er ist definiert als ein Produkt von lokalen Beiträgen, die mit den Primstellen zu tun haben, an denen Y schlechte Reduktion hat. Zur Bestimmung dieser lokalen Beiträge ist es i.A. notwendig, die semistabile Reduktion von Y an den schlechten Stellen zu kennen. Ausgangspunkt unseres Vortrags sind neue Methoden zur Berechnung der semistabilen Reduktion von Kurven, die auch eine explizite Berechnung des Führers ermöglichen. Im meinem Vortrag werde ich für eine spezielle Klasse von Kurven vom Geschlecht 3 (den Picard-Kurven) zeigen, wie man mit diesen Methoden untere und obere Schranken für den Exponenten einer Primzahl im Führer beweisen kann. Am Ende werde ich auf das Problem eingehen, alle Kurven einer Familie zu bestimmen, deren Führer unterhalb einer gegebenen Schranke liegt.

Sommersemester 2016

1. Juni 2016, Oberseminar / Abschlussseminar

• Dr. Ayberk Zeytin (Galatasaray University Istanbul): Arithmetic of quadratic extensions
  In this talk we will try to outline a combinatorial study of class groups of quadratic extensions of certain quadratic number fields. The easiest case reduces to the study of 200 year old class number problems of Gauss through which the main ideas will be explained. Should time permits one application of the above theory will be explained which relates Gauss' problems to Lang conjectures.

13. Juli 2016, Oberseminar / Abschlussseminar

• Hironori Shiga (Chiba University Japan): A visualization of Shimura's complex multiplication theorem via hypergeometric modular functions

• Prof. Lawrence Ein (University of Illinois at Chicago): Title & Abstract: t.b.a.

Wintersemester 2015/16

23., 24., 25.02.2016 Vortragsreihe im Oberseminar

• Prof. Dr. Florian Pop: Eine kurze Einführung in die birationale anabelsche Geometrie: Galois Theorie und Bewertungen.

19.11.2015, Vortrag im Oberseminar

• Dr. Martin Ulirsch (Universität Bonn): Logarithmic structures, Artin fans, Kato fans, and tropicalization.

Sommersemester 2015

16.7.2015, Vortrag im Oberseminar

• Carlos Rito (Porto): Two surfaces with canonical map of degrees 16 and 24
  It is known since Beauville (1979) that if the canonical image $φ(S)$ of an algebraic surface of general type S is a surface, then the degree d of the canonical map φ satisfies d≤ 36. Lower bounds hold for irregular surfaces, in particular q=2 => d≤18. So far all known examples satisfy q>0, d≤ 8 or q=0, d≤16. In this talk I will describe the construction of an example with q=2, d=16 and of an example with q=0, d=24.

17.06.2015, Vortrag im Oberseminar

• Henrik Bachmann (Universität Hamburg): Multiple zeta values and regularised multiple Eisenstein series
  Multiple zeta values (MZV) can be seen as a multiple version of the Riemann zeta values appearing in different areas of mathematics and theoretical physics. The product of these real numbers can be expressed in two different ways, the so called stuffle and shuffle product, which yields a large family of linear relations. MZV have several connections to modular forms for the full modular group where one of them is given by multiple Eisenstein series (MES) which can be seen as a multiple version of the classical Eisenstein series. MES are holomorphic functions in the upper half-plane with a Fourier expansion whose constant term is given by the corresponding MZV. By definition the multiple Eisenstein series functions also fulfill the stuffle product. In the talk I want to explain a recent result on two different regularisation of these series. We discuss the dimorphic structure of these regularised MES which is very close but, because of the existence of cusp forms for the modular group, different to that of MZV.

10.06.2015, Vortrag im Oberseminar

• Robert Kucharczyk (Universität Bonn): On congruence subgroups of triangle groups
  Only finitely many among the Fuchsian triangle groups are arithmetic. Still, also the non-arithmetic ones share many important structures and properties with arithmetic groups. In this talk one example of this principle will be presented: there is a natural way to define congruence subgroups of triangle groups. The quotients of the upper half plane by such congruence subgroups are called triangular modular curves. Using rigid modular embeddings into Shimura varieties it is possible to construct canonical models of all triangular modular curves over certain number fields, thereby significantly generalising canonical models for classical modular curves. This is joint work with John Voight (Dartmouth College).

Wintersemester 2014/15

13.11.2014, Vortrag im Oberseminar

• Alexander Ivanov (TU München): Anabelian properties of arithmetic curves
  The Isom-conjecture of Grothendieck concerns the fact, that the isomorphism class of certain types of varieties is uniquely determined by their étale fundamental groups. While in the case of affine curves over a finite field the Isom-conjecture was solved by Tamagawa, up to now almost nothing is known for arithmetic curves. This is mainly because we lack an analogue of Lefschetz's fixed point theorem but also because the fundamental group of an arithmetic curve is still poorly understood. We discuss a new method to overcome at least the first difficulty, i.e., how to reconstruct the decomposition subgroups of points inside the fundamental group (assuming some still unknown properties of it), using some additional data and the Tsfasman-Vladut theorem, but avoiding any cohomology theory.

02.10.2014, Vortrag im Oberseminar

• Bernd Sturmfels (UC Berkeley): Moduli of Tropical Plane Curves
  We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus g, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with g interior lattice points. It has dimension 2g+1 unless g lower or equal to 3 or g=7. We compute these spaces explicitly for small g.

Sommersemester 2014

11.07.2014, Vorträge im Oberseminar 

• Kirsten Wickelgren (Georgia Tech): Motivic desuspension
  Certain problems such as classifying manifolds up to cobordism are stable in the sense that they are solved in categories where it is possible to desuspend. Other problems, such as classifying algebraic vector bundles on schemes, require analogous unstable information. The EHP sequence in algebraic topology is a tool for turning stable information into unstable information. We will discuss the situation from algebraic topology, and pesent an EHP sequence in A^1 homotopy theory of schemes. The part of the talk which is new is joint work with Ben Williams.
• Joseph Rabinoff (Georgia Tech): Jacobians of curves and Jacobians of graphs
  The Jacobian of a Riemann surface is a complex torus equipped with some extra structure, a canonical polarization. The Jacobian of a graph with edge lengths is a real torus, also equipped with a canonical polarization. I'll introduce both kinds of Jacobians, and show how the two theories are strongly analogous. I'll also discuss the deep connections between the two objects when the ground field is non-Archimedean, and indicate some applications in number theory.
  
  

03.07.2014, Vortrag im Oberseminar

• Jochen Gärtner (Universität Heidelberg): On Riemann's existence theorem for arithmetic Galois groups

28.05.2014, Vortrag im Oberseminar

• Daniele Turchetti (Jussieu): Lifting Galois covers to characteristic zero with non-Archimedean analytic geometry

Wintersemester 2013/14

12.12.2013, Vortrag im Oberseminar

• Gareth Jones (University of Southampton): Quasiplatonic Riemann surfaces: symmetries and chirality
  A Riemann surface is quasiplatonic if it carries a regular dessin, or equivalently, is uniformised by a normal subgroup of finite index in a triangle group $\Delta(l,m,n)$. I shall prove a generalisation of a 1992 conjecture of David Singerman, namely that for each non-spherical type $(l,m,n)$ there are regular dessins which are chiral (not isomorphic to their mirror images). The proof used the action of automorphism groups on spaces of differentials and on homology. A compact Riemann surface has a symmetry (anticonformal involution) if and only if the associated algebraic curve is defined over the real field. I shall outline a corrected version of a 1974 theorem of Singerman characterising those quasiplatonic surfaces which possess symmetries.

28.11.2013, Vortrag im Oberseminar

• Dr. Myriam Finster (Karlsruher Institut für Technologie): Translationsüberlagerungen und Veechgruppen
  Eine Translationsfläche ist eine 2-dimensionale kompakte Mannigfaltigkeit X zusammen mit einem Translationsatlas auf X\S, wobei S eine endliche Menge von konischen Singularitäten ist. Die Ableitungen der affinen Selbstabbildungen von X bilden die Veechgruppe der Translationsfläche.
  Ich interessiere mich für die Frage, welche Untergruppen der Veechgruppe einer primitiven Translationsfläche sich als Veechgruppe einer Translationsüberlagerung der Fläche wiederfinden lassen. In meinem Vortrag werde ich dazu Kongruenzuntergruppen von Veechgruppen primitiver Translationsflächen definieren. Für viele primitive Basisflächen werde ich zeigen, dass Kongruenzuntergruppen, die als Stabilisatorgruppen unter der Aktion auf der absoluten Homologie mit Einträgen in Z/aZ vorkommen, immer die Veechgruppe einer geeigneten Translationsüberlagerung sind. Das ist eine Verallgemeinerung eines entsprechenden Resultats über Origamis von Gabriela Weitze-Schmithüsen.

31.10.2013, Vortrag im Oberseminar

• Ralph Morrison (Berkeley / MPI Bonn): Algorithms for Mumford curves over the p-adics

14.10.2013, Vortrag

• PD Dr. Alex Küronya (Budapest University of Technology and Economics): "q-ampleness"

Sommersemester 2013

13.06.2013, Vortrag im Oberseminar Algebra und Geometrie

• Prof. Gabino Gonzalez-Diez (UAM Madrid): The action of the absolute Galois group on dessins d'enfants
  
  

Wintersemester 2012/2013

18.01.2013, Vortrag im mathematischen Kolloquium

• Prof. Laurent Bartholdi (Universität Göttingen): Growth of groups and semigroups
  The growth function of a finitely generated group counts the number of elements of the group that can be written with at most n generators. This function depends on the choice of generating set, but only mildly. I will describe the first groups of intermediate growth for which the growth function is known; and how one can construct groups of non-uniform exponential growth. Growth can also be defined for semigroups. In that case, there exist much more constructions, though one does still not know exactly which growth functions may (asymptotically) occur as the growth of a group or a semigroup. I will show that there are semigroups with almost arbitrarily prescribed growth between n^(log n) and 2^n; and that there are groups with almost arbitrarily prescribed growth between 2^(n^0.76) and 2^n. These are joint works with Agata Smoktunowicz and Anna Erschler.

 
14.12.2012, Vortrag im mathematischen Kolloquium

• Prof. Anna Wienhard (Universität Heidelberg): Geometrie und Dynamik diskreter Untergruppen von halbeinfachen Liegruppen
  In diesem Vortrag werde ich einige geometrische und dynamische Eigenschaften von diskreten Untergruppen in halbeinfachen Liegruppen (z.B. SL(n,R)) diskutieren. Ich werde hierbei insbesondere Untergruppen betrachten, die im Zusammenhang mit höhere Teichmuellertheorie auftreten.

Sommersemester 2012

12.07.2012, Vortrag im Oberseminar Algebra und Geometrie

• Olivier Warin (Uni. Basel): Über x+y+z+w=1 und Höhen
  Zusammenfassung: Link

19.04.2012, Vortrag im Oberseminar Algebra und Geometrie

• Lars Kühne (ETH Zürich): Effective and uniform results of André-Oort type
  The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety is an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. I will discuss a rarely known approach to the AOC that goes back to Yves André himself. Before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie-Zannier approach André's proof was the only known unconditional proof of the AOC for a non-trivial Shimura variety. In my talk, I discuss two different ways to improve André's proof, enabling various effective results of André-Oort type for products of modular curves. Finally, I will discuss some of the obstructions to extending these methods to more complicated Shimura varieties.
  

Wintersemester 2011/2012

21.11.-22.11.2011.  Kolloquium zur algebraischen Geometrie

27.10.2011, Vortrag im Oberseminar Algebra und Geometrie

• Patrik Hubschmid (Uni. Heidelberg): The André-Oort Conjecture for Drinfeld modular varieties
  The André-Oort conjecture states that an irreducible subvariety of a Shimura variety containing a Zariski dense subset of special points is a special subvariety. In this talk, I consider the analogue of this conjecture for Drinfeld modular varieties in the function field case.
  I will first introduce Drinfeld modular varieties and explain the notion of special subvariety. Then I will explain how the methods of Edixhoven, Klingler and Yafaev in the classical case can be adapted to the function field case. This leads to a proof of the conjecture for special points with separable reflex field over the base field. Finally, I will provide an outlook about possible future work to tackle the case of inseparable reflex fields.

Sommersemester 2011

30.08.2011, Vortrag im Oberseminar Algebra und Geometrie

• Johannes Cuno (TU Graz, ehemals Uni Frankfurt): Nichtsphärische Dreiecke von Gruppen: Der Zauber eines Lemmas
  Die meisten Beweise, die in der zweiten Hälfte meiner Diplomarbeit zu finden sind, basieren auf einem einzigen Lemma. Ziel des Vortrags ist, den Zauber dieses Lemmas herauszuarbeiten. Worum geht es genau? Zu Beginn der Neunzigerjahre haben Steve Gersten und John Stallings Dreiecken von  Gruppen eine Krümmung zugeordnet und eine Reihe von Aussagen über Colimites nichtsphärischer Dreiecke von Gruppen bewiesen. Nach einer kurzen Einführung diskutieren wir die Frage, unter welchen Bedingungen der Colimes eines nichtsphärischen Dreiecks von Gruppen entweder eine nichtabelsche freie Untergruppe enthält oder virtuell auflösbar ist.

22.08.2011, Vortrag im Oberseminar Algebra und Geometrie

• David Torres-Teigell (UAM Madridl): Non-homeomorphic conjugate Beauville surfaces
  Abstract 

15.07.2011, Vortrag im Oberseminar Algebra und Geometrie

• Ayberk Zeytin (Ankara): Finding Q rational points on moduli spaces of pointed rational curves
  In this talk, we will first describe two lattices, say $\Lambda$ and $\Lambda'$, closely related to both moduli space pointed rational curve and moduli space of cone metrics. In fact, $\Lambda$, respectively $\Lambda'$, parametrizes non-negatively curved triangulations, resp. quadrangulations, of the sphere. We will describe an idea together with results of Wolfart to obtain $\bar{\QQ}$ rational points on $\mm_{0,8}$ and $\mm_{0,12}$.

16.06.2011, Vortrag im Oberseminar Algebra und Geometrie

• Tara Brough (Kiel): Poly-context-free groups and semilinear sets
• Eine endlich erzeugte Gruppe G heißt k-kontextfrei, wenn das Wortproblem von G der Schnitt von k kontextfrien Sprachen ist und polykontextfrei, wenn  G k-kontextfrei für ein k aus den natürlichen Zahlen ist. Die 1-kontextfreien Sprachen sind (nach einem Resultat von Muller-Schupp 1983 und Dunwoody 1985) genau die endlich erzeugten virtuell freien Gruppen .
  Es wird vermutet, dass die auflösbaren polykontextfreien Gruppen genau die endlich erzeugten virtuell abelschen Gruppen sind. Ich werde meine Fortschritte auf dem Weg zu einem Beweis dieser Vermutung präsentieren und dabei besonders den Zusammenhang zwischen kontextfreien Sprachen und semilinearen Mengen betonen.
  Es werden keine Kenntnisse über kontextfreie Sprachen oder semilineare Mengen vorausgesetzt.
  
  

27.05.2011, Vortrag im mathematischen Kolloquium

• Prof. Gareth Jones (University of Southampton):  Beauville surfaces
  A Beauville surface is a complex algebraic surface isogenous to a higher product, that is, the quotient of the product of two curves of genus at least 2 by a finite group G acting freely on the product. It has unmixed type if G preserves the two curves and their quotients by G are isomorphic to the complex projective line, ramified over three points (so the curves are defined over algebraic number fields, by Belyi'sTheorem). Such surfaces have interesting geometric properties, such as rigidity, while their construction poses some challenging group-theoretic problems. I will report on recent progress to answer two questions: which finite groups G can be used in this context, and which groups can arise as  the automorphism groups of Beauville surfaces? Some of these results are joint work with Yolanda Fuertes, Gabino Gonzalez-Diez and David Torres-Teigell, in Madrid.

19.05.2011 Vortrag im Oberseminar Algebra und Geometrie

• Alex Wright (Chicago, USA): Abelian square-tiled surfaces
  
  

28.04.2011 Vortrag im Oberseminar Algebra und Geometrie

• Cornelius Reinfeldt (McGill, Montreal, Canada):  Limesgruppen und Makanin-Razborov-Diagramme über hyperbolischen Gruppen
  Nach G. Makanin und A. Razborov kann die Lösungsmenge jedes Gleichungssystems über einer freien Gruppe in einem endlichen Baumdiagramm kodiert werden, einem Makanin-Razborov-Diagramm. Davon ausgehend hat Zlil Sela mithilfe von Limesgruppen und der "Rips Machine" die Existenz analoger MR-Diagramme für Gleichungssysteme über torsionsfreien hyperbolischen Gruppen gezeigt. In diesem Vortrag möchte ich einen Überblick liefern über die Methoden des Beweises von Zlil Sela, sowie meiner Arbeit mit Richard Weidmann, die dieses Resultat verallgemeinert und die Existenz von MR-Diagrammen für Gleichungssysteme über beliebigen hyperbolischen Gruppen (mit Torsion) beweist.

Wintersemester 2023/24

Di, 12. März 2024

• Bachelorabschlussvortrag (Raum 309)
  13:00 - 14:00: Celine Pham (Universität Frankfurt):
  Die Entwicklung der Kryptographie: Algorithmen des Quantumcomputings und der codebasierten Post-Quanten-Kryptographie.

Do, 08. Februar 2024

• Bachelorabschlussvortrag (Raum 310)
  14:00 - 15:00: Hellen Gella (Universität Frankfurt):
  Irreduzible Polynome über einem endlichen Körper und primitive Halsketten.

Mo, 22. Januar 2024

• Bachelorabschlussvortrag (Raum 310)
  16:00 - 16:45: Andreas Erter (Universität Frankfurt):
  Dirichlets theorem on primes in an arithmetic progression in the ring of polynomials over a finite field.
• Bachelorabschlussvortrag (Raum 310)
  16:45 - 17:30: Carola Prescher (Universität Frankfurt):
  The Hurwitz genus formula.

Fr, 15. Dezember 2023

• Disputation (Raum 711gr)
  10:30 - 12:00: Stefan Rettenmayr (Universität Frankfurt):
  On analogs of Cremona automorphisms for matroid fans.

Mi, 01. November 2023

• Bachelorabschlussvortrag (Raum 711gr)
  16:00 - 16:45: Luise Kaßner (Universität Frankfurt):
  Tropische Hurwitz-Zahlen.
• Bachelorabschlussvortrag (Raum 711gr)
  16:45 - 17:30: Jonas Glückmann (Universität Frankfurt):
  Hodge-Theorie von Matroiden.

Mo, 23. Oktober 2023

• Bachelorabschlussvortrag (Raum 711kl)
  14:00 - 15:00: Arne Gideon (Universität Frankfurt):
  Fourier-Eigenfunktionen mithilfe von Modulformen.

Sommersemester 2023

Mo, 11. September 2023

• Masterabschlussvortrag (Raum 711kl)
  10:30 - 11:30: Claudia Mrozik (Universität Frankfurt):
  Tropische Kombinatorik des Satzes von Bezout.

Fr, 08. September 2023

• Masterabschlussvortrag (Raum 309)
  10:30 - 11:30: Milan Bogdanovic (Universität Frankfurt):
  Euklidische Zahlkörper.

Mo, 04. September 2023

• Masterabschlussvortrag (Raum 308)
  10:00 - 11:00: Adrian Baumann (Universität Frankfurt):
  Massey-Produkte in der Galoiskohomologie.

Mo, 31. Juli 2023

• Bachelorabschlussvortrag (Raum 309)
  10:00 - 11:00: Simon Kaib (Universität Frankfurt):
  Galoiskorrespondenz für Picard-Vessiot-Erweiterungen.

Fr, 23. Juni 2023

• Masterabschlussvortrag (Raum 309)
  10:15 - 11:15: Leonie Scherer (Universität Frankfurt):
  Die etale Fundamentalgruppe.

Mi, 21. Juni 2023

• Bachelorabschlussvortrag (Raum 711kl)
  14:00 - 15:00: Arne Riester (Universität Frankfurt):
  Das kubische Reziprozitätsgesetz.

Wintersemester 2022/23

Di, 24. Januar 2023

• Disputation (Raum 110)
  16:00 - 17:00: Felix Röhrle (Universität Frankfurt):
  Tropical Covers and their Applications: from the Realizability of Tropical Pluri-Canonical Divisors to the n-gonal Construction.

Mi, 11. Januar 2023

• Bachelorabschlussvortrag (Raum 309)
  14:45 - 15:45: Tobias Fischer (Universität Frankfurt):
  Kurven über endlichen Körpern mit vielen Punkten.

Mi, 14. Dezember 2022

• Bachelorabschlussvortrag (Raum 711gr)
  16:00 - 17:00: Marie Kassner (Universität Frankfurt):
  Der Satz von Belyi. 

Mo, 17. Oktober 2022

• Disputation (Raum 711gr)
  16:00 - 18:00: Theresa Kumpitsch (Universität Frankfurt):
  Outer automorphisms of the absolute Galois group of local fields of mixed characteristic.
  

Sommersemester 2022

Di, 12. April 2022 

• Bachelorabschlussvortrag (Raum 310 + on Zoom) 
  William Bock (Universität Frankfurt): The cohomology ring of the complex Grassmannian.
  

Di, 26. April 2022

• Bachelorabschlussvortrag (Raum 310 + on Zoom)
  Thorger Geiß (Universität Frankfurt): Eine topologische Verallgemeinerung von Kummerflächen.

Di, 03. Mai 2022

• Masterabschlussvortrag (Raum 309 + on Zoom)
  Fabian Vogel (Universität Frankfurt): Intersection theory on toric varieties.
  

Do, 14. Juli 2022

• Bachelorabschlussvortrag (Raum 711gr + on zoom)
  Benjamin Steklov (Universität Frankfurt): Die Golod-Shafarevich Ungleichung.

• Bachelorabschlussvortrag (Raum 711gr + on zoom)
  Xiaoyu Xia (Universität Frankfurt): Die diophantische Dimension eines Körpers und der Satz von Tsen.

Mi, 27. Juli 2022

• Masterabschlussvortrag (bbb)
  Fatima Taoufik (Universität Frankfurt): Der Modulfunktor $\bar{\mathcal{M}}_{0,n}$.

• Disputation (Raum 711gr)
  14:00 - 16:00: Adrian Zorbach (Universität Frankfurt):
  Nilpotent Higgs bundles and profinite vector bundles in p-adic Hodge theory.
  
  

Wintersemester 2021/22

Mi, 16. Februar 2022

• Bachelorabschlussvortrag (on zoom)
  Marlene Eichholtz (Univ. Frankfurt): Stabile Bündel auf kompakten Riemannschen Flächen und der Satz von Narasimhan-Seshadri.

Mi, 23. Februar 2022

• Bachelorabschlussvortrag (Raum 308 + on zoom)
  Milan Bogdanovic (Univ. Frankfurt): Arithmetische Derivationen und abc.

Di, 15. März 2022

• Disputation (Raum 302 Hilbertraum + on BigBlueButton)
  Riccardo Zuffetti (Univ. Frankfurt): Cones of special cycles and unfolding of the Kudla-Millson lift.

December 14, 2023 - First meeting in the Winter Semester 2023/24

Adam Afandi (Universität Münster): Stationary Descendents and the Discriminant Modular Form
Abstract: By using the Gromov-Witten/Hurwitz correspondence, Okounkov and Pandharipande showed that certain generating functions of stationary descendent Gromov-Witten invariants of a smooth elliptic curve are quasimodular forms. In this talk, I will discuss the various ways one can express the discriminant modular form in terms of these generating functions. The motivation behind this calculation is to provide a new perspective on tackling a longstanding conjecture of Lehmer from the middle of the 20th century; Lehmer posited that the Ramanujan tau function (i.e. the Fourier coefficients of the discriminant modular form) never vanishes. The connection with Gromov-Witten invariants allows one to translate Lehmer's conjecture into a combinatorial problem involving characters of the symmetric group and shifted symmetric functions.

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July 7, 2023 - Second meeting in the Summer Semester 2023

Noema Nicolussi (University of Potsdam): Hybrid curves and their moduli spaces   -   cancelled

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May 5, 2023 - First meeting in the Summer Semester 2023

Léonard Pille-Schneider (ENS, Paris): The SYZ conjecture for families of hypersurfaces
Abstract: Let X -> D* be a polarized family of complex Calabi-Yau manifolds, whose complex structure degenerates in the worst possible way. The SYZ conjecture predicts that the fibers X_t, as t ->0, degenerate to a tropical object; and in particular the program of Kontsevich and Soibelman relates it to the Berkovich analytification of X, viewed as a variety over the non-archimedean field of complex Laurent series.

I will explain the ideas of this program and some recent progress in the case of hypersurfaces.

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February 3, 2023 - Second meeting in the Winter Semester 2022/23

Victoria Schleis (Universität Tübingen): Linear degenerate tropical flag matroids
Abstract: Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules. As linear degenerations of flag varieties are difficult to analyze algebraically, we describe them in a combinatorial setting and further investigate their tropical counterparts. In this talk, I will introduce matroidal, polyhedral and tropical analoga and descriptions of linear degenerate flags and their varieties obtained in joint work with Alessio Borzì. To this end, we introduce and study morphisms of valuated matroids. Using techniques from matroid theory, polyhedral geometry and linear tropical geometry, we use the correspondences between the different descriptions to gain insight on the structure of linear degeneration. Further, we analyze the structure of linear degenerate flag varieties in all three settings, and provide some cover relations on the poset of degenerations. For small examples, we relate the observations on cover relations to the flat irreducible locus studied in representation theory.

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November 25, 2022 - First meeting in the Winter Semester 2022/23

Sabrina Pauli (Universität Duisburg-Essen): Quadratically enriched tropical intersections 1
Abstract:  Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. Results from motivic homotopy theory allow to study questions in enumerative geometry over an arbitrary field k. In these two talks we present one of the first examples of how to use tropical geometry for questions in enuemrative geometry over k, namely a proof of the quadratically enriched Bézout's theorem for tropical curves.

In the first talk we explain what we mean by the "quadratic enrichment", that is we define the necessary notions of enumerative geometry over arbitrary fields valued in the Grothendieck-Witt ring of quadratic forms over k.

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July 8, 2022 - Third meeting in the Summer Semester 2022

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June 10, 2022 - Second meeting in the Summer Semester 2022

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May 13, 2022 - First meeting in the Summer Semester 2022

February 18, 2022 - Second meeting in the Winter Semester 2021/22

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January 21, 2022 - First meeting in the Winter Semester 2021/22

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June 25, 2021 - Third meeting in the Summer Semester 2021

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May 28, 2021 - Second meeting in the Summer Semester 2021

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April 30, 2021 - First meeting in the Summer Semester 2021

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March 12, 2021 - Fourth meeting in the Winter Semester 2020/21

Abstract: The tropical moduli space $\Delta_{g,n}$ is a topological space that parametrizes isomorphism classes of $n$-marked stable tropical curves of genus $g$ with total volume 1. Its reduced rational homology has a natural structure of $S_n$-representations induced by permuting markings. In this talk, we focus on $\Delta_{2,n}$ and compute the characters of these $S_n$-representations for $n$ up to 8. We use the fact that $\Delta_{2,n}$ is a symmetric $\Delta$-complex, a concept introduced by Chan, Glatius, and Payne. The computation is done in SageMath.

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February 19, 2021 - Third meeting in the Winter Semester 2020/21

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January 22, 2021 - Second meeting in the Winter Semester 2020/21

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December 4, 2020 - First meeting in Winter Semester 2020/21

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June 26, 2020 - Third meeting in Summer Semester 2020

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May 29, 2020 - Second meeting in Summer Semester 2020

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April 24, 2020 - First meeting in Summer Semester 2020

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January 24, 2020 - Second meeting in Winter Semester 2019/20

Karl Christ (Ben-Gurion University):  Severi problem and tropical geometry
Abstract: The classical Severi problem is to show that the space of reduced and irreducible plane curves of fixed geometric genus and degree is irreducible. In case of characteristic zero, this longstanding problem was settled by Harris in 1986. In the first part of my talk I will give a brief overview of the ideas involved. Then, I will describe a tropical approach to studying degenerations of plane curves, which is the main ingredient to a new proof of irreducibility obtained in collaboration with Xiang He and Ilya Tyomkin. The main feature of the construction is that it works in positive characteristic, where the other known techniques fail.

Oliver Lorscheid (IMPA Rio de Janeiro/MPI Bonn): Towards a cohomological understanding of the tropical Riemann Roch theorem
Abstract: In this talk, we outline a program of developing a cohomological understanding of the tropical Riemann Roch theorem and discuss the first established steps in detail. In particular, we highlight the role of the tropical hyperfield and explain why ordered blue schemes provide a satisfying framework for tropical scheme theory.
In the last part of the talk, we turn to the notion of matroid bundles, which we hope to be the right tool to set up sheaf cohomology for tropical schemes. This is based on a joint work with Matthew Baker.

Diane Maclagan (University of Warwick): Connectivity of tropical varieties
Abstract: The structure theorem for tropical geometry states that the tropicalization of an irreducible subvariety of the algebraic torus over an algebraically closed field is the support of a pure polyhedral complex that is connected through codimension one. This means that the hypergraph whose vertices correspond to facets of the complex, and whose hyperedges correspond to the ridges, is connected. In this talk I will discuss joint work with Josephine Yu showing that this hypergraph is in fact d-connected (when the complex has no lineality space). This can be thought of as a generalization of Balinski's theorem on the d-connectivity of the edge graph of a d-polytope. A key ingredient of the proof is a toric Bertini theorem of Fuchs, Mantova, and Zannier, plus additions of Amoroso and Sombra.

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October 31, 2019 - First meeting in Winter Semester 2019/20

Enrica Mazzon (Max-Planck-Institute Bonn): Tropical affine manifolds in mirror symmetry
Abstract: Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometrical objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties. In this talk I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert. In particular, I will focus on the construction of a so-called “tropical affine manifold" using methods of non-archimedean geometry, and the guiding example will be the case of K3 surfaces and some hyper-Kähler varieties. This is based on a joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.

Christoph Goldner (Tübingen): Tropical mirror symmetry for ExP^1
Abstract: We recall some results of tropical mirror symmetry that relate the generating series of tropical Gromov-Witten invariants of an elliptic curve E to sums of Feynman integrals. After that, we present an approach to tropical mirror symmetry in case of ExP^1. The approach is based on the floor decomposition of tropical curves which is a degeneration technique that allows us to apply the results of the elliptic curve case. The new results are joint work with Janko Böhm and Hannah Markwig.

Sam Payne (University of Texas, Austin): Local h-vectors
Abstract: tba

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July 5, 2019 - Second meeting in Summer Semester 2019

Schedule:

Madeline Brandt (University of California at Berkeley): Matroids and their Dressians
Abstract: In this talk we will explore Dressians of matroids. Dressians have many lives: they parametrize tropical linear spaces, their points induce regular matroid subdivisions of the matroid polytope, they parametrize valuations of a given matroid, and they are a tropical prevariety formed from certain Plücker equations. We show that initial matroids correspond to cells in regular matroid subdivisions of matroid polytopes, and we characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. If time permits, we will also discuss an ongoing project extending these ideas to flag matroids.

Dhruv Ranganathan (University of Cambridge): Tropical curves, stable maps, and singularities in genus one
Abstract: In the early days of tropical geometry, Speyer identified an extremely subtle combinatorial condition that distinguished tropical elliptic space curves from arbitrary balanced genus one graphs. Just before this, Vakil and Zinger gave a very explicit desingularization of the moduli space of elliptic curves in projective space, with remarkable applications. Just after this, Smyth constructed new compactifications of moduli spaces of pointed elliptic curves, using worse-than-nodal singularities, as part of the Hasset-Keel program. A decade on, we understand these three results as part of a single story involving logarithmic structures and their tropicalizations. I will discuss this picture and how the unified framework extends all three results. This is joint work with Keli Santos-Parker and Jonathan Wise. 

Yoav Len (Georgia Institute of Technology): Algebraic and Tropical Prym varieties
Abstract: My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian vari- eties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concern- ing Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results con- cerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci. This is joint work with Martin Ulirsch.

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June 7, 2019 - First meeting in Summer Semester 2019

Margarida Melo (Università degli studi Roma Tre): Combinatorics and moduli of line bundles on stable curves.
Abstract: The moduli space of line bundles on smooth curves of given genus, the so called universal Jacobian, has a number of different compactifications over the moduli space of stable curves. These compactificatons have very interesting combinatorial properties, which can be used to describe their geometry. In the talk I will explain different features and applications of these interesting objects, focusing on properties which have a natural tropical counterpart.

Farbod Shokrieh (University of Copenhagen): Heights and moments of abelian varieties
Abstract: We give a formula which, for a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the N\'eron-Tate height of $(A,\lambda)$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a role.
(Based on joint works with Robin de Jong.)