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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.
    Zeit und Ort: 11:00-12:00 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Mi, 22. Mai 2019 Oberseminar Algebra und Geometrie

  • Alexander Betts (MPIM Bonn)
    Non-abelian Kummer maps for curves

    The Q -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.
    Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

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.
    Zeit und Ort: 15:00-16:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10
  • 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.
    Zeit und Ort: 16:30-17:30 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Mi, 12. Juni 2019 Oberseminar Algebra und Geometrie

  • Dr. Samouil Molcho (Hebrew University of Jerusalem)
    The Logarithmic Picard Group and its Tropicalization
    Abstract: 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. 
    Zeit und Ort: 16:00-17:30 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

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
    Abstract:  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.
    Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

Mi, 26. Juni 2019 Oberseminar Algebra und Geometrie

  • Prof. Dr. Jonathan Wise (University of Colorado, Boulder) Complete moduli of line bundles and divisors
    Abstract:  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.
  • Zeit und Ort: 16:00-17:30 Uhr im Raum 711 (groß), Robert-Mayerstr. 10

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.
    Zeit und Ort: 16:00-18:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10

Wintersemester 2018/19

Mi, 07. November 2018
  • Kristin Shaw, Universität Oslo (im Oberseminar Algebra und Geometrie)
    Chern-Schwartz-MacPherson classes of matroids
    Chern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes of the tangent bundle to singular and non-complete algebraic varieties. In this talk, I will provide a combinatorial analogue of CSM classes for matroids, motivated by the geometric case of hyperplane arrangements. The CSM classes of matroids are polyhedral fans which are Minkowski weights. One goal for defining these classes is to express matroid invariants using the language of algebraic geometry and in turn use geometric intuition to study the properties of these invariants. Moreover, CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón. 
    Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10
Mi, 21. November 2018 Abschlussseminar
  • Philipp Habegger (Universität Basel)
    On the Bogomolov Conjecture over Function Fields in Characteristic 0

    On an abelian variety A defined over a number field a suitable canonical height vanishes precisely on the points of finite order. The distribution of such points on a subvariety X of A is well-understand thanks to work of Raynaud from the 1980s: they lie on a finite number of components of algebraic subgroups of A contained in X. The Bogomolov Conjecture expects points of sufficiently small canonical height to behave similarly. It was first proved by Ullmo and Zhang in the 1990s. These objects make sense when the base field is a function field; much progress was made in this setting by Cinkir, Faber, Gubler, Moriwaki, Yamaki and others. However, the function field version of the Bogomolov Conjecture remains open in general. I will report on recent work in collaboration with Cantat, Gao, and Xie in characteristic 0.
    Zeit und Ort: ab 16:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10
Do, 29. November 2018 Oberseminar
  • Stefan Rettenmayr (Universität Bonn)
    Titel: Mumford–Tate-Gruppen und Breuil–Kisin-Moduln
    Zeit und Ort: ab 14:15 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10
Mi, 16. Januar 2019 Oberseminar
  • Dmitry Sustretov (HSE Moskau)
    Gromov-Hausdorff limits of curves with flat metrics and
    non-Archimedean geometry

    In this talk I will state and sketch the proof of the following
    result. Let X_t be a holomorphic family of smooth compact complex
    curves of genus >=1 over C^\times, and let Ω be a relative
    holomorphic 1-form on the total space of the family. Assume that the
    action of the monodromy on H^1(X_t) has a Jordan block of size
    2. Consider the pseudo-Kahler metric on X_t with the Kahler form
    i/2 Ω \wedge \bar Ω and further rescale it so that the diameter
    of X_t is constantly 1.  The Gromov-Hausdorff limit of X_t as t tends
    to 0 is a metric graph which can be described in terms of the
    non-Archimedean analytic space over C((t))^alg associated to X. More
    precisely, it is a quotient of the skeleton of any semi-stable model
    of this space by an equivalence relation that depends on Ω and is
    related to the weight function introduced by Kontsevich and Soibelman
    and further studied by Mustata, Nicaise and Xu.
    Zeit und Ort: ab 14:30 Uhr im Raum 308, Robert-Mayerstr. 6-8

Sommersemester 2018

Mi, 23. Mai 2018 Oberseminar Algebra und Geometrie

  • Dr. Emre Sertöz (Max-Planck-Institut MiS, Leipzig):
    Computing and using periods of hypersurfaces
    The periods of a smooth complex projective variety X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods.
    Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

Mi, 20. Juni 2018 Oberseminar Algebra und Geometrie

  • Florian Pop (UPenn, Philadelphia):
    Recovering canonical inertia generators
    Birational anabelian geometry is about (canonical) reconstruction of function fields K from Galois theoretical information. As a major step in the strategies to tackle the problem one recovers the divisor group Div(X) of "nice" models X of the function field K. It turns out that recovering Div(X) is equivalent to recovering "canonical" inertia generators. The aim of this talk is to explain the terms in detail, and report on work in progress concerning the recovering of canonical inertia generators.
    Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

Mi, 27. Juni 2018 Oberseminar Algebra und Geometrie

  • Angelo Lopez (Università degli Studi Roma Tre, Rom):
    Extremal cycles and diagonals in symmetric product of curves The d-fold symmetric product $C_d$ of a smooth curve C is a  variety with very interesting geometry. There are, for example, several open questions on the cones of ample divisors, connected to classical conjectures, such as Nagata's. In the talk I will report on some recent work on the cones of nef and pseudoeffective n-cycles on $C_d$, by highlighting the fundamental role played by the diagonals.
    This is in collaboration with F. Bastianelli, A. Kouvidakis and F. Viviani.
    Zeit und Ort: 14:00-16:00 Uhr im Raum 711 (klein), Robert-Mayerstr. 10

Di, 03. Juli 2018 Forschungs- und Oberseminar

  • Dmitry Zakharov (Central Michigan University)
    The double ramification cycle, relations in the tautological ring, and tropical geometry
    The moduli space $\mathcal{M}_{g,n}$ parametrizes smooth algebraic curves of genus g with n marked points, and the Deligne—Mumford moduli space $\overline{\mathcal{M}}_{g,n}$ compactifies $\mathcal{M}_{g,n}$ by adding curves with nodal singularities. The spaces $\mathcal{M}_{g,n}$ and $\overline{\mathcal{M}}_{g,n}$ are the subject of a large body of work, however, their geometry is still far from being completely understood. The double ramification cycle is a family of codimension g loci in $\mathcal{M}_{g,n}$, parametrizing curves admitting a meromorphic function with prescribed zeroes and poles. A natural question is to construct its compactification in $\overline{\mathcal{M}}_{g,n}$, and to compute its class in the Chow or cohomology rings. Recently, a formula for the double ramification cycle compactified via relative stable maps was proved by Janda, Pandharipande, Pixton and Zvonkine. A related family of relations in the Chow ring of $\overline{\mathcal{M}}_{g,n}$ were proved by Clader and Janda. I will discuss the consequences of the relations of Clader and Janda, and show that they naturally reproduce classical vanishing results in the tautological ring of $\overline{\mathcal{M}}_{g,n}$. Furthermore, they give an effective algorithm for computing boundary formulas for classes that vanish on $\mathcal{M}_{g,n}$. I will also discuss upcoming work on a tropical analogue of the double ramification cycle. Joint work with Emily Clader, Samuel Grushevsky, Felix Janda and Martin Ulirsch.
    Zeit und Ort: 16:00 c.t. im Raum 309, Robert-Mayerstr. 6

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.
    Zeit und Ort: 14:15-15:15 Uhr in Raum 711 (groß), Robert-Mayerstr. 10

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(A1k,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.
    Zeit und Ort: 14:15-15:15 Uhr in Raum 711 (groß), Robert-Mayerstr. 10

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.
    Zeit und Ort: 14:00-16:00 Uhr in Raum 711 (groß), Robert-Mayerstr. 10

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 OK its ring of integers, compute the étale cohomology ring H*(Spec OK,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 {Ki} such that Aut(PSL(2,q2)), for q an odd prime power, cannot be realized as an unramified Galois group over Ki, 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.
    Zeit und Ort: 16:00-17:00 Uhr in Raum 711 (klein), Robert-Mayerstr. 10

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 Hq(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.
  • 15:45-16:45 Uhr: 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.
    Zeit und Ort: 14:15-16:45 Uhr in Raum 711 (groß), Robert-Mayerstr. 10

Sommersemester 2017

Mi, 21. Juni 2017  Oberseminar

  • Martin Ulirsch (University of Michigan):
    A moduli stack of tropical curves
    The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not functions as a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.
    In this talk I am going to give an introduction to these fascinating moduli spaces and discuss recent work  with Renzo Cavalieri, Melody Chan, and Jonathan Wise (arXiv 1704.03806), where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this 2-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose a way of describing the process of tropicalization via logarithmic geometry in the sense of Kato-Illusie using the theory of Artin fans. Finally, given time, I will also report on an ongoing  follow-up project (joint with Margarida Melo, Filippo Viviani, and Jonathan Wise) that uses these techniques to construct a universal Picard variety in logarithmic and tropical geometry.
    Zeit und Ort: 14:15 im Raum 308, Robert-Mayer Str. 6-8.

Do, 29. Juni 2017  Oberseminar

  • Piotr Achinger (IHES):
    Wild ramification and K(π, 1) spaces
    I will sketch the proof that every connected affine scheme in positive characteristic is a K(π, 1) space for the etale topology. The key technical ingredient is a “Bertini-type” statement regarding the wild ramification of ℓ-adic local systems on affine spaces. Its proof uses in an essential way recent advances in higher ramification theory due to T. Saito. Time permitting, I will discuss some "anabelian" and "irregular" ramifications of the result.
    Zeit und Ort: 15:00 im Raum 711 (groß), Robert-Mayer-Str. 10.

Wintersemester 2016/17

19. Oktober 2016 Oberseminar

  • Ariyan Javanpeykar (Universität Mainz):
    The Lang-Vojta conjecture and integral points on the moduli of smooth hypersurfaces
    Abstract: 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.
    Zeit und Ort: 16:00 im Raum 711 (klein), Robert-Mayer-Str. 10.

09. November 2016 Oberseminar

  • Daniel Greb (Universität Duisburg-Essen):
    The Miyaoka-Yau Inequality and uniformisation of canonical models
    Abstract: 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).
    Zeit und Ort: 15:00 im Raum 711 (klein), Robert-Mayer-Str. 10.

16. November 2016 Oberseminar

  • Arne Smeets (University of Leuven):
    Pseudo-split varieties and arithmetic surjectivity
    Abstract: 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(kv) → Y(kv) 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(kv) → Y(kv) 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.)
    Zeit und Ort: 17:00 im Raum 711 (klein), Robert-Mayer-Str. 10.

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.
    Zeit und Ort: 17:00 im Raum 711 (klein), Robert-Mayer-Str. 10.


Sommersemester 2016

1. Juni 2016 Oberseminar / Abschlussseminar

  • Dr. Ayberk Zeytin (Galatasaray University Istanbul):
    Arithmetic of quadratic extensions
    Abstract: 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.
    Zeit und Ort: 16:00 im Raum 711 (klein), Robert-Mayer-Str. 10

13. Juli 2016 Oberseminar / Abschlussseminar

  • Hironori Shiga (Chiba University Japan):
    A visualization of Shimura's complex multiplication theorem via hypergeometric modular functions
    Zeit und Ort: : 16:00 im Raum 711 (klein), Robert-Mayer-Str. 10.
  • Prof. Lawrence Ein (University of Illinois at Chicago):
    Title & Abstract: t.b.a.
    Zeit und Ort: : 17:00 im Raum 711 (klein), Robert-Mayer-Str. 10.


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.

    Zeit und Ort: 10:00 im Raum 309, Robert-Mayer-Str. 6-8.

19.11.2015 Vortrag im Oberseminar

  • Dr. Martin Ulirsch (Universität Bonn): Logarithmic structures, Artin fans, Kato fans, and tropicalization.
    Zeit und Ort: 15:15 im Raum 110, Robert-Mayer-Str. 10.


Sommersemester 2015

16.7.2015 Vortrag im Oberseminar

  • Carlos Rito (Porto): Two surfaces with canonical map of degrees 16 and 24
    Abstract: 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.
    Zeit und Ort: 15:30-16:30 im Raum 711 (gr), Robert-Mayer-Str. 10.

17.06.2015 Vortrag im Oberseminar

  • Henrik Bachmann (Universität Hamburg): Multiple zeta values and regularised multiple Eisenstein series.
    Abstract: 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.
    Zeit und Ort: 15:30 - 16:30 Uhr im Raum 404, Robert-Mayer-Str. 10.

10.06.2015 Vortrag im Oberseminar

  • Robert Kucharczyk (Universität Bonn): On congruence subgroups of triangle groups.
    Abstract: 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).
    Zeit und Ort: 17 - 18 Uhr im Raum 404, Robert-Mayer-Str. 10.


Wintersemester 2014/15

13.11.2014 Vortrag im Oberseminar (Vorsicht: neuer Termin/eine Woche verschoben, und st eine Stunde früher)

  • Alexander Ivanov (TU München)
    Anabelian properties of arithmetic curves
    Abstract: 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.
    Zeit und Ort: 15 Uhr s.t. im Raum 711, Robert-Mayer-Str. 10.

02.10.2014 Vortrag im Oberseminar

  • Bernd Sturmfels (UC Berkeley)
    Moduli of Tropical Plane Curves
    Abstract: 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.
    Zeit und Ort: 16 Uhr c.t. (Kaffee/Tee ab 15:30 Uhr) im Raum 110, Robert-Mayer-Str. 10.

Sommersemester 2014

11.07.14 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.
    Zeit und Ort: 10 Uhr ct, Raum 612, Robert-Mayer-Str. 10
  • 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.
    Zeit und Ort: 12 Uhr st!, Raum 612, Robert-Mayer-Str. 10

03.07.14 Vortrag im Oberseminar

28.05.14 Vortrag im Oberseminar


Wintersemester 2013/14

12.12.13 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.
    Zeit und Ort: 15:45 Uhr s.t., Raum 711 (groß), Robert-Mayer-Str. 10

28.11.13 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.
    Zeit und Ort: 14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

31.10.13 Vortrag im Oberseminar

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

    Zeit und Ort: 16 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

14.10.13 Vortrag

  • PD Dr. Alex Küronya (Budapest University of Technology and Economics)
    "q-ampleness"
    Zeit und Ort: 16 Uhr (c.t.) Raum 404, Robert-Mayer-Str. 10



Sommersemester 2013


13.06.13 Vortrag im Oberseminar Algebra und Geometrie

  • Prof. Gabino Gonzalez-Diez (UAM Madrid):
    The action of the absolute Galois group on dessins d'enfants
    Zeit und Ort: 16 Uhr (c.t.) Raum 711 (groß), Robert-Mayer-Str. 10


Wintersemester 2012/2013

18.01.2013 Vortrag im mathematischen Kolloquium

  • Prof Laurent Bartholdi (Universität Göttingen)
    Growth of groups and semigroups
    Abstract: 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
    Abstract: 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



12.07.12 Vortrag im Oberseminar Algebra und Geometrie

  • Olivier Warin (Uni. Basel)
    Über x+y+z+w=1 und Höhen
    Zusammenfassung: Link
    Zeit und Ort: 14-15 Uhr (s.t.) Raum 308, Robert-Mayer-Str. 6-8

19.04.12 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragender: Lars Kühne (ETH Zürich)
    Titel: Effective and uniform results of André-Oort type
    Zusammenfassung: 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.
    Zeit und Ort: (Achtung: neue Zeit) 14 Uhr (c.t.) Raum 308, Robert-Mayer-Str. 6-8



Wintersemester 2011/2012

21.11.-22.11.11Kolloquium zur algebraischen Geometrie

  • Ort Montag:   309  ("Ecksaal", Robert-Mayer-Str. 6-8)
    Dienstag: 308 ("großer Hörsaal", Robert-Mayer-Str. 6-8

27.10.11 Vortrag im Oberseminar Algebra und Geometrie

  • Zeit und Ort: 16 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)
    Vortragende: Patrik Hubschmid (Uni. Heidelberg)
    Titel: The André-Oort Conjecture for Drinfeld modular varieties
    Abstract: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

30.08.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragende: Johannes Cuno (TU Graz, ehemals Uni Frankfurt)
    Titel: Nichtsphärische Dreiecke von Gruppen: Der Zauber eines Lemmas.
    Abstract: 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.
    Zeit und Ort: 14 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)

 

22.08.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragende: David Torres-Teigell (UAM Madridl)
    Titel: Non-homeomorphic conjugate Beauville surfaces.
    Abstract
    Zeit und Ort: 16 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)

 
15.07.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragender: Ayberk Zeytin (Ankara)
    Titel: Finding Q rational points on moduli spaces of pointed
    rational curves

    Abstract: 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 $\mm_{0,12}$.
    Zeit und Ort: 14 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)

16.06.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragende: Tara Brough (Kiel)
    Titel: Poly-context-free groups and semilinear sets.
    Abstract: 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.
    Zeit und Ort: 16 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)

27.05.2011 Vortrag im mathematischen Kolloquium

  • Vortragender: Prof. Gareth Jones (University of Southampton)
    Titel: Beauville surfaces
    Abstract: 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.
    Zeit und Ort: 16 Uhr in Hörsaal 711 groß (Robert-Mayer-Straße 10)

19.05.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragender Alex Wright (Chicago, USA)
    Titel: Abelian square-tiled surfaces.
    Zeit und Ort: 16 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)

28.04.11 Vortrag im Oberseminar Algebra und Geometrie

  • Vortragender Cornelius Reinfeldt (McGill, Montreal, Canada)
    Titel: Limesgruppen und Makanin-Razborov-Diagramme über hyperbolischen Gruppen
    Abstract:
    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.
    Zeit und Ort: 16 Uhr im großen Hörsaal (Raum 308, Robert-Mayer-Str. 6-8)