Gastvorträge

Wintersemester 2021/22

Mi, 20. Oktober 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Remi Reboulet (Universite Grenoble Alpes)
Plurisubharmonic geodesics in non-Archimedean geometry

Mi, 10. November 2021, Oberseminar Algebra und Geometrie (on Zoom)

• Yujie Xu (Harvard University)
On normalization in the integral models of Shimura varieties of Hodge type

Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type constructed by Kisin (resp. Kisin-Pappas). I will talk about recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models.

Mi, 1. Dezember 2021, Oberseminar Algebra und Geometrie (on Zoom)

• David Reutter (MPI Bonn)
Semisimple topological quantum field theories and exotic smooth structure

A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TQFT see? After introducing topological quantum field theories and sketching a number of examples, I will outline an answer to this question for even-dimensional semisimple’ TQFT. Such theories can at most see the stable diffeomorphism type of a manifold and conversely, if two sufficiently finite manifolds are not stably diffeomorphic then they can be distinguished by semisimple field theories. In this context, semisimplicity' is a certain algebraic condition satisfied by all currently known examples of linear algebraic TQFT in more than two dimensions, and two 2n-manifolds are said to be stably diffeomorphic if they become diffeomorphic after connected sum with sufficiently many copies of S^n x S^n. If time permits, I will discuss various implications, such as the fact that 4d oriented semisimple TFT cannot see smooth structure, while unoriented ones can. This is based on arXiv:2001.02288 and ongoing joint work with Christopher Schommer-Pries.

Mi, 12. Januar 2022, Oberseminar Algebra und Geometrie (on Zoom)

• Oishee Banerjee (University of Bonn)
Filtration of cohomology via symmetric (semi)simplicial spaces

Inspired by Deligne’s use of the simplicial theory of hypercoverings in defining mixed Hodge structures we replace the indexing category ∆ by the symmetric simplicial category ∆S and study (a class of) ∆S-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration. For ∆S-hypercoverings we construct a spectral sequence, somewhat like the Cˇech-to-derived category spectral sequence. The advantage of working on ∆S is that all of the combinatorial com- plexities that come with working on ∆ are bypassed, giving simpler, unified proof of known results like the computation of (in some cases, stable) singular cohomol- ogy (with rational coefficients) and étale cohomology (with Q_l coefficients) of the moduli space of degree n maps C to a projective space , C a smooth projective curve of genus g, of unordered configuration spaces, that of the moduli space of smooth sections of a fixed gdr that is m-very ample etc.

Mi, 26. Januar 2022, Oberseminar Algebra und Geometrie (on Zoom)

• Tyler Kelly (Birmingham)
An Open Enumerative Theory for Landau-Ginzburg models.

Landau-Ginzburg models consist of a pair (W, G) where W is a potential (that is, a complex valued regular function from a quasi-affine variety X) and G is a group acting on X so that W is invariant. In the context of mirror symmetry, oftentimes they can be viewed as a noncommutative symplectic deformation of a symplectic manifold. Over the past couple of decades there has been work in establishing an enumerative theory for a Landau-Ginzburg model, akin to Gromov-Witten theory. Recently, a few of us have aimed to create an open enumerative theory for Landau-Ginzburg models. In the end, we can construct the mirror Landau-Ginzburg model's potential function as a generating function of open enumerative invariants. This provides the Landau-Ginzburg analogue to Maslov index two discs / tropical discs for a symplectic manifold. This is joint work with Mark Gross and Ran Tessler.

Mi, 2. Februar 2022, Oberseminar Algebra und Geometrie (on Zoom)

• Christin Bibby (Louisiane State University)
A generating function approach to new representation stability phenomena in orbit configuration spaces

As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces: using the notion of twisted commutative algebras, which essentially categorify exponential generating functions. This idea allows for a factorization of the orbit configuration space “generating function” into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it reveals a sequence of primary, secondary, and higher representation stability phenomena. This is joint work with Nir Gadish.

Mi, 9. Februar 2022, Oberseminar Algebra und Geometrie (on Zoom)

• Pieter Belmans (Université de Luxemburg)
Graph potentials, TQFTs and mirror partners

In a joint work with Sergey Galkin and Swarnava Mukhopadhyay we introduced a class of Laurent polynomials associated to decorated trivalent graphs which we called graph potentials. These Laurent polynomials satisfy interesting symmetry and compatibility properties, leading to the construction of a topological quantum field theory which efficiently computes the classical periods as the partition function.
Under mirror symmetry graph potentials are related to moduli spaces of rank 2 bundles (with fixed determinant of odd degree) on a curve of genus $g\geq 2$, which is a class of Fano varieties of dimension $3g-3$. I will discuss how enumerative mirror symmetry relates classical periods to quantum periods in this setting. Time permitting I will touch upon aspects of homological mirror symmetry for these Fano varieties and their mirror partners.

Mi, 16. Februar 2022, Oberseminar Algebra und Geometrie (on Zoom)

• Petru Constantinescu (MPIM Bonn)
Dissipation of Correlations of Automorphic Forms

Mass equidistribution of eigenfunctions is a central topic in quantum chaos and number theory. In this talk we highlight a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight.

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

Mi, 4. Nov. 2020, Oberseminar Algebra und Geometrie (on Zoom):

• 16:00-17:00: Ming-Hao Quek (Brown University): Resolution of singularities via stacks and weighted blowings up
One feature of Hironaka's resolution of singularities over a field k of characteristic zero is that the process is independent of choices of local embeddings, and hence, this allows us to reduce to the case of embedded resolution of singularities (namely, resolving the singularities of a reduced, finite type -scheme embedded in a smooth -scheme). I will explain recent work by Abramovich-Temkin-Włodarczyk and me, which revisits this topic of embedded resolution of singularities by considering stack-theoretic weighted blowings up. Doing so forces us to consider embeddings in (log) smooth Deligne-Mumford stacks over, instead of embeddings in a smooth -scheme. The result is a simpler and faster procedure to Hironaka's resolution of singularities in characteristic zero, although the end result is a smooth stack instead of a smooth scheme. We resolve that final problem by applying a de-stackification theorem due to Bergh-Rydh.

Mi, 18. Nov. 2020, Oberseminar Algebra und Geometrie (on Zoom):

• 16:00-17:00: Beatrice Pozzetti (University of Heidelberg): The real spectrum compactification of higher rank Teichmüller spaces
After introducing and motivating the study of higher rank Teichmüller spaces, interesting connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups, I will discuss joint work with Burger Iozzi and Parreau in which we use ideas from real algebraic geometry as well as non-Archimedean techniques to compactify these spaces and to give geometric characterizations of the boundary points.

Mi, 2. Dez. 2020, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Martin Schwald (University of Essen):
Title: On the definition of irreducible holomorphic symplectic manifolds and their singular analogues

Abstract: We show that in the definition of IHSM being simply connected can be replaced by vanishing irregularity. This fits well with the theory of singular symplectic varieties. The proof uses the decomposition theorem for compact Kähler manifolds with trivial canonical bundle as well as the representation theory of finite groups to analyze quotients of complex tori.

Mi, 16. Dez. 2020, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Yajnaseni Dutta (University of Bonn): Holomorphic 1-forms and geometry
Abstract: In this talk I will discuss various geometric consequences of the existence of zeros global holomorphic 1-forms on smooth projective varieties. Such geometry has been indicated by a plethora of results. I will present some old and new results in this direction.  Then I will discuss two sets of such 1-forms that arise out of both the generic vanishing theory and the decomposition theorem and present a new connection between the two. This connection gives us further geometric properties of the variety. This is an on-going joint work with Feng Hao and Yongqiang Liu.

Mi., 20. Jan. 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Dori Bejleri (Harvard University): Compact moduli of higher dimensional varieties
• The Deligne-Mumford space of pointed stable curves and its cousin the space of stable maps are central objects in algebraic geometry with deep connections to many other fields. The higher dimensional analogue is the moduli space of stable log varieties or stable pairs. The existence of compact moduli spaces of such stable pairs in all dimensions is one of the crowning achievements of the last several decades of progress on the minimal model program. However, little is known about the structure of these moduli spaces in general and even the most basic computations of e.g. cohomology or enumerative invariants appear out of reach. In this talk I will give an introduction to the theory of stable log varieties and describe some recently developed tools for studying these spaces with a view toward possible applications.

Mi., 27. Jan. 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: John Cristian Ottem (Univ. Oslo): Enriques surface fibrations of even index​.
• Abstract: I will explain a geometric construction of an Enriques surface fibration over P^1 of even index. This answers a question of Colliot-Thelene and Voisin, and provides new counterexamples to the Integral Hodge conjecture. This is joint work with Fumiaki Suzuki.

Di., 2. Feb. 2021, Interdisciplinary Seminar (on Zoom)

• 15:00 - 17:00 Interdisciplinary Seminar on Topology in Condensed Matter Physics
Abstract: The idea of this seminar is to start off with an overview on the concepts from group theory and topology that enter the physical classification of electronic states in condensed matter. Moreover, we present some open problems and limitations of these tools.
Organization: K. Zantout, R. Valentí and T. Weth

Mi, 3. Feb. 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Daniele Agostini (MPI Leipzig): On the irrationality of moduli spaces of K3 surfaces.
Abstract: In this talk, we consider quantitative measures of irrationality for moduli spaces of polarized K3 surfaces of genus g. We show that, for infinitely many examples, the degree of irrationality is bounded polynomially in terms of g, so that these spaces become more irrational, but not too fast. The key insight is that the irrationality is bounded by the coefficients of a certain modular form of weight 11. This is joint work with Ignacio Barros and Kuan-Wen Lai.
Mi, 24. Feb. 2021, Oberseminar Algebra und Geometrie (on Zoom)
• 16:00-17:00: Eleonore Faber (University of Leeds): Grassmannian categories of infinite rank and rings of countable Cohen-Macaulay type
• Abstract: We construct a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over a hypersurface singularity. This gives an infinite rank analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank. In a special case, when the hypersurface singularity is a curve of countable Cohen-Macaulay type, our category has a combinatorial model by an ''infinity-gon'' and we can determine triangulations of this infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

Mi, 10. March 2021, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Sourav Das (Tata Institute of Fundamental Research, Mumbai): Log-symplectic structure on a degeneration of moduli of Higgs bundles
Given a one parameter degeneration of a smooth projective curve to a nodal curve Balaji et. al. constructed a semistable degeneration of the moduli of Higgs bundles. In this talk, I will show that there exists a relative log-symplectic form on the total space of the degeneration whose restriction to the generic fibre is the known symplectic form. If time permits, I will also show that the special fibre is an example of an integrable system with normal-crossing singularities.

Sommersemester 2020

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

• 17:30-18:30: Hannah Larson (Stanford University):
Vector bundles on P^1 bundles

Abstract: Every vector bundle on P^1 splits as a direct sum of line bundles. Given a vector bundle E on a P^1 bundle PW --> B, the base B is stratified by subvarieties defined by the condition that the restriction of E to the fibers has a certain splitting type. It is natural to ask for the classes of the closures of these strata in the Chow ring of B. In joint work with Ravi Vakil, we answer this question through a study of the moduli stack of vector bundles on P^1 bundles. In describing this moduli space, we discover an algebraic version of Bott periodicity. I will also discuss applications of this work to the Brill-Noether theory of curves of low gonality.

Mi, 20. Mai 2020, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Emanuel Reinecke (University of Michigan):
The cohomology of the moduli space of curves at infinite level
• Abstract: By work of Harer, the Betti cohomology of the moduli space of smooth, complex curves of genus g > 1 vanishes in degrees above 4g - 5. In my talk, I give a new perspective on this result which is inspired by recent developments in p-adic geometry. The approach also yields statements about moduli of stable curves and curves of compact type that are not covered by Harer's methods.

Mi, 17. Juni 2020, Oberseminar Algebra und Geometrie (on Zoom)

• 16:00-17:00: Frederik Benirschke (Stony Brook University):

The boundary of linear subvarieties

Moduli spaces of meromorphic differential forms on Riemann surfaces, also known as strata, have a distinguished set of coordinates with linear transition functions, given by the periods of the differential; so called period coordinates. A special class of subvarieties of strata are linear subvarieties, which are algebraic subvarieties of the strata, given by linear equations in period coordinates. In particular, all GL(2,R)-orbit closures are linear subvarieties. We study the closure of linear subvarieties in the compactification of strata given by multi-scale differentials. The boundary of the space of multi-scale differentials has again a distinguished set of coordinates, similar to period coordinates, and we show that the boundary of linear subvarieties is given by linear equations in those coordinates.

Wintersemester 2019/20

Do, 6. Februar 2019 Oberseminar Algebra und Geometrie
• Prof. Dr. Cecília Salgado (MPI Bonn/Universidade Federal do Rio de Janeiro)
Mordell Weil rank jumps and the Hilbert property.

Abstract: Let X be surface endowed with a (non-constant) elliptic fibration with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.
Zeit und Ort: 14:00-16:00 Uhr im Raum 711 (gr.), Robert-Mayer-Str. 10
Do, 19. Dezember 2019 Oberseminar Algebra und Geometrie
• Prof. Dr. Mariá Angélica Cueto (Ohio State University)
Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-known statement "any smooth surface of degree three in P^3 contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work with Anand Deopurkar (arXiv: 1906.08196)
Zeit und Ort: 14:00-16:00 Uhr im Raum 711 (gr.), Robert-Mayerstr. 10
Mi, 18. Dezember 2019 Oberseminar Algebra und Geometrie
• Dr. Chiara Damiolini (Princeton University)
Conformal blocks defined by modules over vertex algebras of CohFT-type

In this talk I will discuss properties of certain vector bundles on the moduli space of stable n-pointed curves which arise from finitely generated admissible modules over certain vertex algebras. I will in particular describe conditions on the vertex algebra that guarantee that the factorization property holds for these vector bundles and discuss its consequences.  This is based on a joint work with A. Gibney and N. Tarasca.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (kl.), Robert-Mayerstr. 10
Mi, 4. Dezember 2019 Oberseminar Algebra und Geometrie
• Dr. Mirko Mauri (MPI Bonn)
Dual complexes of log Calabi-Yau pairs and Mori fibre spaces

Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu conjectures that the dual complex of a log Calabi-Yau pair should be a sphere or a finite quotient of a sphere. It is natural to ask whether the conjecture holds on the end products of minimal model programs. In this talk, we will validate the conjecture for Mori fibre spaces of Picard rank two
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (kl.), Robert-Mayerstr. 10
Mi, 23. Oktober 2019 Oberseminar Algebra und Geometrie
• Dr. Giulio Bresciani (FU Berlin)
On the (birational) section conjecture over finitely generated fields

We prove that, if the section conjecture holds over number fields, then it holds for every curve X over a field finitely generated over Q with a non-constant morphism to an abelian variety which is defined over a number field. Our method also gives an independent proof of the recent result by Saidi-Tyler of the fact that the birational section conjecture over number fields implies it over fields finitely generated over Q.
Zeit und Ort: 16:00-17:00 Uhr im Raum 711 (kl.), Robert-Mayerstr. 10

Sommersemester 2019

Mi, 03. April 2019 Oberseminar Algebra und Geometrie

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

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):
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
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)