Projects

Part A: Moduli spaces and automorphic forms

M. Möller (Frankfurt)

A large subproject aims at efficiently evaluating intersection numbers in the tautological ring of strata of Abelian differentials. Applications include the Kodaira dimension of strata and their ample cone of strata. This is complemented by a subproject extending a SAGE package for such intersection theory computations. The other subprojects investigate integral models for strata and use compactifications for applications in Teichmüller dynamics.

Available positions are listed here.
M. Möller (Frankfurt), M. Ulirsch (Frankfurt), A. Werner (Frankfurt)

A main goal of our project is the calculation of the top weight cohomology of strata of flat surfaces. For this purpose we aim to develop an explicit combinatorial model for the boundary complex of the compactification of strata. Building on these developments, we intend to prove a correspondence principle for spin Hurwitz numbers. Other subprojects focus on a non-archimedean Siegel upper half plane, the logarithmic description of bordifications of Teichmüller space, and the tropical geometry of a moduli space of Higgs bundles.

Available positions are listed here.

A. Küronya (Frankfurt), M. Ulirsch (Frankfurt)

The main goal of this project is to investigate the interplay between non-archimedean skeletons and Newton–Okounkov bodies. We aim to construct non-archimedean SYZ-fibrations, i.e., a retraction to the essential skeleton, using Newton–Okounkov bodies as a crucial technical ingredient. Central examples that will be studied in detail are the Grassmannian and a free group character variety. Another strain of our project deals with the tropical geometry of multigraded rings and, in particular, the tropical geometry of the weighted Grassmannian.

Available positions are listed here.

J. Bruinier (Darmstadt)

We aim to study different constructions of Green currents for special cycles on orthogonal and unitary Shimura varieties, as well as their applications in the context of Arakelov geometry. In particular, we are interested in the corresponding classes in arithmetic Chow groups for higher codimension cycles, their generating series, and connections to automorphic L-functions. We shall also investigate higher automorphic Green functions and periods of associated meromorphic differentials.

Available positions are listed here.

Y. Li (Darmstadt)

The first two subprojects are concerned with computing Fourier expansions of theta integrals of real-analytic functions on hyperbolic spaces. Applications include giving new constructions of harmonic Maass forms associated to positive definite lattices, and proving new rationality results of theta lifts. The last two subprojects aim to calculate expansions of theta integrals at CM points and higher dimensional boundaries, in order to investigate the arithmetic information they contain.

Available positions are listed here.

N. Scheithauer (Darmstadt)

In the first subproject we want to classify the reflective automorphic products of singular weight and derive the classification the holomorphic vertex operator algebras of central charge 24 from this result. In the second subproject these vertex operator algebras shall be classified geometrically by classifying generalised deep holes. The third subproject is devoted to the construction and study of Hecke operators for vertex operator algebras.

Available positions are listed here.

G. Böckle (Heidelberg)

Building on recent developments, we shall investigate arithmetic-geometric and combi­natorial aspects of cusp forms on certain moduli varieties, uniformized by Drinfeld period domains mostly over function fields and beyond the well-established case of rank 2. To Drinfeld cusp forms of rank at least 3 we shall attach motives, also in the sense of Mornev shtukas, prove an Eichler–Shimura isomorphism, explore motivic weights and work toward a geometric Jacquet–Langlands type correspondence. The link to harmonic co­chains and boundary distributions, also for rank at least 3, shall be further developed. Possible applications are to Hecke-stable filtrations and to higher rank L-invariants.

Available positions are listed here.

J. Bruinier (Darmstadt), M. Möller (Frankfurt)

This project studies two classes of geodesic cycles in orthogonal Shimura varieties, special cycles and Kobayashi geodesics. For special cycles we intend to construct extensions to toroidal compactifications that are compatible with rational relations. We shall also prove injectivity results for the geometric theta correspondence of Kudla–Millson. This has applications to the subproject investigating the cones generated by the classes of special cycles. A complementary project uses derivatives of theta functions to construct new examples of Kobayashi geodesics.

Available positions are listed here.

A. Küronya (Frankfurt), J. Stix (Frankfurt)

The main goal of our project is to study the effective positivity of line bundles on varieties with special topology. The main focus will lie on non-vanishing and global generation of line bundles, following the conjectures of Kawamata and Fujita. We will start investigating special cases such as ball quotients and varieties with finite morphisms to abelian varieties. We will search for useful half-uniformizations, motivated by recent work on the Shafa­revich conjecture and the situation for abelian varieties and smooth curves.

Available positions are listed here.

Part B: Galois representations and étale invariants

A. Schmidt (Heidelberg), J. Stix (Frankfurt)

The main objective is to deepen our understanding of anabelian phenomena in higher dimension; in particular, to find more anabelian varieties over absolute finitely generated fields of characteristic zero and to obtain first results in positive characteristics. Moreover, we plan to investigate the motivic character of the étale homotopy type, i.e., the question to what extend the 𝔸1-homotopy type of a variety over an absolutely finitely generated fields is determined by its étale homotopy type.

Available positions are listed here.

J. Stix (Frankfurt)

This project studies Galois sections for hyperbolic curves over number fields with p-adic methods. We will study profinite/pro-p analogs of Selmer varieties to better understand non-abelian first cohomology sets. We will adapt the results of Lawrence–Venkatesh for p-adic period maps to the case of Selmer sections, aiming at finiteness results for the local components that can occur in Selmer sections. We will also develop the modularity method for Galois sections and apply it to the Fermat curve.

Available positions are listed here.

D. van Straten (Mainz)

The project aims at a detailed arithmetic study of a certain class of rank four local systems that appear in the cohomology of certain families of Calabi–Yau three-folds with three or four singular points. Contrary to the well-studied 14 hypergeometric cases, these local systems are non-rigid. The precise monodromy groups will be determined, Euler factors for the L-functions of the fibres will be computed and classical and Siegel modular properties and congruences of the fibre motives will be studied. Various methods will be tried to ascertain whether the current list of examples is complete.

Available positions are listed here.

G. Böckle (Heidelberg)

Locally, framed universal deformation rings for any residual mod p representation of the absolute Galois group of a p-adic field to GL(n) are being considered. We shall establish geometric properties in order to prove Zariski-density of crystalline points and study analogous questions for more general reductive groups. Globally, automorphic forms with conjugate self-twists are a main focus. We shall construct a corresponding non-connected monodromy group and use it to explore big image questions in p-adic families and adelically, in small rank cases, and in particular for GSp(4). Again for GSp(4), also an analog of a conjecture of Greenberg shall be pursued.

Available positions are listed here.

O. Venjakob (Heidelberg)

The goal of the first subproject consists of constructing epsilon-isomorphisms (Local Tamagawa Number Conjecture) in new cases related to the setting of Lubin–Tate formal groups. The second subproject concerns the systematic study of families of (ϕ,Γ)-modules parametrised by adic spaces. Within the third subproject we would like to extend the work of Fukaya–Kato to families of Galois representations parametrised by adic spaces. Finally we would like to construct new Lubin–Tate regulator maps as part of the fourth subproject.

Available positions are listed here.

J. Ludwig (Heidelberg)

In this project we study endoscopic aspects of Langlands functoriality in the p-adic Lang­lands programme. The aim of the first subproject is to construct new examples of p-adic endoscopic automorphic forms for classical groups such as symplectic, orthogonal or unitary groups. In the second subproject we will study the geometry of eigenvarieties at endoscopic points. In the third subproject we will investigate consequences of our previous work on p-adic L-packets of the group SL(2) for the p-adic local Langlands correspondence for SL(2).

Available positions are listed here.

T. Richarz (Darmstadt)

This project investigates a categorical approach towards a motivic Langlands decom­position over global function fields. The first subproject investigates extensions of the motivic Satake equivalence to include motivic constant term functors, twisted reductive groups and relative versions over Beilinson–Drinfeld affine Grassmannians. The second subproject aims at constructing derived versions of Drinfeld’s lemma for different co­homology theories with a view towards a motivic Drinfeld lemma. In the third subproject we work towards a motivic (categorical) Langlands decomposition for abelian and non-abelian reductive groups, with possible applications to questions of independence of l“.

Available positions are listed here.

Part C: Cohomological structure and degeneration in positive characteristic

K. Hübner (Heidelberg), A. Schmidt (Heidelberg)

The goal of this project is to advance the theory of tame cohomology and tame homotopy of schemes and adic spaces. First the necessary cohomological machinery, such as base change properties should be developed further. This involves proper and smooth base change for adic spaces and schemes. Moreover, we want to investigate tame homotopy groups.  A dream result would be a long exact homotopy sequence for tame homotopy groups in a smooth fibration. Another objective is the connection to the 𝔸1-homotopy theory of Morel and Voevodsky.

Available positions are listed here.

M. Blickle (Mainz), G. Böckle (Heidelberg)

The construction of the derived Grothendieck–Serre duality functor incorporating Frobenius actions will be the first step; on the one hand by following the techniques form Bhatt–Lurie but also by using higher categorical methods on the other. Afterwards its compatibility with other cohomological operations has to be verified as a crucial step towards showing the aimed for duality. The investigation of more general coefficient systems, applications to birational geometry and a higher dimensional Euler–Poincaré formula will be approached afterwards.

Available positions are listed here.

M. Blickle (Mainz), T. Wedhorn (Darmstadt)

The project aims to study the total de Rham cohomology of proper syntomic morphisms in positive characteristic p > 0 and, more generally, over certain p-adically complete bases. The goal is to endow the total de Rham cohomology with additional structures extending those of an F-zip to obtain the notion of a derived F-zip. We want to study moduli spaces of derived F-zips to obtain mod p period spaces. These moduli spaces will be locally geometric derived stacks and we plan to study their structure and in particular their singularities. We also plan to study variants for log-syntomic morphisms. Using the pris­matic cohomology we plan to extend these results in order to obtain mixed characteristic period spaces.

Available positions are listed here.

T. Richarz (Darmstadt), E. Viehmann (München), T. Wedhorn (Darmstadt)

The first goal is to calculate the motive of the classifying stack of a reductive group and of the stack of G-zips for a reductive group G defined over a finite field. From this we will build the motive of moduli spaces of local G-shtukas and truncated local G-shtukas, including a generalization to the case of non-constant smooth group schemes G. The third goal is to calculate the motive of the moduli space of truncated G-displays of a fixed type and to apply this to higher Chow groups of Shimura varieties of abelian type.

Available positions are listed here.

T. Wedhorn (Darmstadt)

The project aims to study the Ekedahl–Oort stratification on smooth projective toroidal compactifications of Shimura varieties of Hodge type in positive characteristic and their cycles classes in the tautological ring of a toroidal compactification. As an application, the objective is to obtain a description of the restriction of the tautological ring to the Shimura variety and to construct maximal projective subvarieties of Shimura varieties of Hodge type in positive characteristic. A further goal is to show that many interesting cycle classes, such as special cycles, those of Newton strata or those of central leaves, are also contained in the tautological ring, and to express them as linear combinations of cycle classes of Ekedahl–Oort strata. For this one extends the existing mod-p period maps to p-adic period maps. Finally, it is planned to extend results to the case of parahoric reductions.

Available positions are listed here.

A. Werner (Frankfurt)

The first subproject aims at a systematic study of Mustafin models for varieties, which are defined as the closure of projective varieties in certain models of the projective space. In the second subproject, we plan to study degenerations of certain vector bundles on such Mustafin models of curves. In the third subproject we plan to exploit Scholze’s theory of diamonds and their v-topology to investigate questions on vector bundles and local systems.

Available positions are listed here.