Part A: Moduli spaces and automorphic forms
A01 Teichmüller geometry of the moduli space
A central subproject aims at determining the cohomology of strata of Abelian differentials. The key strategies are integral models in order to gain information via point counting, and the combinatorics of the boundary complex. For this goal and also to determine fundamental groups we will use the comparison to moduli spaces of Bridgeland stability conditions, which are structurally similar. The subprojects aiming at the understanding of the intersection ring of strata and applications to linear submanifolds will be continued.
Available positions are listed here.A02 Non-archimedean and tropical geometry of moduli spaces - terminated 12/2025
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.
A03 Non-archimedean skeletons and Newton–Okounkov bodies
The main goal of this project is to investigate the interplay between non-archimedean skeletons and Newton–Okounkov bodies. The aim is mirror symmetry and, in particular, to construct non-archimedean SYZ-fibrations, i.e., a retraction to the essential skeleton, using Newton–Okounkov bodies as a crucial technical ingredient. The central example that will be studied in detail is the Grassmannian. Another strain of this project deals with the tropical geometry of multigraded rings.
Available positions are listed here.A04 Green currents on Shimura varieties
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.A05 Expansion and rationality of theta integrals - terminated 12/2025
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.
A06 Automorphic forms and vertex operator algebras
In the first subproject we want to show that the reflective orthogonal modular forms of singular weight are in natural bijection to the conjugacy classes in Conway’s group Co0, the orthogonal group of the Leech lattice, with non-trivial fixed-point sublattice. Furthermore, the cusps of the associated modular varieties shall be classified. The second subproject deals with the construction and investigation of Hecke operators for vertex operator algebras.
Available positions are listed here.A07 Rigid meromorphic cocycles on Drinfeld period domains
This project investigates geometric and automorphic aspects related to Drinfeld period domains. We will study open questions on higher rank Drinfeld cusp forms and complete work on the Jacquet-Langlands correspondence. Focussing on automorphic type aspects and motivated by recent seminal work of Darmon and Vonk, we will develop a theory of rigid meromorphic cocycles over global function fields, with an initial focus on GL(2). The project also treats questions on rigid meromorphic cocylces over the p-adic period domain involving p-adic families of automorphic forms.
Available positions are listed here.A08 Geodesic cycles and modular forms
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. Complementarily, we use derivatives of theta functions to construct new examples of Kobayashi geodesics. A new subproject investigates refined distribution statements for intersections of special cycles, based on the properties of the theta lift of Maass forms to genus two and higher.
Available positions are listed here.A09 Effective global generation for uniformized varieties
The main goal of our project is to study the effective positivity of line bundles on varieties with special topology, namely K(π,1) spaces. The main focus will lie on global generation of line bundles, following the conjectures of Fujita. We will investigate 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.A10 Gromov–Witten theory and orthogonal modular forms
The goal of this project is to relate the Gromov–Witten series of K3 and abelian surface fibrations to modular forms for orthogonal groups, which appear naturally in the geometry of moduli spaces of K3 and abelian surfaces. The classical Enriques surface is a basic test case for which we aim to prove general modularity results. On the modular forms side, we intend to study spaces of orthogonal quasimodular forms and regularizations of divergent Borcherds lifts of meromorphic Jacobi forms.
Available positions are listed here.A11 Tropical correspondences for 𝔸1-enumerative geometry
This project aims to develop and extend tropical correspondence theorems in 𝔸1-enumerative geometry. They provide an effective method for carrying out computations. One goal is to define 𝔸1-Hurwitz numbers that generalize classical and real Hurwitz numbers. Furthermore, we aim to analyze the quasimodular structure of generating series for real covers of elliptic curves. Another objective is to extend existing tropical correspondence theorems for curves on surfaces. As an ambitious goal, we seek to explore how these methods can be extended to count curves in higher-dimensional varieties.
Available positions are listed here.A12 Algebraic surgery and enumerative geometry
The overall goal is to develop a motivic analogue of the theory of surgery for manifolds, allowing a systematic use of degeneration methods, especially in applications to A¹-enumerative geometry and residual intersection formulas therein. Since a precise definition of motivic surgery is still missing, we will first investigate its computational consequences, in particular in Witt theory. Witt theory is SL-oriented and η-periodic. These are two funda-mental properties that often simplify computations. Therefore, we will also study the role of η-periodic and SL-oriented cohomology theories, with a particular focus on η-periodic SL-cobordism.
Available positions are listed here.Part B: Galois representations and étale invariants
B01 Higher dimensional anabelian geometry
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. We will in particular investigate relative good Artin neighbourhoods over an arbitrary base. The étale homotopy type should be made more accessible for anabelian geometry by making available the benefits of its interpretation as a profinite anima.
Available positions are listed here.B02 Galois representations in anabelian geometry
This project studies Galois sections for hyperbolic curves over number fields with p-adic methods. We so far adapted the p-adic period map of Lawrence–Venkatesh to the case of Selmer sections, and we will continue to use this tool aiming for uniformity of bounds for the local components that can occur in Selmer sections. Arboreal Galois representations will be studied as combinatorial analogues of the linear Galois representations attached to Galois sections arising form local systems on the curve. In a third part of the project, we will further develop the modularity method for Galois sections and apply it to the Fermat curve.
Available positions are listed here.B03 Motivic local systems of Calabi–Yau-type - terminated 12/2025
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.
B04 Images of Galois representations and deformations
Locally, over p-adic fields we shall construct minimally ramified deformation rings for ℓ-adic representations for ℓ≠p, and classify irreducible mod p representations for ℓ=p, in both cases valued in reductive groups. Motivated by the Bloch–Kato conjecture for the adjoint motive of automorphic forms, certain invariants and the Cohen–Macaulayness of local deformation rings will be studied, and some new modularity results shall be explored. A further focus on the global side are p-adic families and compatible systems of automorphic Galois representations with conjugate self-twists valued in GL(n), as well as an analog of a conjecture for Greenberg for GSp(4). Related abstract big image questions lead naturally to particular questions on group cohomology.
Available positions are listed here.B05 Iwasawa cohomology of Galois representations
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, duality results for the cohomology of (ϕ,Γ)-modules in the context of condensed mathematics are to be shown. Finally we would like to construct new Lubin–Tate regulator maps as part of the fourth subproject.
Available positions are listed here.B06 L-packets of p-adic automorphic forms
In this project we investigate endoscopic aspects of Langlands functoriality in the p-adic Langlands programme with the guiding question: What is a p-adic L-packet? We study the local geometry of eigenvarieties at endoscopic points with a particular emphasis on symplectic groups. The long term goal of this project is a conceptual understanding of the interplay of endoscopy, the local geometry of eigenvarieties and classicality of p-adic automorphic forms.
Available positions are listed here.B07 Algebraic cobordism in geometric Langlands
This project investigates the geometry of moduli spaces, such as the affine Grassmannian, within the Langlands program using motivic methods. We aim to extend the motivic Satake equivalence established in the first funding period to generalized cohomology theories such as K-theory, Morava E-theory, and algebraic cobordism. Key goals include advancing foundations in equivariant motivic homotopy theory, establishing a spectrally enriched Satake equivalence, and exploring applications in representation theory, particularly connections between chromatic homotopy theory and modular representation theory.
Available positions are listed here.Part C: Cohomological structure and degeneration in positive characteristic
C01 Tame cohomology of schemes and adic spaces
C02 Duality with Frobenius and 𝔽p-étale cohomology
Building on the progress achieved during the first funding period, the next step is to construct a Grothendieck–Serre duality and a Riemann–Hilbert correspondence for general morphisms of F-finite noetherian schemes, which are compatible with the three developed cohomological operations. Furthermore, a central objective will be to investigate more general coefficient systems and to explore the connection between mod p theory and p-adic theories, with parallel investigations addressing applications in birational geometry, commutative algebra, and arithmetic geometry.
Available positions are listed here.C03 Derived and prismatic F-zips
The project investigates the derived de Rham cohomology of proper morphisms in positive characteristic p > 0 and, more generally, over certain p-adically complete bases. The aim is to equip the completed derived de Rham cohomology with additional structures that extend those of an F-zip in order to obtain the notion of a derived F-zip. For this purpose we want to define geometrizations in the sense of Bhatt–Lurie to obtain functorialities, with which we then use characterize singularities by properties of the associated derived F-Zips by studying resolutions of singularities.
Available positions are listed here.C04 Motives for shtukas and Shimura varieties
The project extends the definition of truncated local shtukas to the global setting in the function field case and studies these spaces via geometrization in analogy with the local case. We aim to obtain explicit Traverso bounds. Moreover, we study truncated period maps for integral models of Shimura varieties with good reduction and their toroidal compactifications using syntomic realizations. To extend these ideas to bad reduction integral models we work towards a uniform moduli description for local models.
Available positions are listed here.C05 Strata and tautological classes for compactifications of Shimura varieties - terminated 12/2025
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.
C06 p-adic non-abelian Hodge theory
This project studies p-adic analogues of the Simpson correspondence using v-vector bundles, a new class of vector bundles on p-adic varieties. Our first goal is to use moduli spaces of v-vector bundles to make progress on an open question of Faltings about the p-adic Simpson correspondence for curves. Second, we aim for a moduli-theoretic description of v-vector bundles in arithmetic situations towards applications to the p-adic Local Langlands programme. Third, we apply v-vector bundles to study new analogues of non-Kähler manifolds in p-adic geometry.
Available positions are listed here.C07 Motivic homotopy in p-adic cohomology
The aim of this project will be to establish more connections between the study of motivic homotopy theory and tame cohomology as defined by Hübner-Schmidt. We hope to get more properties of tame motivic cohomology and tame motives (rigidity, six operations, Bott elements etc.) and further study the connection with (P1,∞)-invariant theories for logarithmic schemes over general bases (focusing on prismatic and syntomic cohomology) and possible applications to more refined motivic filtrations of K-theory. A fundamental example will be the study of the tame cohomology of log de Rham-Witt forms, and more general examples will come from the study of the cohomology of reciprocity sheaves of Kahn–Saito–Yamazaki.
Available positions are listed here.C08 K-theory, regularity, and homotopy invariance
In this project we aim to quantify the failure of homotopy invariance for algebraic K-theory of a ring or scheme. A key ingredient is an analysis of differential forms and trace invariants in the cdh topology. These new insights should enable progress on Vorst’s conjecture on a K-theoretic characterization of regularity. A further goal is the study of K-theory of adic spaces. In order to investigate homotopy invariance, we will study regularity conditions on adic spaces based on suitable dévissage theorems. Finally, we will develop a motivic framework to give a universal characterization of the K-theory of adic spaces.
Available positions are listed here.