Rigid local systems and (some) finite groups

• N. Katz (Princeton)

Room: Centre de Conférences Marilyn et James Simons We will discuss rigid local systems on the affine line in positive characteristic whose monodromy groups are certain finite groups of Lie type.

Character formulas in the modular representation theory of algebraic groups

• G. Williamson (University of Sydney)

Room: Centre de Conférences Marilyn et James Simons I will review present two formulas for the characters of representations of reductive algebraic groups in positive characteristic p. Both formulas involve certain polynomials ("p-Kazhdan-Lusztig polynomials") which measure the failure of the decomposition theorem for resolutions of affine Schubert varieties, when the sheaf coefficients are of characteristic p. These formulas are proven under mild bounds on p (e.g. p ≥ 2 × Coxeter number) and are expected to hold uniformly. (This is joint work with S. Riche, with parts also joint with P. Achar and S. Makisumi.)

Stratifying reductive groups

• G. Lusztig (MIT)

Room: Centre de Conférences Marilyn et James Simons We define a decomposition of a reductive group into finitely many strata. The largest stratum is the set of regular elements, the smallest stratum is the centre.

Hilbert's thirteenth problem and the moduli space of abelian varieties

• M. Kisin (Harvard)

Room: Centre de Conférences Marilyn et James Simons The (multi-valued) solution of a general polynomial of degree n is a priori a function of n-1 variables. Hilbert's thirteenth problem and its variants ask when such functions can be written as a composite of functions in a smaller number of variables. I will explain some progress on this question which uses the geometry of A_g. This is joint work with Benson Farb and Jesse Wolfson.

Prisms and deformations of de Rham cohomology

• B. Bhatt (University of Michigan)

Room: Centre de Conférences Marilyn et James Simons Prisms are generalizations of perfectoid rings to a setting where "Frobenius need not be an isomorphism". I will explain the definition and use it to construct a prismatic site for any scheme. The resulting prismatic cohomology often gives a one-parameter deformation of de Rham cohomology. For instance, it recovers the recently constructed A_{inf}-cohomology for smooth schemes over perfectoid rings (and thus crystalline cohomology when in characteristic p). A relative variant yields cohomological Breuil-Kisin modules, and related ideas also give a co-ordinate free construction of q-de Rham cohomology. Joint work with Peter Scholze.

p-adic K-theory of p-adic rings

• P. Scholze (Universität Bonn)

Room: Centre de Conférences Marilyn et James Simons The original proof of Grothendieck's purity conjecture in étale cohomology (the Thomason-Gabber theorem) relies on results on l-adic K-theory and its relation to étale cohomology when l is invertible. Using recent advances of Clausen-Mathew-Morrow and joint work with Bhatt and Morrow, our understanding in the complementary case of p-adic K-theory of p-adic rings has reached a similar level. In particular, we can express p-adic étale K-theory in terms of the cohomology theories of integral p-adic Hodge theory, such as the prismatic cohomology discussed in Bhatt's talk. Depending on time, I may indicate some possible applications.

Perfectoid Cohen-Macaulay rings and homological aspects of commutative algebra in mixed characteristic

• Y. André (CNRS & IMJ-PRG)

Room: Centre de Conférences Marilyn et James Simons The homological turn in commutative algebra due to Auslander and Serre was pushed forward by Peskine and Szpiro with a systematic use of the Frobenius functor, which led to tight closure theory, a powerful instrument developed by Hochster and Huneke to study singularities in characteristic p. We shall report on recent advances in the mixed characteristic case, where perfectoid Cohen-Macaulay algebras play the role of absolute integral closures in characteristic p, and lead to a mixed characteristic analog of tight closure theory.

Perfectoid spaces and log-regular rings

• L. Ramero (Université de Lille I)

Room: Centre de Conférences Marilyn et James Simons I will present a generalization of Scholze's perfectoid spaces that includes the limits of certain very ramified towers of log-regular rings. This is part of an on-going joint work with Ofer Gabber.

Arakelov geometry on degenerating curves

• G. Faltings (MPIM)

Room: Centre de Conférences Marilyn et James Simons We investigate the asymptotic of Arakelov Green functions and metrics, and of the delta-function, if a smooth Riemann surface degenerates to a stable curve.

Hochschild and cyclic homology of log schemes

• M. Olsson (UC Berkeley)

Room: Centre de Conférences Marilyn et James Simons I will discuss an approach to extending the notions of Hochschild and cyclic homology from schemes to log schemes. The approach is based on a more general theory for morphisms of algebraic stacks.

Log Drinfeld modules and moduli spaces

• K. Kato (University of Chicago)

Room: Centre de Conférences Marilyn et James Simons We construct toroidal compactifications of the moduli space of Drinfeld modules of rank d with N-level structure. We obtain them as the moduli spaces of log Drinfeld modules of rank d with N-level structure. The theory of toroidal compactifications was announced by Pink long ago (using the works of Fujiwara) but the details are not yet published. We follow the ideas of Pink adding log studies and considering cone decompositions related to simplices of Bruhat-Tits buildings. This is a joint work with T. Fukaya and R. Sharifi.

On relative log de Rham-Witt complex

• A. Shiho (University of Tokyo)

Room: Centre de Conférences Marilyn et James Simons The notion of relative log de Rham-Witt complex, which is the log version of relative de Rham-Witt complex of Langer-Zink, is defined by Matsuue. In this talk, we give the comparison theorem between relative log de Rham-Witt cohomology and relative log crystalline cohomology for log smooth saturated morphism of fs log schemes satisfying certain condition on which p is nilpotent. Our result generalizes most of the previously known results by Illusie, Hyodo-Kato, Langer-Zink and Matsuue. This is a joint work with Kazuki Hirayama.

"fundamental local equivalence" for quantum geometric Langlands

• D. Gaitsgory (Harvard)

Room: Centre de Conférences Marilyn et James Simons The key role in the usual geometric Langlands is played by the geometric Satake equivalence, which says that the category of spherical perverse sheaves on the affine Grassmannian Gr_G of the group G is equivalent to the category Rep(G^L) of algebraic representations of the Langlands dual G^L. Despite its importance, the above statement is rather fragile; for example it holds only at the level of abelian categories, but fails at the derived level. A more robust assertion can be formulated by replacing the spherical category by the Whittaker category. In this talk we will introduce the quantum context, which amounts to replacing sheaves on Gr_G by sheaves twisted by a certain gerbe (the latter is the quantum parameter q). It turns out that the Whittaker variant of Geometric Satake admits a deformation, where on the Langlands dual side, the category Rep(G^L) gets deformed to the category of modules over the quantum group, whose quantum parameter is the same q. The construction of the equivalence between the two sides relies on the description of a certain remarkable perverse sheaf on the configuration space of colored divisors, which encodes the combinatorics of the Cartan matrix.

Perverse equivariant sheaves on loop Lie algebras, and affine Springer theory

• Y. Varshavsky (Hebrew University of Jerusalem)

Room: Centre de Conférences Marilyn et James Simons This is a joint work with A. Bouthier and D. Kazhdan. Let G be a connected reductive group, and let LG be the corresponding loop group. Our main goal is to construct a "perverse" t-structure on the derived category of Ad LG-equivariant sheaves on LG and to show that the affine Grothendieck-Springer sheaf belongs to its core. More precisely, we construct the t-structure on the derived category of LG-equivariant sheaves supported on bounded regular semi-simple elements of LG, and we only consider its Lie algebra analog.

Spreading-out for families of rigid analytic spaces (joint work with Brian Conrad)

• O. Gabber (CNRS & IHES)

Room: Centre de Conférences Marilyn et James Simons

On the classification of p-healthy regular schemes

• A. Vasiu (Binghamton University)

Room: Centre de Conférences Marilyn et James Simons A regular local ring R of dimension at least 2 and mixed characteristic (0,p) is called p-healthy if each p-divisible group over the the punctured spectrum of R extends to a p-divisible group over Spec R. In the book of Faltings and Chai, it has been claimed that, in the current language, each such R is p-healthy. A counterexample due to Raynaud, recollected by Ogus, and worked out by Gabber in 1992 shows that this claim is far from being true. After several erroneous attempts by different specialists, just in 2010, Zink and Vasiu were able to generalize Gabber's counterexample, to show the existence of plenty of p-healthy regular rings of dimension 2 and to provide a first correct proof of the uniqueness of integral canonical models of Shimura varieties. As two joint papers with Gabber, we report on a complete classification of p-healthy regular rings of dimension 2 which are henselian and have perfect residue fields and on the very first examples of p-healthy regular schemes of arbitrary dimension greater than 2. These examples in dimension greater than 2 correct several errors in the literature and provide a new (second) proof of the uniqueness of integral canonical models of Shimura varieties.

Characteristic cycle of constructible sheaves and restriction to curves

• T. Saito (University of Tokyo)

Room: Centre de Conférences Marilyn et James Simons The characteristic cycle of a constructible sheaf is determined by its rank and the conductor of the restrictions to curves. We also discuss compatibility with proper push forward.

Deligne's geometrical approach to the product formula for l-adic epsilon factors

• F. Orgogozo (CNRS & École polytechnique)

Room: Centre de Conférences Marilyn et James Simons Around 1985, Gérard Laumon gave a proof of the local factorization of the determinant of the Frobenius map acting on the cohomology of a curve. Shortly before Laumon's work, however, Deligne had developed a different approach to the problem, which is not well known today. In this talk, I would like to introduce Deligne's method to a wider audience, in the hope that its full potential has yet to be exploited. Its starting point is Deligne's symmetric Künneth formula and the acyclicity properties of the Abel-Jacobi morphism, which led to the first proof of the result in the tame case. (Understanding Deligne's method is one part of an ongoing project with Joël Riou.)

Compatible systems along the boundary

• W. Zheng (Morningside Center of Mathematics)

Room: Centre de Conférences Marilyn et James Simons A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field, based on Gabber's theorem on compatible systems. This has applications to the equicharacteristic case of some classical conjectures on l-independence. I will also discuss the relationship with compatible wild ramification. This is joint work with Qing Lu.