Seminars Sorted by Series

Let X be a smooth rigid-analytic space over C_p. In contrast to algebraic geometry, it turns out that there are many pro-etale Q_p local systems on X that do not admit any Z_p-lattice. Furthermore, cohomology of these local systems often fail to be...

Arthur proposed a description of automorphic forms in terms of tempered automorphic forms for centralizers of SL2 homomorphisms. I will explain a point of view on the Arthur parameterization in the setting of function fields coming from relative...

Gromov--Witten invariants and Welschinger invariants count curves over the complex and real numbers. In joint work with J. Kass, M. Levine, and J. Solomon, we gave arithmetically meaningful counts of rational curves on smooth del Pezzo surfaces over...

Let X be a complex algebraic variety, and X à D a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local...

I will talk about a proof of local-global compatibility at p for higher coherent cohomology mod p in weight one for Hilbert modular varieties at an unramified prime, assuming we are not in middle degree. I will discuss some key ingredients to the...

If X is a smooth proper rigid variety over C_p and L is a Z_p-local system on X, the cohomology groups H*(X,L) are finitely generated Z_p-modules by a basic result of Scholze. If L is merely a Q_p-local system, its cohomology groups are still finite...

The inverse Galois problem asks for finite group G, whether G is a finite Galois extension of the rational numbers. Malle’s conjecture is a quantitative version of this problem, giving an asymptotic prediction of how many such extensions exist with...

TBD

I will explain how several different moduli spaces of bundles on a smooth projective curve have abelian motives. Our starting point is a formula for the motive of the stack of vector bundles on the curve in Voevodsky's category of motives with...

Classical Fourier theory describes measures on a locally compact abelian group in terms of functions on its Pontryagin dual. In this talk, I will explain an analogous theory for p-divisible rigid analytic groups (in the sense of Fargues) that...

I will explain a connection between relative Langlands duality and geometric Langlands on real forms of the projective line (i.e. the real projective line or the twistor P1), then explain recent results using this to answer some questions in...

Recent work of Drinfeld, Bhatt, and Lurie provides a “geometrization” of the theory of prismatic cohomology, where, for a p-complete commutative ring R, one produces various algebraic stacks (“prismatizations”) whose coherent cohomology identifies...

The notion of singular support has its origin in the theory of partial differential equations, and was introduced to the world of constructible sheaves by M. Kashiwara and P. Schapira in the 1970s. It is a basic invariant (like the support), and...

A result of Jan Nekovář says that the Galois action on p-adic intersection cohomology of Hilbert modular varieties with coefficients in automorphic local systems is semisimple. We will explain a new proof of this result for the non-CM part of the...

Let A be an abelian variety over a number field E ⊂ ℂ. We prove that, after replacing E by a finite extension, the action of Gal(Ē/E) on the ℓ‑adic Tate modules of A gives rise to a strongly compatible system of ℓ‑adic representations valued in the...

I will start by reviewing various Hodge-theoretic invariants of complex singularities. After that, I will discuss how to study such singularities using methods of positive characteristic such as Cartier operators. This is based on joint work with...

The Gysin map, or the wrong-way map, is a classical construction that has been available for various cohomology theories and in 𝐀¹-homotopy theory. In this talk, I will give a construction of it based on 𝐏¹-homotopy theory, so that it specializes to...

In the past decade, $p$-adic Hodge theory has been transformed by the discovery of perfectoid spaces and the Fargues–Fontaine curve. One area, however, that has remained almost untouched by these breakthroughs is the theory of $p$-adic differential...

The talk will be devoted to the problem of finiteness of the de Rham cohomology of holonomic D-modules, with the emphasis on the nonarchimedean setting. First, I will briefly recall what holonomic D-modules are and why they are important. Next, I...

The regulator of a number field is defined in terms of values of the logarithm function evaluated at units of this number field. Similarly, one can define the p-adic regulator of a number field in terms of the p-adic logarithm, and this p-adic...

A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. Simpson conjectured that for a reductive group G, rigid G-local systems...

The classical limit of the global geometric Langlands correspondence is a conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. This "global" statement can sometimes be approached by global tools...

Motivated by the instances of Grothendieck’s generalized Hodge conjecture which have a purely algebraic expression, we study the restriction map in cohomology from a smooth projective variety $X$ to an affine $U$. Starting from $X$ being defined...

The Shafarevich conjecture is concerned with finiteness results for families of g-dimensional principally polarized abelian varieties over a base B. Famously, Faltings settled the arithmetic case of B=O_{K,S}. In the case where B is a curve over a...

By an observation of Efimov and Scholze, Efimov's theorem on the rigidity of localizing motives allows one to construct refinements of topological Hochschild homology and its variants. In this talk, I'll explain how this refinement works, sketch an...

$p$-adic character varieties are rigid analytic moduli spaces of $p$-adic representations of fundamental groups. For curves, I will explain how these varieties lead to a non-abelian generalisation of the $p$-adic Hodge-Tate decomposition, by way of...

The compactly supported cohomology groups of the locally symmetric space for $GL_n(Z)$ form, as n varies the $E_1$ page of a spectral sequence. This spectral sequence was introduced by Quillen in 1973 in his proof of finite generation of $K$-groups...

A cohomology class of the group GL_n(Q_p) gives rise to a characteristic class of Q_p-local systems on algebraic varieties or topological spaces. It turns out that all rational primitive cohomology classes (in degrees >1) give vanishing...

In this talk, I will describe a variant of the category of derived rings that we call the category of synthetic $E_{\infty}$-rings. The category of derived rings may be viewed as the derived category of the category of discrete commutative rings...

We describe the obstruction to lift Frobenius on certain Shimura varieties and apply this to the theory of companion forms for classical modular forms (Gross, Coleman-Voloch, Faltings-Jordan) and also in higher dimension.

Schubert varieties are known to be Frobenius split in positive characteristic by the work of Mehta and Ramanathan. More recently, Bhatt showed that the full flag variety for $GL_n$ is a splinter by entirely different methods. In this talk, we will...

A top degree differential form \omega on a smooth algebraic variety X over a local field K gives rise to a (real valued) measure on X(K). The Serre duality yields a natural isomorphism between 1-dimensional spaces of global top degree forms on an...

The Arithmetic Quantum Unique Ergodicity (AQUE) conjecture predicts that the $L^2$ mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact...

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of...

Given a compact hyperbolic surface together with a suitable choice of orthonormal basis of Laplace eigenforms, one can consider two natural spectral invariants: 1) the Laplace spectrum $\Lambda$, and 2) the 3-tensor $C_{ijk}$ representing pointwise...

Teichmuller dynamics give us a nonhomogeneous example of an action of SL_2(R) on a space H_g preserving a finite measure. This space is related to the moduli space of genus g curves. The SL_2(R) action on H_g has a complicated behavior: McMullen...