Seminars Sorted by Series

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...

Let $X/S$ be a regular scheme over the spectrum of a discrete valuation ring. Bloch conjectured a formula relating algebraic differential forms to the total dimension of the vanishing cohomology of $X/S$. In this talk, I will explain a proof of this...

In a seminal paper from 2011, Wilfried Schmid and Kari Vilonen made a bold proposal that apparently intractable questions in the representation theory of real reductive groups could be resolved with the help of natural Hodge structures coming from...

The main goal of this talk is to state a conjectural formula for the compactly supported cohomology of  $\Gamma$ \ $G$ / $K$ when $G$ is a semisimple real group with maximal compact $K$, and $\Gamma$ is a congruence subgroup. The formula gives a...

For any smooth projective curve over a finite field, we construct the p-adic analytic moduli stack of isocrystals and study its geometry. This is the crystalline analogue of the moduli of integrable connections. Notably, even though it is a...

We use Teichmüller theory to construct a new geometric model for the classifying space of the mapping class group of a three-dimensional handlebody. Two consequences are obtained: (i) Chan-Galatius-Payne have shown that the homology of Kontsevich's...

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...

Let w be a word in a free group. A few years ago, Magee and I, relying on a work of Calegari, discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier...

In this talk we will discuss some aspects of the question: given a group, what is the range of Furstenberg entropy of ergodic stationary actions of it? For the special linear group and its lattices, constraints on this spectrum come from Nevo-Zimmer...

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple planar graphs on n vertices, answering a question from 1969. The proof uses a connection with continued fractions and advances towards Zaremba’s...

We will discuss recent results towards the quantum unique ergodicity conjecture of Rudnick and Sarnak, concerning the distribution of Hecke--Maass forms on hyperbolic arithmetic manifolds. The conjecture was resolved for congruence surfaces by...

 We prove ''reasonable'' quantitative bounds for sets in $\mathbb{Z}^2$ avoiding the ${polynomial}$ ${corner}$ ${configuration}$ $(x,y), (x+P(z),y), (x,y+P(z))$, where $P$ is any fixed integer-coefficient polynomial with an integer root of...

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the difference between the largest and second-largest eigenvalues—measures the expansion properties of a...

A common heuristic in Diophantine approximation asserts that different arithmetic expansions (such as digit and continued-fraction expansions) should be independent of one another. In some cases, these elementary heuristics have led to surprisingly...

We discuss what a "typical" short curve on a random large genus hyperbolic surface looks like. In particular, for each $L$, there are finitely many curves of length at most $L$. We find length scales at which such a curve chosen at random is highly...

The Bergelson conjecture from 1996 asserts that the multilinear polynomial ergodic averages with commuting transformations converge pointwise almost everywhere in any measure-preserving system. This problem was recently solved affirmatively for...

Every braid can be thought of as a homeomorphism of a punctured disc. Morally, the more complicated a braid is, the more dynamics is contained in the corresponding homeomorphism, which one can quantify using topological entropy. In particular, one...

The plan of the talk is to describe joint work with A. Brown and Z. Wang on the smooth classification of actions of lattices in $\mathrm{SL}(n,\mathbb{R})$ on $n-1$ dimensional manifolds. The method is also amenable to show rigidity for some...