Seminars Sorted by Series

Gamow introduced the liquid drop model in 1928 as a model for the nucleus. I will discuss some recent work (with Ian Ruohoniemi) concerning roundness of minimizers. 

An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by...

New results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field are presented. This includes a family of double well potentials containing examples for which the low-energy eigenvalue splitting...

The lateral periodic condition has been imposed canonically and investigated extensively in the study of a channel flow, which unfortunately is not compatible with either the celebrated Reynolds' experiment or the self-similar Blasius boundary layer...

Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the...

In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. However, uniqueness within the broader class of solutions with L^p vorticity remains a key unresolved...

Let $M(n,m)$ denote the real $m\times n$ matrices. A continuous function $f\colon M(n,m)\to \R$ is called {\it Morrey quasi-convex} if  $$\int_{\R^n}(f(A+\nabla\vf(x))-f(A))\,dx\ge 0$$ for each smooth, compactly supported $\vf\colon\R^n\to\R^m$ and...

I will describe a new bipartite Euler system-type construction system for GSp_4 and its inner forms, based on the special cycles appearing in the Kudla program (for instance, Shimura curves on Siegel threefolds). This leads to new results towards...

Sen's theorem on the ramification of a p-adic analytic Galois extension of p-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing p-adic Hodge theory for general valuation...

By Deligne's Hodge theory, the integral cohomology groups H^n(X^h, Z) of the C-analytification of a separated scheme X of finite type over C are provided with a mixed Hodge structure, functorial in X. Given a non-Archimedean field K isomorphic to...

Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the...

Local Langlands correspondence (LLC) is a conjectural finite-to-one map from representations of a p-adic reductive group G to the set of L-parameters for G. Recently there have been two major advances in this area: Kaletha's characterization of the...

Topology of the Hitchin system has been studied for decades, and interesting connections were found to orbital integrals, non-abelian Hodge theory, mirror symmetry etc. I will explain that a large part of the symmetries in these geometries above are...

Studying the \’etale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence. 
While the structure of the rational cohomology groups is predicted conjectures of Kottwitz and...

The analytic de Rham stack is a new construction in Analytic Geometry whose theory of quasi-coherent sheaves encodes a notion of p-adic D-modules. It has the virtue that can be defined even under lack of differentials (eg. for perfectoid spaces or...

Kazhdan-Lusztig theory provides a pattern of applying tools of algebraic geometry,
such as the theory of Frobenius or Hodge weights, to numerical problems of representation theory.
These techniques have been used in representation theory over a field...

Ohta described the ordinary part of the 'etale cohomology of towers of modular curves in terms of Hida families. Ohta's approach crucially depended on the one-dimensional nature of modular curves. In this talk, I will present joint work with Chris...

We show how the language of motivic non-archimedean homotopy theory can be used to define p-adic cohomology theories and prove new results about them. For example, we show how to define solid relative rigid cohomology and deduce a version of

I will discuss a theorem obtained in joint work with Wyss and Ziegler, which is devoted to describing the Frobenius traces for the IC sheaf on moduli space of objects in symmetric abelian categories linear over a finite field. The formula is...

Pretannakian categories, that is k-linear abelian categories with finite dimensional Hom groups, given with a commutative and associative tensor product, with a unit object (such that End(1)=k) and duals, can be viewed as generalizations of linear...

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.