Seminars Sorted by Series

In the mid 1980s, Mark Green and I conjectured that one could read off the gonality of an algebraic curve $C$ from the syzygies among the equations defining any one sufficiently positive embedding of $C$. Ein and I recently noticed that a small...

In the course of constructing the Langlands correspondence for GL(2) over a function field, Drinfeld discovered a surprising fact about the interaction between étale fundamental groups and products of schemes in characteristic p. We state this...

The Sato–Tate conjecture, originally stated for elliptic curves on 1963, predicts the equidistribution of the normalized Frobenius traces with respect to the Sato–Tate measure, given by the pushforward of the Haar measure on SU(2). We would like to...

Does a smooth proper variety in positive characteristic know the Hodge number of its liftings? The answer is "of course not". However, it's not that easy to come up with a counter-example. In this talk, I will first introduce the background of this...

This talk will focus on geometric realizations of vertex algebras. The Virasoro uniformization provides an incarnation of the Virasoro algebra in the tangent space of the Hodge line bundle on moduli of curves with marked points and local coordinates...

Computing volumes of moduli spaces has significance in many fields. For instance, Witten's conjecture regarding intersection numbers on the Deligne–Mumford moduli space of stable Riemann surfaces has a fascinating connection to the Weil–Petersson...

A significant step in the Pila–Zannier approach to the André–Oort conjecture is a geometric transcendence result for the uniformization map of modular curves. I will discuss joint work with François Loeser. We prove an analogue of this result in non...

Kazhdan–Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the...

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties...

To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme X. Under various assumptions Faltings, Brodmann, and Kawasaki built the sought Cohen–Macaulay modifications without...

The p-curvature conjecture is an analogue of the Hasse principle for arithmetic differential equations. I will discuss this conjecture in the context of families of algebraic varieties, and among other things, demonstrate a proof of the p-curvature...

The Langlands and Fontaine–Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such...

Achar recently introduced a "nearby cycles formalism" in the framework of chain complexes of parity sheaves. In this talk, I'll explain joint work with Achar in which we compute the output of this functor in two related settings. The first is affine...

Motivation. Isogenies with a given polarized abelian variety give moduli points in a Hecke orbit. In characteristic zero any Hecke orbit is dense in this moduli space. What can be said about the Zariski closure of a Hecke orbit in positive...

Motivic homotopy theory is a powerful framework for understanding algebraic varieties and the associated algebraic objects. Moreover, computations in the motivic stable homotopy category can also assist in the computation of stable homotopy groups...

P-adic non abelian Hodge theory, also known as the p-adic Simpson correspondence, aims at describing p-adic local systems on a smooth rigid analytic variety in terms of Higgs bundles. I will explain in this talk why the « Hodge-Tate stacks »...

Given a connected reductive group G and a Levi subgroup M, Braverman-Gaistgory and Laumon constructed geometric Eisenstein functors which take Hecke eigensheaves on the moduli stack $Bun_{M}$ of M-bundles on a curve to eigensheaves on the moduli...

We study spectral invariants of torsion type on Calabi-Yau manifolds of dimension three. We define a new holomorphic invariant modifying the so-called BCOV torsion and explore its applications in the study of moduli space of Calabi-Yau manifolds...

If we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. Being Lagrangian is preserved along MCF in Kahler-Einstein manifolds. We can thus try to construct special...

Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f...

This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measure on the manifold of Riemannian metrics with the fixed volume form. We show that diameter and Laplace eigenvalue and volume entropy...

We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens...

We discuss recently established criteria for the formation of extended states on tree graphs in the presence of disorder. These criteria have the surprising implication that for bounded random potentials, as in the Anderson model, in the weak...

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of...

We consider a passive scalar field under the action of pumping, diffusion and advection by a smooth flow with a Lagrangian chaos. We present theoretical arguments showing that scalar statistics is not conformal invariant and formulate new effective...

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the...

In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of...

A proposal for QED in half space is made. Starting from the well known principle of mirror charges in electrostatics, we formulate boundary conditions for electromagnetic fields and charge carrying currents both in the classical and the quantum...

The Kuramoto synchronization model is the reference model for synchronization phenomena in biology (and, to a certain extent, also in other fields). The model is formulated as a dynamical system of interacting plane rotators. Variations of it...

Macdonald processes are probability measures on sequences of partitions de?ned in terms of nonnegative specializations of the Macdonald symmetric functions and two parameters q, t ? [0,1). Utilizing the Macdonald difference operators we prove...

In 2002, J.B. Conrey and K. Soundararajan showed that there are infinitely many Dirichlet $L$-functions which do not vanish on the critical segment. Let $\mathcal{G}$ be the family of Dirichlet $L$-functions they considered. Following their work, a...

Suppose E is an elliptic curve defined over a number field K, and p is a prime where E has good ordinary reduction. The usual methods of Iwasawa theory give a single Iwasawa module from which one can recover the Selmer groups of E over all finite...

In this talk, I will give applications of Kedlaya's recent results on p-adic differential equations. In particular, I will give a new proof of Colmez-Fontaine's theorem which describes semistable p-adic representations.

For a number field K and a (compact) p-adic Lie groups G, the inverse Galois problem asks whether G can be realized as the Galois group of an extension of K. Already in the case that G is zero-dimensional, this is too difficult. So I propose to...

In this talk, I'd like to describe my recent work with P.Sarnak concerning the equidistribution properties for Hecke igenforms on the modular surface. We evaluate asymptotically the variance for the equidistribution by means of the trace formula and...

Serre conjectured that all continuous, irreducible, odd $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ arise from modular forms. If $\rho$ is modular, then proven refinements provide recipes for the possible weights and levels of...

Let $f$ be a primitive cusp form of weight at least 2, and let $\rho_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of $\rho_f$ to a decomposition group at $p$ is `upper...