Seminars Sorted by Series

The ring of symmetric functions has a linear basis of Schur functions $s_{\lambda}$ indexed by partitions $\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq 0 )$. Littlewood-Richardson coefficients $c^{\nu}_{\lambda, \mu}$ are the structure...

Abstract: In this talk I will construct a “quasisymmetric flag variety”, a subvariety of the complete type A  flag variety built by adapting the BGG geometric construction of divided differences to the newly introduced “quasisymmetric divided...

We present a combinatorial approach to the Deligne-Mumford-Knudsen compactification of the moduli space of n distinct points on the projective line $P^1$. The idea is to choose a totally symmetric embedding of the orbits of generic points into a...

A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject...

Given a triple (X,π,s) consisting of a homogeneous space X=G/P, a dominant weight π giving a projective embedding of X, and a reduced expression s for the minimal coset representative of w_0 in the parabolic quotient W/W_P, we construct a polytope...

Suppose given a class of finite combinatorial structures, such as graphs or total orders. Nate Harman and I recently introduced a notion of measure in this context: this is a rule assigning a number to each structure such that some axioms are...

Configuration spaces of points in graphs are nonsmooth analogs of braid arrangements, appearing in robotics applications and in theory of moduli spaces of tropical curves. While their cohomology is extremely difficult to understand, and depends on...

In part 1, I will survey the history of total positivity, beginning in the 1930's with the introduction of totally positive matrices, which turn out to have surprising linear-algebraic and combinatorial properties. I will discuss some modern...

I will show that any Schubert or Richardson variety R in a flag manifold G/P is equivariantly rigid and convex. Equivariantly rigid means that R is uniquely determined by its equivariant cohomology class, and convex means that R contains any torus...

The long-standing F-Conjecture asserts that there is a very simple description for the closed cone of effective curves on the moduli space M_{g,n}\bar of stable n-pointed curves of genus g as being determined by a finite collection of so-called F...

The Khovanskii-Teissier inequality provides the fundamental log-concavity property of intersection numbers of divisors of algebraic varieties, extending the Alexandrov-Fenchel inequality of convex geometry. In this talk I will explain, and attempt...

I will present a (cohomological) Fourier-Mukai transform for tropical Abelian varieties and give some applications, including a (generalized) Poincaré formula (for non-degenerate line bundles on tropical Abelian varieties).

Based on joint work with...

Abstract: I will discuss work in progress developing the theory of log prismatic cohomology and related theories using the Cartier-Witt stack.   

Given a family of motives, the de Rham realization (a certain vector bundle with integrable "Gauss-Manin" connection) can be compared to the crystalline realizations for various primes p, but the resulting Frobenius structures cannot be directly...

Let X be a proper, smooth rigid space and G a commutative rigid group. We study the relationship between G-representations of the fundamental group of X and G-Higgs bundles on X. This is joint work with Ben Heuer and Mingjia Zhang.

We develop on a new strategy based on point-set topology, which allows us to produce a purely p-adic statement for the crystallinity properties of rigid flat connections. 

Joint with Michael Groechenig.

Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. In this talk, we will describe a construction which extends this "depth zero" picture to...

Let $E$ be a finite degree extension of $Qp$. Given a mod p representation of the absolute Galois group of E we construct a sheaf on a punctured absolute Banach-Colmez space that should give the first step in the construction of the mod p local...

A theorem of Borel says that any holomorphic map from a complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the (poly) punctured disc to the Baily...

Let $f:Y \rightarrow X$ be a finite covering map of complex algebraic varieties. The essential dimension of f is the smallest integer e such that, birationally, f arises as the pullback of a covering $Y^{'} \rightarrow X^{'}$ of dimension e, via a...

The minimal model program for 3-folds has been developed only in characteristics $p \geq 5$. A key difficulty at small primes is that the singularities occurring in the minimal model program need not be Cohen-Macaulay, as they are in characteristic...

I will talk about (very much in progress) joint work with Mark Kisin on a Hodge—Newton style inequality for the mod p Breuil—Kisin modules arising from crystalline Galois representations.

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced...

The recent work of Drinfeld and Bhatt-Lurie led to a new geometric approach to p-adic cohomology theories, analogously to what was done earlier in Hodge theory by Simpson. This stacky perspective gives in particular a new approach to p-adic non...

In joint work in progress with Anschütz and Le Bras we aim to construct a 6-functor formalism for quasicoherent sheaves on the relative Fargues-Fontaine curve over rigid-analytic varieties (and even general v-stacks), providing new insights into the...

In chromatic homotopy theory, an object like the sphere spectrum $S^0$ is studied by means of its "localizations", much as an abelian group can be localized at each prime p.  Remarkably, the "primes" $K(n)$ in the homotopy setting correspond to...

A smooth, oriented n-manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard n-sphere. In dimensions $n>4$, one often studies the group Θn of homotopy spheres up to orientation-preserving...

Abstract: Since the original conjectures of Beilinson and Lichtenbaum in the 80s, several versions of motivic cohomology have been introduced and developed, notably by Voevodsky. Most classically, Bloch's higher Chow groups provide the accepted...

Abstract: Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively.  In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of...

Abstract: The p-adic Simpson correspondence aims to give a non-abelian generalisation of the Hodge-Tate decomposition. Following an idea of Faltings, it should relate pro-étale vector bundles on smooth rigid spaces over Cp to Higgs bundles. In this...

Abstract: Recently the work of Fargues--Scholze provides a geometrization of the local Langlands conjecture. It is natural to ask if in this context any form of local-global compatibility can be stated/verified. We discuss some expectations and...

Abstract: In recent work with Antieau and Nikolaus we use prismatic cohomology to compute algebraic K-theory of Z/pn and similar rings. Our approach is based on a new description of absolute prismatic cohomology, which can be made completely...

Abstract: Given an étale Zp-local system of rank n on an algebraic variety X, continuous cohomology classes of the group GLn(Zp) give rise to classes in (absolute) étale cohomology of the variety with coefficients in Qp. These characteristic classes...

Abstract: We study prismatic crystals and their cohomology by using q-Higgs modules (= a q-analogue of p-connections). When the base is lying over the q-crystalline prism, they are locally described in terms of q-Higgs modules and the associated...

Abstract: Chromatic homotopy theory relates certain important questions in homotopy theory to the theory of formal groups. 

Recent advancements in p-adic geometry can be thus used to study questions in homotopy theory. I will discuss how this...

Abstract: We will explain how to construct a Kirillov model for Emerton's completed cohomology of the tower of modular curves. The trickiest part is to prove injectivity of this model.  This is joint work with Shanwen Wang.

Abstract: I will discuss the formulation and sketch the proofs of duality theorems for the geometric and arithmetic p-adic pro-étale cohomology of Stein spaces. This is based on a joint work with Pierre Colmez and Sally Gilles.

Abstract: Let p be a prime number. Emerton introduced the p-adically completed cohomology, which admits a representation of some p-adic group and can be thought of as some spaces of p-adic automorphic forms. In this talk, I want to explain that for...

Abstract: For a reductive group $G$, its $B_{d}R^{+}$-affine Grassmannian is defined as the étale (equivalently, v-) sheafification of the presheaf quotient $LG/L^{+}G$ of the $B_{d}R$-loop group $LG$ by the $B_{d}R^{+}$-loop subgroup $L^{+}G$. We...

Abstract: The key principle in Grothendieck's algebraic geometry is that every commutative ring be considered as the ring of functions on some geometric object. Clausen and Scholze have introduced a categorification of algebraic and analytic...

Abstract: I'll give an exposition of the theory of "multiplicative polynomial laws," introduced by Roby, and how (following a suggestion of Scholze) they can be applied to the theory of commutative (flat) group schemes. This talk will feature more...

This NSF-funded workshop supported the participation of underrepresented groups in the mathematical sciences.  Talks featured senior professionals about their experiences in academia, industry, and government, as well as current research by young...

Abstract: The twenty years between when I started college and my first visit to IAS saw great changes in opportunities for women in mathematics, as in most professions. Concurrent with this was the birth of a new field of global or geometric...

Abstract: Being a scientist or mathematician can be challenging, and being a minority mathematician even more so. My path hasn’t always been easy, but it has been worthwhile and fulfilling. However, in some sense, I believe that I ended up where I...