Seminars Sorted by Series

The main tool in Ngô's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For $\mathrm{GL}(n)$ and a divisor of degree $ > 2g-2$, the theorem says...

The Andre-Oort conjecture describes the expected distribution of special points on Shimura varieties (typically: the distribution in the moduli space of principally polarized Abelian varieties of points corresponding to Abelian varieties with...

In the theory of algebraic cycles, the archimedean height is a real number attached to null-homologous non-intersecting algebraic cycles on a smooth projective variety \(X\) whose dimensions add up to one less than the dimension of \(X\). For...

This will be an expository talk, mostly drawn from the literature and with emphasis on the several parameter case of degenerating families of algebraic varieties.

A decade ago, John Morgan and I classified all integral weight 3 VHS of Hodge type (1,1,1,1) which can underly a family of Calabi-Yau three folds over the thrice-punctured sphere, subject to conditions on monodromy coming from mirror symmetry. There...

In this talk, I will begin by recalling the classification of normal functions over $\mathcal M_{g,n}$, the moduli space of $n$-pointed smooth projective curves of genus $g$. I'll then explain how they can be used to resolve a question of Eliashberg...

A famous result of Borel says that the cohomology of $\mathcal A_g$ stabilizes. This was generalized to the Satake compactification by Charney and Lee. In this talk we will discuss whether the result can also be extended to toroidal...

In this talk I will present some joint work with Jeff Achter concerning the problem of determining when the cohomology of a smooth projective variety over the rational numbers can be modeled by an abelian variety. The primary motivation is a problem...

A Cartwright-Steger surface is a complex ball quotient by a certain arithmetic cocompact group associated to the cyclotomic field $Q(e^{2\pi i/12})$, its numerical invariants are with $c_1^2 = 3c_2 = 9, p_g = q = 1$. It is a cyclic degree 3 cover of...

Inspired by the elementary notion of graph chordality, we introduce the notion of toric chordality, which naturally gives a tool to to study the geometry and combinatorics of cohomology classes of toric varieties and the weight algebras of polytopes...

We will first quickly recall basic facts on the tree of SL_2 over a field K with a discrete valuation v, following Serre's book. We will then generalize the geometric interpretation given in that book for curves to a higher dimensional situation...

For low degree K3 surfaces there are several way of constructing and compactifying the moduli space (via period maps, via GIT, or via KSBA). In the case of degree 2 K3 surface, the relationship between various compactifications is well understood by...

We will first quickly recall basic facts on the tree of SL_2 over a field K with a discrete valuation v, following Serre's book. We will then generalize the geometric interpretation given in that book for curves to a higher dimensional situation...

The Torelli map associates to a genus g curve its Jacobian - a $g$-dimensional principally polarized abelian variety. It turns out, by the works of Mumford and Namikawa in the 1970s (resp. Alexeev and Brunyate in 2010s), that the Torelli map extends...

Tropical geometry gives a new approach to understanding old questions about algebraic curves and their moduli spaces, synthesizing techniques that range from Berkovich spaces to elementary combinatorics. I will discuss an outline of this method...

We study the restriction map to the closed fiber for the Chow group of zero-cycles over a complete discrete valuation ring. It turns out that, for proper families of varieties and for certain finite coefficients, the restriction map is an...

I will first discuss a relation between the cohomology groups (with rational coefficients) of the compactified Jacobian and those of the Hilbert schemes of a projective irreducible curve $C$ with planar singularities, which extends the classical...

KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli space, by showing...

Let $X$ be a smooth canonically polarised surface defined over an algebraically closed field of characteristic 2. In this talk I will present some results about the geometry of $X$ in the case when the automorphism scheme $\mathrm{Aut}(X)$ of $X$ is...

We discuss developments in motivic homotopy theory over the last ten years, including structural aspects, the role of Postnikov towers, oriented theories and quadratic forms.

The minimal log discrepancy is a measure of singularities of pairs. While akin to the log canonical threshold, it turns out to be much more difficult to study, with many questions still open. I will discuss a question about the boundedness of...

De Fernex and Hacon associated a multiplier ideal sheaf to a pair $(X, \mathfrak a^c)$ consisting of a normal variety and a closed subscheme, which generalizes the usual notion where the canonical divisor $K_X$ is assumed to be Q-Cartier. I will...

In this talk, I will present the recent joint work with Yi Zhu on $A^1$-connectedness for quasi-projective varieties. The theory of $A^1$-connectedness for quasi-projective varieties is an analogue of rationally connectedness for projective...

I will try to tell the story of the projective line minus three points from the point of view of periods, and if time permits, discuss some open problems.

Starting from an example in which the Hitchin correspondence can be written down explicitly, we look at what might be said relating the incidence complex of the boundary of the character variety, and the Hitchin map.

This is joint work with G .Garkusha. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic variety $X$, the framed motive $M_{fr}(X)$ is associated...

I will try to tell the story of the projective line minus three points from the point of view of periods, and if time permits, discuss some open problems.

Let $R$ be a regular semi-local domain, containing a field. Let $G$ be a reductive group scheme over $R$. We prove that a principal $G$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. If the regular semi-local domain $R$...

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, I will discuss an interesting relation between the derived categories of a cyclic cover of $Y$ and its branch divisor. As examples, I will describe the cases of...

It is long expected that to study the representation theory of the double loop group and the corresponding Lie algebra, the third cohomology group should play an important role. The idea is that the third cohomology classes are naturally realized...

In this talk we plan to define and study the spherical Hecke algebra for (untwisted) affine Kac-Moody groups over a local non-archimedian field. We shall prove a generalization of the Satake isomorphism for these algebras, relating it to integrable...

In this general survey talk, we will describe an approach to doing homotopy theory within Univalent Foundations. Whereas classical homotopy theory may be described as "analytic", our approach is synthetic in the sense that, in ``homotopy type theory...