Algebraic Geometry
Organizer: Ravi Vakil
Upcoming Events
In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities.…
I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions. One can view this as a “higher” version of the Mordell-Weil group for families of elliptic curves, giving rise to sections of “higher” Jacobian…
Fano varieties are one of the three building blocks of algebraic varieties.In this talk, we will discuss how to describe a general n-dimensional Fano variety.Although there is no consensus on how to answer to this question, we will explore some new invariants motivated by…
Past Events
The moduli spaces of one-dimensional sheaves on the projective plane have been studied through their connections to enumerative geometry and representation theory. In this talk, I will explain a systematic approach to study their cohomology rings, using notably tautological relations of…
We give a new proof, along with some generalizations, of a folklore theorem - attributed to Laurent Lafforgue - that a rigid matroid (i.e., a matroid whose base polytope is indecomposable) has only finitely many projective equivalence classes of representations over any given field. A key…
I will talk about a new algebra of operations on polynomials which has the property
$T_iT_j=T_jT_{i+1}$ for $i>j$ and a family of polynomials dual to them called forest polynomials. This family of operations plays the exact role for quasisymmetric polynomials and forest polynomials as…
We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new…
Given a nondegenerate smooth variety X in P^n, let S(X) (resp. T(X)) be the subvariety of the Grassmannian Gr(2, n+1) consisting of secant (resp. tangent) lines to X. I will give closed-form formulae for the classes of S(X) and T(X) in the Chow ring of Gr(2,n+1) in terms of the “higher…
The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can consider a complementary perspective - given a smooth projective variety whose nonrationality is known, how "irrational" is it? I will survey…
I will describe some connections between arithmetic geometry of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For a principally polarized abelian variety, we show an identity relating the Faltings height and the Néron--Tate height (of a symmetric effective divisor…
Given a compact Riemann surface, nonabelian Hodge theory relates topological and algebro-geometric objects associated to it. Specifically, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field.…
A theorem of Donaldson and Sun asserts that the metric tangent cone of a smoothable Kähler–Einstein Fano variety underlies some algebraic structure, and they conjecture that the metric tangent cone only depends on the algebraic structure of the singularity. Later Li and Xu extend this…
Vertex operator algebras (VOAs) are generalizations of commutative associative algebras and of Lie algebras. As I will illustrate, there are a number of interesting examples of VOAs that come from moduli spaces, and striking instances where the VOA formalism has been used to solve problems about…