Algebraic Geometry

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…

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…

The well-known Adams conjecture in topology is a theorem about compactifications of real vector bundles on CW-complexes, which has important implications for analyzing stable homotopy groups of spheres. In the talk we will discuss an algebro-geometric version of this statement, which tackles…

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.…

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…