On the parametric variation of the solution spaces of an A-hypergeometric system.

We survey the variation with respect to parameters of the solution space of an A-hypergeometric system, including joint works with Barrera, Fern´andez-Fern´andez, Forsg˚ard, and Matusevich.

Moduli spaces of rational curves on Fano hypersurfaces.

I will talk about the geometry of moduli spaces of rational curves and linear subvarieties on smooth Fano hypersurfaces and discuss the dimension and irreducibility of these moduli spaces. I will also talk about some non-positivity results for spaces of rational curves on Fano hypersurfaces of low index.

Singularities and syzgies of secant varieties of non-singular projective curves

Let C be a smooth projective curve of genus g and C is embedded into the projective space by a very ample complete linear system |L|. For each positive integer k, we consider the k-th secant varieity \( Sec_k (C,L) \), We show that if \( \deg L \ge 2g+1+2k\). Then \( Sec_k(C,L) \) is projectively Cohen Macaulary. Furthermore, \( Sec_k(C, L)\) has normal Du Bois singularities. This is joint work with Wenbo Niu and Jinhyung Park.

Metabelian Galois Representations

We are used to working with Galois representations associated to elliptic curves by considering the action of the absolute Galois group on torsion points. However there is a slightly more exotic way to view this construction once we realize that the Tate module of an elliptic curve is just the abelianization of the ´etale fundamental group of the punctured torus. In this talk, we discuss how to construct a class of Galois representations by considering covers of elliptic curves which are branched over one point. We discuss how this is related to the question of surjectivity of certain Galois representation, and how to construct representations with image isomorphic to the holomorph of the quaternions. We will not assume extensive knowledge of ´etale cohomology. This is joint work with Rachel Davis

Projective threefolds admitting nowhere vanishing holomorphic 1-forms.

A celebrated result of Popa and Schnell states that on a smooth complex projective variety of general type any holomorphic 1-form has at least one zero, so the existence of a holomorphic 1-form without zeros forces the variety to be of special type. In this talk I will give a full classification of smooth projective threefolds which admit nowhere vanishing holomorphic 1-forms. This is a joint work with S. Schreieder.

Surfaces generating the space of Hodge classes in the cohomology of theta divisors of dimension 4.

For the theta divisor of a generic principally polarized abelian variety of dimension 5, the space of degree 4 Hodge classes is of dimension 7. We produce 27 surfaces, obtained as special subvarieties for Prym structures on the abelian variety, which generate this 7-dimensional space over the rational numbers. We show that the sublattice generated by the cohomology classes of these surfaces is (up to a multiple of 2) isomorphic to the Picard lattice of a generic cubic surface. This is joint work with Jonathan Conder and Edward Dewey.

Hodge theory of degenerations.

The asymptotics and monodromy of periods in degenerating families of algebraic varieties are encountered in many settings — for example, in comparing various compactifications of moduli, in computing limits of invariants of algebraic cycles, and in topological string theory. In this talk, based on joint work with Radu Laza and Morihiko Saito, we shall describe several tools (building on classical work of Milnor, Deligne, and Clemens) for comparing the Hodge theory of singular fibers to that of their nearby fibers, and touch on some relations to birational geometry.

Measuring the complexity of hypersurface singularities in algebraic geometry and topology

I will discuss two different ways to measure the complexity of singularities of a (globally-defined) complex hypersurface. The first is derived via (Hodge-theoretic) characteristic classes of singular complex algebraic varieties, while the second is provided by the multiplier ideals. I will also point out a natural connection between these two points of view. (Joint work with Morihiko Saito and Joerg Schuermann.)

Hodge filtration, minimal exponent, and local vanishing

I will discuss a circle of ideas relating Saito’s minimal exponent of a singularity, the Hodge filtration on the localization along a regular function, the V-filtration of Malgrange and Kashiwara, and local vanishing for differential forms with log poles. This is based on joint work with Mihnea Popa

Hypergeometric Motives

IThis is a report on ongoing joint work with Deepam Patel. We will discuss some analogues of Grothendieck's comparison theorems that have been obtained in this context, and an application to local monodromy.

On extension of the motivic cohomology to singular k-schemes

In this talk, I will take about a recent new approach on the question of extending the motivic cohomology theory on smooth k-schemes to singular ones. I will first briefly explain what motivic cohomology is (or should be), and sketch the idea of the new construction.

I will discuss some results obtained in the process, and some potential applications as well. If time permits, some works in progress will be mentioned, too. Some of them are jointly studied with Sinan Ünver.

Minimal exponents of singularities

The minimal exponent of a function is the negative of the largest root of its reduced Bernstein-Sato polynomial. It refines the notion of log canonical threshold, and it is related (sometimes conjecturally) to other interesting invariants. I will describe some results towards understanding minimal exponents, based on viewing them in the context of D-modules and Hodge theory on one hand, and birational geometry on the other. This is joint work with Mircea Mustata, whose lecture will present further joint results regarding minimal exponents.

Cohomology jump loci in geometry and topology

Twenty-some-odd years ago, Donu Arapura wrote the paper ``Geometry of cohomology support loci for local systems." To this day, this seminal paper provides inspiration and guides the work of geometers and topologists alike. I will discuss some recent developments in the theory, especially in regards to duality and finiteness properties of spaces and groups, and I will outline some applications to complex algebraic geometry and low-dimensional topology.

Weight filtration on Hodge modules coming from toric embeddings.

We describe the weight filtration on the Hodge module that arises as direct image of the structure sheaf of the d-torus under the toric (monomial) map induced by the d by n integer matrix A, if the semigroup of A is saturated. The graded parts are combinatorial ad not arithmetic, and express the failure of the cone of A to be simplicial.

Desingularization of schemes and morphisms.

We review some recent results on desingularization of schemes, and morphisms. In a joint paper by Abramovich Temkin Wlodarczyk we give a proof of canonical resolution of singularities of the varieties in charactersitic zero, which is functorial with respect to logarithmically smooth morphisms. The results and the method are extended in the subsequent paper to the canonical desingularization of morphisms. The method requires use of more general centers with monomials in the logarithmic structure allowing fractional exponents. This can be further extended in a recent paper on the canonical desingularization which uses weighted blow-ups. Such algorithm uses natural weights associated with a sequence of maximal contacts and leads to the direct improvement of the resolution invariant. On the other hand when combined functorial toroidal desingularization with the author’s recent result on the canonical desingularization of locally binomial varieties, one proves a canonical desingularization of varieties except of toroidal locus. It modifies a variety without affecting toroidal singularities. The resulting singularities are exactly the same as the ones in the untouched toroidal locus. The result gives, in particular, a toroidal compactification of toroidal varieties.

Torelli locus and locally symmetric subspaces of a Siegel modular variety

The purpose of the talk is to explain some geometric approaches to study whether it is possible to realize a locally symmetric space as a totally geodesic subvariety in the Torelli locus of a Siegel modular variety. Restricting to locally Hermitian symmetric subvarieties of a Siegel modular variety, the problem is related to conjectures of Coleman and Oort.