The problem of completing period mappings with geometric & Hodge-theoretic meaningful boundary data has been long-standing since Griffiths' question in 1970. In this talk I will survey my recent works joint with Colleen Robles which provided an answer for any period mappings coming from a variation of polarized Hodge structures over a quasi-projective surface. I will also talk about some applications including the construction of generalized Neron models and computing generic degree of period maps.

For even some of the smallest and most well-understood finite groups, classifying indecomposable representations over a field of positive characteristic is impossible. Since the development of support varieties in the 1980s, one rougher attempt to understand these categories of representations is to classify indecomposable objects up to a suitable equivalence; formally, this goal amounts to classifying the thick ideals of the category, and a full classification for finite groups was given by Benson—Carlson—Rickard. Tensor-triangular geometry, initiated in the early 2000s by Paul Balmer, gives a vast generalization of these techniques, and produces a topological space, the Balmer spectrum, to any tensor triangulated category; these categories have a tensor product which behaves in a similar way to the tensor product of vector spaces, and the Balmer spectrum is analogous to the prime spectrum of a commutative ring, where the tensor product plays the role of multiplication. We will discuss some recent progress in extending the Benson—Carlson—Rickard theorem to representation categories of finite-dimensional Hopf algebras, which is joint with Nakano and Yakimov.

The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle​ are algebraic $1$-cycle associated to a smooth algebraic curve with a chosen base point. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus $\ge 3$. Given a pointed algebraic curve, there is no general method to determine whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem.

Schur functors are fundamental objects sitting at the intersection of representation theory, combinatorics, and algebraic geometry. There are many ways to try to generalize such objects, but one perspective is to view the classical Jacobi--Trudi identity as saying that Schur functors are built "with respect to" the symmetric algebra. This leads to the following question: given an arbitrary algebra A, do there exist Schur functors "with respect to" the algebra A? In this talk, I'll make this question more precise, and discuss an answer to this question which has deep connections to both the equivariant analogues of positivity in combinatorics and Boij--Soederberg theory of non-regular rings in commutative algebra. This is joint work with Steven V. Sam.

TBA