Abstract: The complexity is an invariant of log pairs that was shown by Brown-McKernan-Svaldi-Zong to characterize toric varieties. More precisely, they showed that toric Calabi-Yau pairs minimize the complexity among all Calabi-Yau pairs. I will discuss two works that study this invariant further. The first, joint with Fernando Figueroa (Northwestern), identifies all minimizers of the complexity and studies their birational geometry. The second, joint with Jennifer Li (Princeton) and José Yáñez (UCLA) studies other geometric consequences of small complexity and provides a criterion in terms of the complexity for a variety to be cluster type.

Abstract: To what extent do Chow-valued Chern classes determine the isomorphism class of an algebraic vector bundle? In this talk, I'll discuss some progress on this question for algebraic vector bundles of rank 2 over smooth affine fourfolds. These results imply some concrete cohomological classification results (e.g., over the complex numbers, there are exactly 9 isomorphism classes of rank 2 vector bundles over the complement of a smooth degree 3 hypersurface in P^4). I'll also highlight some possible computations that, if completed, would shed further light on this problem. This is joint work with Thomas Brazelton and Tariq Syed.

Abstract: Let F be a homogeneous polynomial over a field of characteristic p>0 with an isolated singularity and let X be (the compactification of) its fiber away from the origin. We show that there is a connection between the "roots of the b-function" of F and the monodromy eigenvalues on the slope zero part of the crystalline cohomology of X. This is joint work in progress with Hiroki Kato and Daichi Takeuchi.

Abstract: A useful vanishing theorem for understanding characteristic zero singularities is Grauert-Riemenschneider vanishing, which asserts that if f: Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of \omega_Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational. As one could expect, this vanishing theorem fails in positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, answering a question of Blickle, Esnault, Chatzistamatiaou and Rülling. We will also discuss applications to singularities.

Abstract: We study asymptotic behavior of the stability thresholds of a big line bundle, and prove explicit bounds on the error terms. This answers Jin-Rubinstein-Tian's questions affirmatively. A key step in our proof is to show that the stability thresholds of a big line bundle can always be computed by quasi-monomial valuations. This generalizes Blum-Jonsson's result on the stability thresholds of an ample line bundle.

Abstract: Classical inequalities such as the Loomis-Whitney and Alexandrov-Fenchel inequalities reveal deep geometric properties of convex bodies. In this talk, we explore their analogs in two parallel settings: Hermitian matrices and divisors on projective varieties. We introduce the notion of volume polynomials associated to convex bodies, Hermitian matrices, and divisor classes, and investigate the relationships among them. These connections lead to new inequalities for mixed volumes, mixed discriminants, and intersection numbers. This is joint work with June Huh, Mateusz Michałek and Jian Xiao.

Abstract: Arising in the construction of ramified double-covers of Fano varieties, as well as the investigation of Fano four-folds, Casagrande-Druel varieties are conic bundles $Y$ realized as certain blow-ups of projective bundles over their base $V$ with discriminant locus $B$. These varieties are Fano under minor additional assumptions. We show that a Casagrande-Druel variety $Y$ satisfying an additional proportionality condition, is K-poly/semistable if and only if the pair $(V,aB)$ is K-poly/semistable for an explicit coefficient $a$, and that Casagrande-Druel variety failing this proportionality condition is K-unstable. Using this we then construct the K-moduli spaces of Casagrande-Druel varieties by directly relating the K-moduli of such Fano varieties to the K-moduli spaces of the base pairs $(V,aB)$.