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: TBA

Abstract: TBA