Abstract: Given a normal integral scheme of finite type over a mixed characteristic complete DVR or a perfect field of characteristic p, one can define the notion of a test ideal. This sheaf is used to characterize a class of mild singularities known as splinter singularities, which are analogous to rational singularities in characteristic 0. In equal characteristics, it is a well-known result that test and multiplier ideals are stable under small perturbations. In this talk, I will explain how to extend this stability result to the mixed characteristic setting and discuss some of its applications. Time permitting, I will also outline the key ideas and tools from p-adic geometry that underlie the proof. This is based on joint work in progress with Bhargav Bhatt and Linquan Ma.

Abstract: A recent manuscript written by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.

Abstract: Hilbert--Kunz multiplicity and relative F-rational signature are invariants on Noetherian local rings of prime characteristic that provide quantitative measures of singularities. I will introduce the n-Witt--Hilbert--Kunz multiplicity and the relative n-quasi-F-rational signature as generalizations and refinements of these notions and discuss their fundamental properties.

Abstract: Bridgeland stability conditions provide a powerful tool for studying complexes of coherent sheaves on algebraic varieties. I will discuss some recent progress on their existence, moduli spaces of stable objects, Bogomolov-type inequalities, and related applications.

Abstract: In noncommutative algebraic geometry, we say that two smooth proper k-varieties X and Y are derived equivalent if we have an (exact k-linear) equivalence of their derived categories D^b_{coh}(X) \simeq D^b_{coh}(Y). One fundamental question in the field is what kind of properties of varieties are preserved under derived equivalence. For example, when X and Y are projective, a conjecture of Orlov predicts that derived equivalence preserves rational Chow motives, so derived equivalent complex varieties are predicted to have equal Hodge numbers in particular. This talk concerns the unexpected connection of derived equivalence to the property of liftability, which was raised in a question of Lieblich. The question asks: for varieties over a perfect field k of positive characteristic, is the liftability to W_2(k) (or to characteristic 0 etc.) preserved under derived equivalence? In this talk, I will present my forthcoming work which answers the infinitesimal version of this question under one cohomological vanishing condition by studying the derived deformation theory of twisted derived categories. This is a "twisted" extension of the work by Moore (also obtained independently in my unpublished joint work with Perry) which answers the question with two additional vanishing assumptions.

Abstract: This talk outlines the classification of cluster-type pairs on P^3 , presenting joint work in progress with J. Moraga. We will first review the foundational definitions of cluster-type pairs and explain the established results for del Pezzo surfaces. We then survey recent advances in cluster-type varieties for Fano threefolds developed by Alves da Silva, Figueroa, Moraga, and Yáñez. To conclude, we will see the classification of cluster type pairs on P^３which is an orthogonal perspective to this framework, building on prior investigations by Alves da Silva, Figueroa, and Moraga.

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Abstract: TBA