Abstract: For $\Sigma_{g,n}$ a genus $g$ surface with $n$ punctures, we study the character variety parameterizing representations of $\pi_1(\Sigma_{g,n})$. This character variety has a natural action of the fundamental group of the moduli space of curves. In joint work with Josh Lam and Daniel Litt, we aim to describe the points with finite orbit under this action, which we call canonical representations. Although in some sense this topic is a continuation of my talk in the Johns Hopkins number theory seminar last year, no prior knowledge will be assumed.

Abstract: Let $Y$ be a smooth projective threefold admitting a $K3$ fibration $f: Y \rightarrow \mathbb{P}^{1}$ with $-K_{Y} = f^{\ast} \mathcal{O}(1)$. Then the extremal rays of the cone of curves of $Y$ in the region $K_{Y} < 0$ are of two types: the blowup of a smooth curve (Type A), or a conic bundle (Type B). I show that (1) the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces of the movable cone of Type B and (2) the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces of the movable cone of Type A if $H^{3}(Y, \mathbb{C}) = 0$. (The pseudoautomorphism group acts with finitely many orbits on faces of the movable cone corresponding to faces of the cone of curves in the hyperplane $K_{Y} = 0$ by work of Kawamata). This result is implied by the Kawamata-Morrison-Totaro cone conjecture.

Abstract: A landmark result of Birkar, Prokhorov, and Shramov shows that automorphism groups of Fano (or more generally rationally connected) varieties over C of a fixed dimension are uniformly Jordan. This means in particular that there is some upper bound on the size of symmetric groups acting faithfully on rationally connected varieties of fixed dimension. We give the first effective asymptotic bound on these symmetric group actions, as well as optimal bounds in all dimensions for special classes, such as Fano weighted complete intersections and toric varieties. Finally, we show that klt Fano fourfolds with maximal symmetric actions are bounded, establishing a link between boundedness and large group actions. This talk is based on joint work with Lena Ji and Joaquín Moraga.

Abstract: The coregularity is an invariant that measures a specific type of combinatorial complexity of a pair. We will start this talk by defining this invariant and giving some examples. Then, we will explain how results about complements of Fano varieties of bounded dimension are still valid for Fano varieties of bounded coregularity. Finally, we will show some structural theorems about the orbifold fundamental groups of log Calabi-Yau pairs with bounded coregularity, resembling results for log Fano pairs. This talk is based on previous works with S. Filipazzi, J. Moraga and J. Peng as well as work in progress with L. Braun.

Abstract: In this talk, I will discuss my recent research that establishes the minimal model program for Q-factorial foliated dlt algebraically integrable foliations and log canonical generalized pairs. This is joint work with G. Chen, J. Han, and L. Xie.

Abstract: Log Fano cone singularities are generalizations of affine cones over Fano varieties. Motivated by the study of canonical metrics on Fano varieties, there is a local K-stability theory characterizing the existence of Ricci-flat K\"ahler cone metrics on log Fano cone singularities. In this talk, we aim to give a non-Archimedean characterization of local K-stability. As an application, we will see that to test local K-semistability, it suffices to test special test configurations.

Abstract: Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a p-power order commutative group scheme over a perfect field of characteristic p, in terms of its Dieudonné module. This is a joint work in preparation with Dmitry Kubrak and Shubhodip Mondal.

Abstract: In this talk, I will explain some results about factorizing a birational map between Mori fibre spaces into Sarkisov links, such that the invariants of Sarkisov will decrease. I will explain that such problem is closely related to the termination of certain log flips in lower dimension.

Abstract:

Abstract: In this talk, I will present a notion of unipotent homotopy theory for schemes, which is based on Toen's work on affine stacks. I will discuss some general properties of the resulting unipotent homotopy group schemes and explain how over a field of characteristic p>0, they often recover the unipotent completion of the Nori fundamental group scheme, p-adic etale homotopy groups, and Artin-Mazur formal groups. As examples, we will see computations in the case of curves, abelian varieties, and Calabi-Yau varieties. Joint work with Shubhodip Mondal.

Abstract: Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic. We introduce a mixed characteristic version of the multiplier / test ideal using the p-adic Riemann-Hilbert functor of Bhatt-Lurie. Under mild finiteness assumptions, we show that this version of test ideal commutes with localization and can be computed by a single alteration up to small perturbation. This is based on joint work in progress with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.