Abstract: In this talk, we discuss the relationship between the Castelnuovo–Mumford regularity, which measures algebraic complexity, and the geometric configurations of zero-dimensional schemes. If Γ is a zero-dimensional scheme, then its Castelnuovo–Mumford regularity can be understood as the number 1+d, where d is the smallest degree for which the restrictions of homogeneous polynomials of degree d are sufficient to express all regular functions on Γ. If the scheme is reduced, this is equivalent to determining the degree of interpolating polynomials. 

As our main result, we describe the geometric configuration of Γ when its regularity is close to the known upper bound \ceiling((d-n-1)/t(Γ))+2, where t(Γ) denotes the smallest integer t such that Γ admits a (t+2)-secant t(Γ)-dimensional space. This is joint work with Euisung Park.

Abstract: Recently, progress has been made in the minimal model program (MMP) for foliated varieties. A natural question arises: do foliations with some fixed invariants form a bounded family? In the first part, I will begin with an introduction on the foliations and MMP. In the latter part, I will talk on the boundedness results on foliations, including foliated surfaces of general type and Fano toric foliations.  

We demonstrate the generic invariance of the Fano type property in cases where the volumes of anti-canonical divisors of Fano type fibers are a constant over a Zariski-dense subset, or the Fano type fibers are dimension $2$. Additionally, paralleling this theorem, we establish a conjecture by Schwede and Smith under the condition that the volumes of anti-canonical divisors remain constant in the reduction mod $p$.

Part 1: The Hitchin system plays a pivotal role in modern mathematics, forming a bridge between integrable systems, gauge theory and mirror/Langlands dualities. In this first talk I will review the mirror symmetry phenomenon for the Langlands dual pair of Hitchin systems, as initiated by Hausel–Thaddeus, including both the SYZ dual abelian fibration story and the topological mirror symmetry predictions. Then I will discuss more recent advances—such as the results of Gröchenig-Wyss-Ziegler via p-adic integration and of Maulik-Shen using Ngô's support theorem. Finally I will outline our recent joint work (with B. Wang and X. Wen) extending some of those ideas to parabolic Hitchin systems.

Part 2: In this second talk I will delve into mirror symmetry phenomena for parabolic Hitchin systems. We will see how mirror symmetry connects the geometry of the Hitchin fibration with classical results such as Springer and Lusztig theorems on nilpotent orbits and their duals. Exploiting these connections, in collaboration with B. Wang and X. Wen we have proved the SYZ and topological mirror symmetry statements for parabolic Hitchin systems of types B and C. In closing I will report on our recent progress for the type D case, where we confirm a physical conjecture of Tachikawa.

The $F$-signature is one of the key tools for understanding singularities in positive characteristic. It has been studied that one can define the $F$-signature function on ample divisors via the $F$-signature of the section ring of the divisor. In this talk, I will show that the $F$-signature function $s_X(L)$ can be naturally extended to all big divisor classes. Furthermore, we will extend the theory to ideal pairs and prove that $s_X(L)$ is invariant under proper birational maps. Finally, I will present a lower bound for $s_X(L)$ given by the product of the volume function and the Frobenius-alpha-invariant, which ensures the positivity of $s_X(L)$ on big divisors. This is joint work with S. Pande (University of Utah). 

A Calabi-Yau fibration is a fibration of projective varieties X → Z such that the canonical bundle K_X is numerically trivial over Z. The central question is: under what conditions does the total space of such a fibration belong to a bounded family? Motivated by this, we investigate fibrations whose bases and general fibers are themselves bounded. We show that, after fixing natural invariants, the total spaces are bounded in codimension one. Furthermore, when the general fibers have vanishing irregularity, the total spaces are in fact bounded. These results have further applications to the study of stable minimal models and fibered Calabi–Yau varieties. This is based on the joint work with Xiaowei Jiang and Junpeng Jiao.