Abstract: A classical theorem of C. Jordan asserts that finite subgroups in a general linear group over a field of characteristic zero contains normal abelian subgroups of bounded index. In general, a group G has Jordan property, if any finite subgroup of G contains a normal abelian subgroup of index at most J, where J is a constant only depending on G. J.P. Serre proves Cremona group of rank 2 has Jordan property, and he conjectures Cremona group of any rank has Jordan property. The conjecture is proved by Prokhorov-Shramov and Birkar. In this talk, we give explicit bounds for Cremona group of rank 2 in odd characteristic. This is a joint work with C. Shramov.

Abstract: The Mordell-Lang conjecture, now a theorem, describes the structure of rational points on subvarieties of semiabelian varieties. It states that for a semiabelian variety G over an algebraically closed field K of characteristic zero and a finitely generated subgroup Γ of G(K), the intersection X(K)∩Γ of any subvariety X⊆G with Γ is a finite union of cosets of algebraic subgroups. A fundamental uniformity question asks whether the number and the structure of these cosets can be bounded exclusively in terms of the dimension of G, the degree d of X, and the rank r of Γ.
    In this work we establish such a uniform bound. More precisely, we prove that there exists a constant c=c(dim G,d)^r such that X∩Γ is contained in at most c translates of algebraic subgroups of G.
    The proof combines several novel ingredients. We first define the Betti map associated with a family of semiabelian varieties using the mixed Hodge structure, which eventually gives a non-degeneracy condition for family of subvarieties in semiabelian varieties, via functional transcendence. We establish an equidistribution result for the Galois orbits of small points in a family of semiabelian varieties. Finally, we apply techniques from Diophantine approximation to control the number of algebraic points of large heights.
    Our theorem unifies and extends earlier work on abelian varieties and tori. Due to the twist given by the extension type, our case deal with the negativity of canonical line bundle and canonical currents which are absent in previous works. This is a joint work in progress with Zhaobo Han and Jiawei Yu.

Abstract: Given a variety of general type, the slope is the ratio between its canonical volume and its Euler characteristic. One classical topic in Algebraic Geometry is the search for lower and upper bounds for the slope. In the case of irregular surfaces, one breakthrough moment has been Pardini's proof of the Severi inequality. I will start by recalling this argument, then will describe other results in higher dimension. In particular, I will introduce the methods developed in collaboration with Barja and Pardini and discuss the case when the eventual map is generically birational, giving some new inequalities.

Abstract: Rationality problems for conic bundles have been well studied over surfaces. In this talk, we generalize an etale cohomology diagram from the case of conic bundles to Brauer-Severi surface bundles over rational 3-folds . We use this generalization to prove a sufficient condition for a Brauer-Severi surface bundle to be not stably rational.

Abstract: We give a criterion for slope-stability of the syzygy bundle of a globally generated ample line bundle on a smooth projective variety of Picard number 1 in terms of Hilbert polynomial. As applications, we prove the stability of syzygy bundles on many varieties, such as smooth Fano or Calabi--Yau complete intersections, hyperkähler varieties. This is a joint work with Peng Ren.

Abstract: Moduli spaces of one-dimensional sheaves on smooth projective surfaces have attracted considerable attention recently, due to their rich geometry and connections to Gopakumar-Vafa theory. In this talk, I will introduce an approach to studying the intersection cohomology of these possibly singular moduli spaces, via a family version of Macdonald’s formula by Maulik-Yun and Migliorini-Shende. With this approach, I will present some recent progress on the intersection Betti numbers and perverse Hodge numbers of these spaces. In particular, I will show that these invariants stabilize under suitable conditions. This talk is based on two joint works: arXiv:2511.18426 with Fei Si, and arXiv:2406.10004 with Weite Pi, Junliang Shen, and Fei Si.

Abstract: In this talk, I will discuss the autoduality of compactified Prym varieties associated with étale double covers of integral curves with planar singularities. I will explain the construction of a Poincaré sheaf on the compactified Prym variety and show that the associated integral transform defines a derived autoequivalence. As an application, I will explain how these results place the Laza--Saccà--Voisin (LSV) fibration in a general framework for certain abelian fibrations due to Maulik--Shen--Yin. This yields the motivic decomposition conjecture for the LSV fibration and a multiplicative motivic perverse filtration.

Abstract: In recent years, uniform vector bundles, namely those whose splitting type is independent of the chosen line, have been extensively studied. In this report, I will discuss two central problems for uniform bundles on a generalized Grassmannian X.
    (1) Determine the splitting threshold for uniform bundles on X.
    (2) Classify all unsplit uniform bundles of minimal ranks on X.
Furthermore, I will introduce our latest progress on these two problems, which is based on the joint work with Duo Li and Yanjie Li.

Abstract: Applying an approximation result on surfaces in our earlier work, we prove the degeneracy of integral points for the complement of three divisors on many surfaces over function fields. This generalizes work of Corvaja-Zannier, and Capuano- Turchet. Along the proof of these Diophantine results, we also show that certain open surfaces are of log general type. This is joint work with Aaron Levin amd Zheng Xiao.