Time：November 2-3, 2023
Venue：E4-233, Yungu Campus, Westlake University
Thursday, November 2
9:00-9:50
Speaker: Jiawei Yu, Peking University
Title: An explicit uniform geometric Mordell conjecture
Abstract: We introduce some recent progress on the uniform Mordell conjecture by Dimitrov-Gao-Gabegger, Kühne, Yuan, Looper-Silverman-Wilms and Yu. We also discuss Yu's proof, which is based on Diophantine approximation and admissible adelic line bundles.
10:00-10:50
Speaker: Chengyuan Yang, Peking University
Title: Rationality of Néron-Tate height over function fields
Abstract: We prove that over function fields, the Néron-Tate height of any subvariety is a rational number. We use the induction formula and treat the canonical metric by theta functions.
11:00-11:50
Speaker: Yinchong Song, Peking University
Title: Asymptotic Behavior of Zhang--Kawazumi's phi-invariants
Abstract: In this talk, we will first give a brief introduction of Yuan--Zhang's adelic line bundles over quasi-projective varieties. Then we will descuss some asymptotic behavior of Green functions of adelic line bundles. To deal with Zhang--Kawazumi's phi-invariants, we first show a continuity property of non-archimedean phi-invariants for metrized graphs, and then deduce the asymptotic behavior of phi-invariants for Riemann surfaces.
14:00-14:50
Speaker: Yao Li, Westlake University
Title: Categorification of Harder-Narasimhan Theory
Abstract: The notion of Harder-Narasimhan filtration was firstly introduced by Harder and Narasimhan in the setting of vector bundles on a non-singular projective curve. Curiously, analogous constructions have been discovered in other branches of mathematics which motivate categorical constructions of Harder-Narasimhan filtration. In this talk, we will introduce a categorical construction of Harder-Narasimhan filtration via slope method which does not need a degree function. With a theorem of existence and uniqueness of Harder-Narasimhan filtration in our categorical setting, we give a categorical interpretation of Stuhler-Grayson filtration in the case of non-necessarily Hermitian normed lattice. Moreover, we will also give a sufficient condition to make the length of Harder-Narasimhan filtrations be an invariant under different linearization.
15:00-15:50
Speaker: Weronika Czerniawska, Westlake University
Title: Fundamentals of Adelic Harmonic Analysis
Abstract: In 1950 John Tate proved functional equation and analytic continuation of the Dedekind zeta function using harmonic analysis on the ring of adeles of a number field. I will present a new presentation of adelic harmonic analysis and discuss some of its simple generalisations.
Friday, November 3
9:00-9:50
Speaker: Ruoyi Guo, Peking University
Title: An integration formula of Chern forms on quasi-projective varieties
Abstract: There is a famous formula that the integration of Chern forms of hermitian line bundles equals the algebraic intersection number of the underlying line bundles. I generalize it to a formula on a quasi-projective variety over a complete valuation field which might be archimedean or non-archimedean. I will introduce the proof and some applications of the generalized formula.
10:00-10:50
Speaker: Guoquan Gao, Peking University
Title: Geometric Bombieri-Lang conjecture for subvarieties of Abelian varieties in characteristic zero
Abstract: The Bombieri-Lang conjecture is a high-dimensional generalization of the Mordell conjecture, and it's geometric analogue can be formulated over function fields. Recently, Junyi Xie and Xinyi Yuan gave a new idea for the Geometric Bombieri-Lang conjecture and proved the conjecture for finite covers of abelian varieties in characteristic zero with some extra assumptions. In this talk, based on their idea, I will give a new approach to the conjecture for subvarieties of abelian varieties in characteristic zero, which is actually known (by other methods). It is worth mentioning that this result cannot be deduced directly from Xie and Yuan's results because of their extra assumptions.
14:00-14:50
Speaker: Quan Xu, China Jiliang University
Title: A new class group over a Noetherian and separated scheme
Abstract: As is known, the Grothendieck K-group is the origin of K-theory. In this talk, by analogue to Grothendieck K-groups, we try to define a kind of new class group. Also, compared with Grothendieck K-groups, some properties of the new class group will be determined. At present, we are looking for its possible application. This topic is in progress with B.Yu.
15:00-15:50
Speaker: Mounir Hajli, Shanghai Jiao Tong University
Title: On arithmetic Hilbert-Samuel functions on toric varieties
Abstract: This talk is devoted to the study of some arithmetic Hilbert-Samuel func- tions that arise naturally on toric varieties. We expect that the asymptotic expansion is given in terms of the canonical height of the subvariety.
