北京雁栖湖应用数学研究院

2025-09-18

09:15-10:15 唐国平

Mahler measure, integral $K_2$ and Beilinson’s conjecture of curves over number fields

We will talk about several Mahler measure identities involving families of two-variable polynomials defining curves of arbitrary genus, by means of their integral $K_2$. As an application, we can obtain some relations between the Mahler measure of non-tempered polynomials defining elliptic curves of conductor 14, 15, 24, 48, 54 and corresponding L-values. We construct $g$ independent (integral) elements in the kernel of the tame symbol on several families of curves with genus $g = 1, 2, 4, 7$. Furthermore, we prove that there exist non-torsion divisors $P − Q$ with $P, Q$ in the divisorial support of these $K_2$ elements when $g = 1, 2$, which is potentially different from previous constructions in literature.

10:30-11:30 盛茂

Nonlinear Hodge theory in positive characteristic

I will report our recent progress in nonlinear Hodge theory in positive characteristic.

14:45-15:45 田志宇

Kato homology of rationally connected fibrations

Kato homology is the homology of a special type of Gersten complex of (co)homology theories. Motivated by conjectures like geometric Manin conjecture, Cohen-Jones-Segal conjecture, and some previous work of myself and others, I will propose a conjecture about the Kato homology of rationally connected fibrations, and discuss some evidences.

16:00-17:00 胡文传

The structure of Chow varieties

Let $ C_{p,d}(\mathbb{P}^n) $ denote the Chow variety of effective $ p $-cycles of degree $ d $ in the complex projective space $\mathbb{P}^n $. A result established by Chow and van der Waerden demonstrates that $ C_{p,d}(\mathbb{P}^n)$ carries the structure of a closed complex algebraic variety. Chow varieties are fundamental in algebraic geometry, providing a geometric framework for studying algebraic cycles. In this talk, we will first review known results on the structure of $C_{p,d}(\mathbb{P}^n)$ and then talk about our recent progress and questions on Chow varieties, especially the structure on $C_{p,2}(\mathbb{P}^n)$ parameterizing degree-two cycles, which is jointed with Youming Chen.

2025-09-19

09:15-10:15 訚琪峥

D-equivalence conjecture for $K3^{[n]}$

I will explain how to use Markman’s hyperholomorphic bundles to show that birational hyper-Kähler varieties of $K3^{[n]}$ type are derived equivalent. If time allows, I will also discuss some motivic aspects of this result. This is joint work with Davesh Maulik, Junliang Shen, and Ruxuan Zhang.

10:30-11:30 Jie Zhou

Gamma classes and Stokes matrices for Fermat varieties

The Dubrovin conjecture and Gamma conjecture relate the derived category to the quantum cohomology ring for Fano varieties, via the quantum connection. In this talk, I will start by reviewing some background on these conjectures. Then I will report a work in progress joint with Yefeng Shen, in which we compute the Stokes matrix for the quantum connection in the case of Fermat varieties (which can be Fano or of general type) and then relate the Stokes matrix to the Gram matrix for certain collection of objects in the numerical K group.

14:45-15:45 Fangzhou Jin

Milnor-Witt cycle modules and the homotopy t-structure

Milnor-Witt cycle modules are quadratic analogues of Rost cycle modules, which can be used to define Chow-Witt groups using elementary arithmetic operations on residue fields. We introduce Milnor Witt cycle modules over a base scheme and discuss their relations with the homotopy t-structure on the motivic stable homotopy category. This is a joint work with F. Déglise and N. Feld.

16:00-17:00 Xiaowen Hu

On the algebraic K-theory over truncated Witt vectors

Around 2013, Bloch, Kerz, and Esnault proposed a K-theoretic approach to the $p$-adic variational Hodge conjecture, and showed a formal version of this conjecture. In this talk, I will report my recent progress on the infinitesimal version. We start from a computation of certain cyclic homology, which, by a theorem of Brun, suggests a notion of infinitesimal motivic complexes. Then we study the mod $p$ theory and the products, and then show a relative Chern character isomorphism for smooth schemes over $W_n(k)$, where $k$ is a perfect field with characteristic $p$, which implies a result in infinitesimal deformations in algebraic K-theory. Compared to the continuous K-theory, a new feature in the infinitesimal theory is the appearance of some extra mod $p$ multiplicative generators, which I call basic elements.

2025-09-20

09:15-10:15 刘杭

K2 of curves, regulators and special value of L-functions

In this talk, we focus on Beilinson's conjecture for K2 groups of curves, a conjecture that profoundly illuminates the intrinsic connections between algebraic K-theory and the special values of L-functions. We present explicit constructions of families of elliptic curves defined over cubic or quartic fields, with three and four "integral" elements in their K2 groups respectively—precisely the number of such elements predicted by Beilinson's conjecture. Additionally, we propose new conjectures regarding the relationship between the Beilinson regulator integral and the special values of L-functions of modular forms for strongly modular curves. Finally, we provide evidence of these conjectures for some curves constructed above.

10:30-11:30 杨南君

Witt group of nondyadic curves

Witt group of real algebraic curves has been studied since Knebusch in 1970s. But few results are known if the base field is non-Archimedean except the hyperelliptic case by works of Parimala, Arason et al.. In this talk, we compute the Witt group of smooth proper curves over nondyadic local fields with $char ≠ 2$ by reduction, with a general study of the existence of Theta characteristics.

14:45-15:45 Heng Xie

Hermitian K-theory of Lagrangian Grassmannians via  reducible Gorenstein models

We construct a family of moduli spaces, called generalized Lagrangian flag schemes, that are reducible Gorenstein (hence singular), and that admit well-behaved pushforward and pullback operations in Hermitian K-theory. These schemes arise naturally in our computations. Using them, we prove that the Hermitian K-theory of a Lagrangian Grassmannian over a regular base splits as a direct sum of copies of the base's (Hermitian) K-theory, indexed by certain shifted Young diagrams. The isomorphism is realized via pullback to each generalized Lagrangian flag scheme followed by pushforward to the Lagrangian Grassmannian. This yields an unusual example in which both the base and the target are regular schemes,  while the intermediate reducible Gorenstein models remain sufficient to allow explicit computations in Hermitian K-theory of regular schemes. This is joint work with Tao Huang.

16:00-17:00 杜鹏

Isotropic points in the Balmer spectrum of stable motivic homotopy categories

I will discuss the tensor-triangulated geometry of the stable motivic homotopy category $\mathcal{SH}(k)$ and a big family of the so-called isotropic realisation functors, parameterized by the choices of a Morava K-theory and an extension of the base field $k$ (of characteristic zero). By studying the target category of such an isotropic realisation functor, we are able to construct the so-called isotropic Morava points of the Balmer spectrum $Spc(SH(k)^c)$ of the stable motivic homotopy category $\mathcal{SH}(k)$. Based on joint work with A. Vishik.

2025-09-21

09:15-10:15 古星

The singular and motivic cohomology of $BPGL(n;\mathbb{C})$ over complex numbers

I will present some results on the integral and mod $p$ cohomology of $BPGL(p^m;C)$, the classifying space of the complex projective linear group $PGL(p^m;C)$ of rank pm, (or equivalently, the classifying space of $PU(p^m)$, the projective unitary group of rank $p^m$), where $p$ is an odd prime number. I will also indicate how these results provide information on the the motivic cohomology of $B_{et}PGL(p^m;\mathbb{C})$, the etale classifying space of the projective linear group over $\mathbb{C}$. In particular, I will present a collection of nontrivial p-torsion classes in the Chow ring of $BPGL(p^m;\mathbb{C})$.

10:30-11:30 孙鹏

Chow groups and unramified cohomology of quadrics

In the 1990's, Karpenko computed Chow groups of quadrics based on the work of Swan from the K-theoretic point of view. In a series of papers, Kahn, Rost and Sujatha studied unramified cohomology of quadrics based on the work of Karpenko and tools from motivic cohomology. However, their results are valid with the exception of characteristic two. In this talk, I will explain the main ideas and show how these results can be adopted in characteristic two. These are based on joint works with Yong Hu and Ahmed Laghribi.

14:45-15:45 杨森

Algebraic K-theory, Hodge numbers and Chow groups

Let $X$ be a smooth projective variety over a number field $k$. We use algebraic K-theory to investigate deformations of the Chow group $CH^p(X)$. Provided that certain Hodge numbers of $X$ are trivial, for any local artinian $k$-algebras, we verify the pro-representability of the formal completion of $CH^p(X)$. This generalizes the work on the pro-representability of Chow groups by Bloch, Stienstra, Mackall et al.

Nanjun Yang: yangnanjun@bimsa.cn, Jin Cao: caojin@ustb.edu.cn