召集人：方复全（首都师范大学、南方科技大学）、Stavros Garoufalidis（南方科技大学）、王国祯（复旦大学）、徐宙利（美国加利福尼亚大学圣迭戈分校、加利福尼亚大学洛杉矶分校）

时间：2024.09.08—2024.09.14

Workshop on Computer-assisted Research in Geometry and Topology

MONDAY 9 SEP

TUESDAY 10 SEP

WEDBESDAY 11 SEP

THURSDAY 12 SEP

FRIDAY 13 SEP

 Speakers & Short Talks

Invited Speakers (A–Z)

Haibao Duan

Chinese Academy of Sciences

Make Schubert calculus calculable

Hilbert’s 15th problem is an inspiring and far-reaching one. It promotes the enumerative geometry of the 19th century growing into the algebraic geometry founded by Van der Waerden and Andr´e Weil, and makes Schubert calculus integrated into many branches of mathematics. However, despite great many achievements in the 20th century, the part of the problem of finding an effective rule performing the calculus has been stagnant for a long time, notably, the Schubert’s problem of characteristics, or the Weil’s problem on the intersection theory of flag manifolds G/P , where G is a compact connected Lie group and P a parabolic subgroup.

This talk surveys the background, content, and solution of the 15th problem. Our main results are a unified formula for the characteristics, and a systematic description of the intersection theory of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples.

The integral Weyl invariants of compact Lie groups

Let G be a compact connected Lie group with classifying space B_G and Weyl group W . We introduce a new method (algorithm) to construct the ring of integral Weyl-invariants of G from the rational cohomology of the classifying space B_G.

Guchuan Li

Peking University

Examples of Periodic Families in Homotopy Groups of C-Motivic Spheres

Periodic phenomena play important roles in homotopy theory, such as the Bott periodicity theorem and Adams’ image of J . In motivic homotopy theory, there are more periodic families than in the classical case. We define a g-operator on the C-motivic homotopy groups of spheres and prove the existence of some non-trivial g-periodic families within these groups. This talk is based on joint work in progress with Dan Isaksen, Hana Jia Kong, Yangyang Ruan, and Heyi Zhu.

Weinan Lin

Fudan University

Programming in the computation of homotopy groups of CW spectra, I

In this talk, I will introduce some algorithms which can help us compute the homotopy groups of CW spectra. This involves the computation of the starting pages of spectral sequences, the maps between spectral sequences, differentials and extensions.

Programming in the computation of homotopy groups of CW spectra, II

In this talk, I will give some detailed instructions on how to use the program.

Emanuel Scheidegger 

Peking University

Periods from geometry and from modular forms

According to Kontsevich and Zagier, periods are numbers arising as integrals of algebraic functions over domains described by algebraic equations or inequalities with rational coefficients. Periods are ubiquitous in geometry but also in number theory. I will present two areas, complex geometry and modular forms, where they arise and very recent conjectures lead to new relations. I will explain how to compute these periods and how to verify these relations with computer algebra programs.

XiaoLin Danny Shi

University of Washington

Transchromatic phenomena in the equivariant slice spectral sequence

In this talk, we will construct a stratification for the equivariant slice spectral sequence. This stratification is achieved through the localized slice spectral sequences, which compute the geometric fixed points equipped with residual quotient group actions. As an application, we will utilize this stratification to investigate norms of Real bordism theories and their quotients. These quotients hold significant importance in Hill–Hopkins–Ravenel’s resolution of the Kervaire invariant one problem, as well as in the study of fixed points of Lubin–Tate theories by finite subgroups of the Morava stabilizer group. For these theories, the stratification exhibits a transchromatic phenomenon: the slice spectral sequence of a higher height theory is stratified into distinct regions, each isomorphic to the slice spectral sequences of the lower height theories. This provides an inductive approach and various structural insights when computing the fixed points of Lubin–Tate theories.

Zhouli Xu

University of California San Diego & University of California Los Angeles

Homotopy Groups of Spheres

Up to continuous deformations, all based continuous maps between two spheres form an abelian group, and is called a homotopy group of the target sphere. It turns out that determination of these groups is a very hard problem in topology. The structures of the homotopy groups of spheres are closely related to many topics in topology, such as the Hopf invariant problem, the Kervaire invariant problem, and the number of smooth structures on a given sphere.

In this talk, I will review some classical methods of computing these groups, and discuss some recent progress using motivic homotopy theory and other techniques. This is based upon joint work with Bogdan Gheorghe, Dan Isaksen, Weinan Lin and Guozhen Wang.

Computing differentials in the Adams spectral sequence and the Last Kervaire Invariant Problem

I will review classical methods computing differentials in the Adams spectral sequence, and then discuss some recent progress in joint work with Weinan Lin and Guozhen Wang. In particular, I will discuss the fate of h₆², resolving the Last Kervaire Invariant Problem in dimension 126.

Di Yang

University of Science and Technology of China

Enumerative invariants in large genera

In this talk, we study combinatorial properties and asymptotics of certain enumerative invariants. We will mainly focus on the large genus asymptotics of psi-class intersection numbers. These numbers appeared in Witten’s famous conjecture (proved by Kontsevich), and their computations brought a lot of interest recently. In particular, we will give a proof of the polynomiality conjecture regarding the large genus. Other numbers (BGW numbers, r-spin intersection numbers, etc.) that we will mention turn out to have properties similar to psi-class intersection numbers. New applications will also be discussed briefly.

Xuezhi Zhao

Capital Normal University

The ring structures of cohomologies of flag manifolds

The integer cohomology groups of complex flag manifolds are known to be free abelian groups, because all cells of manifolds are even-dimensional. We shall show how to obtain the ring structures of the cohomologies, based on multiplication rule on additive bases. Further computations are also illustrated, such as torsion indices. From arithmetic point of view, our computations include: (i) finding integer solutions to system of linear equations, (ii) solving division of polynomials with integer coefficients. Our concern is: how to avoid appearance of very large integers during computation.

Foling Zou

Chinese Academy of Sciences

An equivariant computation of tmf

Equivariant homotopy and homology theories are invariants of spaces or spectra with group actions. In this talk, we focus on cyclic groups of order 2 and 3, C₂ and C₃. I will start by describing the algebraic structures that equivariant theories carry and in particular the equivariant homology of a point, which is a non-trivial bi-graded ring. Then, I will explain a C₃-equivariant computation of the Borelification of tmf(2). We used a relative Adams spectral sequence, with input from the Hopf algebroid structure of the C₃-equivariant dual Steenrod algebra. This yields an entirely algebraic computation of the 3-local homotopy groups of tmf, the topological modular form spectrum. This is joint work with Jeremy Hahn, Andrew Senger, and Adela Zhang.

Short Talks 

Rixin Fang 

Fudan University

Algebraic K-theory and redshift phenomena

There are many recent progress in algebraic K-theory, as well as many ingredient in the proof of counterexample for telescope conjecture. We will give motivation about redshift conjecture, and present some recent techniques. In the end, we will discuss K-theory of some ring spectra constructed based on truncated Brown–Peterson spectra.

Zhou Fang

Southern University of Science and Technology

Eigenbundles of energy bands and Higgs bundles

In this talk, I will explain what energy bands are and show that Higgs bundles are useful to describe the topology of energy bands.

Wenbo Liao

Chinese University of Hong Kong

An Alexander Polynomial for Spatial Graphs and the Trapezoidal Conjecture

We introduce an Alexander polynomial for spatial graphs by generalizing Kaufmann states sum. Let G⊂S³ be a spatial graph and G be any planar projection of G. We define the Kaufmann states of G and show the state sum is independent of the choice of G so that we get a well-defined invariant of G, called Alexander polynomial. By using this new invariant, we give a necessary condition for spatial graphs being planar graphs and give an intrinsic invariant of graphs. Also, we briefly introduce our recent proof on the trapezoidal conjecture for planar graphs and relate it to the trapezoidal conjecture for alternating knots.

 Xuecai Ma

Westlake University

Derived Moduli Problems and New Cohomology Theories

We will discuss several derived moduli problems in the context of spectral algebraic geometry and their applications in algebraic topology. Inspired by Lurie’s work on conceptual explanations of elliptic cohomology and Morava E-theories, we study certain derived moduli problems, which we refer to as derived level structures, generalizing Katz and Mazur’s results on level structures. By analyzing the affine representability of certain moduli problems associated with derived level structures, we develop new generalized cohomology theories. Specifically, when investigating the derived version of full-level structures, we obtain new spectra which we call Jacquet–Langlands spectra. We then study homotopy fixed points of these spectra under the action of Morava stabilizer groups, which enables us to derive the Langlands duals of Morava E-theories.

Qingrui Qu

Southern University of Science and Technology

Periodicity in classical and motivic homotopy theory

In this talk, I will introduce the concept and current research status of exotic periodicity in motivic homotopy theory. These phenomena are more complicated than the classical chromatic cases and still not fully understood.

Yifan Wu

Southern University of Science and Technology

K(1)-Localized Morava E-Theory and p-Adic Galois Representations

In this talk, I will introduce some basic ideas of p-adic Galois representations, and discuss the potential relationship between them and the K(1)-localized Morava E-theory of height 2.

Participants