Spring Semester 2026

Computer-Assisted Computations of the Adams Spectral Sequence E2-Page

Large-scale computer calculations are playing an increasingly important role in stable homotopy theory. A landmark example is the recent work of Lin, Wang, and Xu on the Kervaire invariant problem, which relied heavily on Weinan Lin's program for computing the Adams spectral sequence at the prime 2.

We extend this computational approach to odd primes. By developing new algorithms tailored to the more complex structure of the odd-primary Steenrod algebra, our program achieves significant improvements in efficiency. This has enabled us to completely determine the algebra structure of the E2-page in a large range. Our results establish a foundational computational resource for future investigations in odd-primary stable homotopy theory. This is joint work in progress with Weinan Lin.

Surgeries on knots and tight contact structures

The existence and nonexistence of tight contact structures on 3-manifolds are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structures, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not smooth (2n-1)-surgery along the T(2,2n+1) torus knot for any positive integer n. Surprisingly, since then no other example of a 3-manifold without tight contact structures has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.

Singular Riemannian Foliations and Their Applications

We first introduce the theory of singular Riemannian foliations (SRFs), which provide a natural geometric generalization of orbit decompositions of isometric group actions. We then discuss how SRFs interact with various geometric and topological problems, including the existence of closed geodesics, curvature constraints, and topological restrictions on the existence of SRFs. Finally, we survey recent progress on these topics.

On the topology and geometry of smooth manifolds looking like CP^3 x S^7

In this talk, we employ modified surgery theory to classify simply connected closed smooth 13-manifolds with cohomology ring isomorphic to that of CP^3 x S^7, up to diffeomorphism, homeomorphism, and homotopy equivalence. These classifications are governed by the first Pontryagin class. As an application, we investigate linear free S^1-actions on S^7 x S^7 whose quotient manifolds belong to the above class. Moreover, we show that any real vector bundle of rank m >= 8 over CP^3 with nonnegative first Pontryagin class, as well as its unit sphere bundle, admits a metric of nonnegative sectional curvature.

Clifford Modules, Spinor Bundles, and Quadratic Forms on Spheres

In the 1960s, Atiyah, Bott, and Shapiro used Clifford modules to construct spinor bundles on spheres. Spinor bundles represent the nontrivial generators in the real K-theory of spheres (aka. KO-theory), which underlies the Bott periodicity. In 1969, Fossum proved that the comparison map from the algebraic K-theory of spheres to their topological KO-theory is surjective, and Swan later proved that this map is in fact an isomorphism. This shows that, to some extent, algebraic and topological real vector bundles agree for spheres. In 1977, Knebusch noted that quadratic forms on vector bundles over spheres are natural objects to study. In this talk, I will give an explicit construction of quadratic forms on spinor bundles and explain how this construction suggests that algebraic vector bundles over spheres, even with quadratic forms, can still correspond to topological real vector bundles.

A lower bound of v_n-self maps of generalized Moore spectra

Chromatic periodicity theorem guarantees the existence of generalized Moore spectra of the form S/(p, v_1^{i_1}, ..., v_n^{i_n}). At small primes, the powers i_k cannot be 1. In this talk, at the prime 2, we present a lower bound of these i_k by determining minimal periodicities of their E_{h_k}^{hG}-homology for each height k. This is joint work in progress with Zhipeng Duan, Guozhen Wang, and Wei Yang.

Embedding spaces of arcs and pseudo-isotopies of 3-manifolds with infinite fundamental groups

In recent years, the embedding spaces of arcs in 4-manifolds have emerged as a significant tool in the study of 4-dimensional topology. Given any 3-manifold Y with an infinite fundamental group, we show that both the topological and smooth mapping class groups of Y x I are abelian groups of infinite rank. This is achieved by constructing an infinite rank subgroup consisting of pseudo-isotopies of Y. Our proofs rely on a detailed analysis of the actions of barbell diffeomorphisms on the spaces of embedded arcs and the configuration spaces of points. This is joint work with Jianfeng Lin and Boyu Zhang.

Profinite transfers in K(n)-local homotopy theory

The classical J-homomorphism is a map from the orthogonal group O(n) to the iterated loop space Omega^n S^n of the sphere S^n. After K(1)-localization, the stable J-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map Sigma^-1 KO_2^wedge to S_K(1) from the C_2-homotopy fixed points (with a twist) to the Z_2^x-homotopy fixed points of the 2-complete complex topological K-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava E-theory by closed subgroups of the Morava stabilizer group in K(n)-local homotopy theory. We introduce two definitions of the profinite transfer maps. One working definition is as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map Sigma^{-n^2}E_n to S_K(n) on homotopy groups is the filtration n^2-line in the homotopy fixed point spectral sequence. A second definition of the profinite transfer maps is based on the 6-functor formalism for smooth representations of p-adic Lie groups by Heyer-Mann. We prove that the two definitions are equivalent.

On the transfer homomorphism in the homology of symmetric groups

The homology of symmetric groups Sigma_n over a finite field F_p has been extensively studied. By utilizing the p-adic expansion of n, this computation fundamentally reduces to studying the homology of G = Sigma_{p^2}. A standard approach relies on the surjective map induced by the inclusion of the wreath product H = Sigma_p wr Sigma_p, which allows every basis element i_*(x) in G to be represented by a basis element x in H. However, the transfer map from G to H has not been thoroughly explored, although Kahn-Priddy provided a complete formula for the p = 2 case. One method to address this is by analyzing the double coset decomposition with respect to the elementary abelian p-subgroup L = C_p x C_p. This results in a complex combinatorial formula mapping the homology of L to H. In this paper, we resolve this combinatorial complexity, proving that the transfer map sends every basis element i_*(x) in G to the corresponding element x in H with a coefficient of 1, up to a linear combination of other elements.

On topological (bi)quandle

A quandle is an algebraic structure analogous to a group. It essentially replaces the associative law in the definition of a group with the distributive law, which reflects its close connection with the Yang-Baxter equation. In this talk, I will give a quick introduction to the basics of quandle theory, with a particular focus on the construction and applications of topological quandles and topological biquandles. The questions raised in this talk may be more than the results it contains, which reflects how much we still remain ignorant about this research topic.

代数拓扑中的乘法结构与机械化计算

自上世纪中叶以来，上同调环与Steenrod代数等乘法结构的引入，极大地深化了我们对流形与空间同伦型的理解。 本报告中，我们将介绍代数拓扑中丰富的乘法理论，阐释乘法结构如何成为撬动高维几何与拓扑问题的最有力杠杆， 以及我们在高效机械化计算稳定同伦范畴中乘法结构的算法工作。

Equivariant bordism and Thom spectra

In this talk, we explore the relation between bordism theories and Thom spectra in equivariant homotopy theory. For a compact Lie group G, we describe a version of G-equivariant geometric bordism with respect to a given stable tangential structure, and we define an equivariant Thom spectrum functor using parametrized homotopy theory. Equivariant bordism admits a Thom-Pontryagin comparison map to the homology theory represented by the associated Thom spectrum. The main result is that the Thom-Pontryagin map is an isomorphism when the connected component of G is central, for example if G is a product of a finite group and a torus. And when G is not of this type, geometric bordism is not represented by a genuine G-spectrum as it fails to admit certain Wirthmuller isomorphisms. This extends work of tom Dieck and Schwede for unoriented bordism to very general tangential structures.

非正曲率度量的模空间的拓扑

给定一个具有非正截面曲率黎曼度量的光滑闭流形M。流形M上所有非正截面曲率黎曼度量构成的空间在M的 自微分同胚群作用下的商空间称为该流形的非正曲率度量的模空间。在本报告中，我们将介绍关于该模空间的 拓扑的一些相关研究进展。

Finite covers of surfaces and the effective Ehrenpreis conjecture

Let M and N denote two closed hyperbolic Riemann surfaces. The Ehrenpreis conjecture asserts that, for any epsilon > 0, there exist finite covers M_1 to M and N_1 to N such that the distance between M_1 and N_1 in the moduli space is less than epsilon. Kahn and Markovic proved this conjecture using the pants gluing idea and exponential mixing of the geodesic flow. A natural question is how large the degrees of these covers must be, which is known as the effective version of the Ehrenpreis conjecture.

In this talk, I will first explain how we can construct finite covers of Riemann surfaces using dynamical methods. Then I will present my results on the effective Ehrenpreis conjecture. I provide a polynomial upper bound on the degree, and show that this bound is optimal by considering the case when M and N are arithmetic. This talk is based on arXiv:2601.02710.

Chromatic redshift for rigid categories

Recent years have seen tremendous progress in realizing Ausoni and Rognes' chromatic redshift philosophy, which predicts roughly that for nice enough ring spectra, algebraic K-theory raises chromatic height by 1. In particular, this philosophy was made precise and spectacularly confirmed for commutative ring spectra. However, K-theory is fundamentally a categorical invariant, and this resolution left hanging the question of the generality in which one could hope a similar kind of redshift for rigid symmetric monoidal stable categories, the kinds of categories that are closest to categories of modules over commutative ring spectra. In this talk, after discussing this background and the relevance of the rigidity condition, I will describe examples that show that redshift fails for rigid categories, showing that the lower bound in redshift is a ring-theoretic phenomenon.

On the simplest simply connected non-spin rational homology 7-spheres that are not 2-connected

Rational homology spheres occupy a distinguished position in topology, encoding torsion phenomena. In low dimensions they already play an important role, and in higher dimensions they provide a testing ground for surgery theory and manifold classification. For simply connected cases, the classification is well understood in dimensions five and six, whereas dimension seven exhibits much richer topology. This is reflected in Milnor's lambda-invariant, originally defined for homotopy 7-spheres, detecting exotic smooth structures on the 7-sphere and extending to all rational homology 7-spheres. In this talk, we classify the simplest non-spin non-2-connected cases, showing that these manifolds are completely determined by the lambda-invariant.

An almost flat spinc manifold bounds

We prove that every almost flat spinc manifold bounds a compact orientable manifold, thereby settling, in the spinc case, a long-standing conjecture of Farrell-Zdravkovska and S. T. Yau. This is a joint work with Fei Han and Weiping Zhang.