Spring 2026 Algebraic topology seminar

Abstract: Recently, Tadayuki Watanabe disproved the Smale Conjecture, which stated that the group $\mathrm{Diff}_\partial(D^4)$ is homotopically trivial. He showed that there are nontrivial homotopy groups $π_q \mathrm{Diff}_\partial(D^4) \otimes \mathbb{Q}$. The first key idea was to exhibit smooth $D^4$-bundles using trivalent graphs equipped with Hopf and Borromean links as their blueprints. The second key in the construction was the use of Kontsevich’s configuration space integral to detect the non-triviality of such bundles.

These ideas were generalized by Watanabe for higher-dimensional disks $D^d$, $d \geq 4$. I will explain some of the ideas and constructions of Watanabe’s work. In our very recent joint work, we used Brunnian links to construct a chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large. Then we use again Kontsevich’s configuration space integral to detect the nontriviality of such homology elements.

In particular, we provide new constructions of non-trivial elements in the homotopy groups $\pi_{8k−10}(B \mathrm{Diff}_\partial(D^{2k})) \otimes \mathbb{Q}$ (for $k \geq 17$) which are derived from well-known cycles in the graph complex.

Abstract: We use Teichmüller theory to construct a new geometric model for the classifying space of the mapping class group of a three-dimensional handlebody. Two consequences are obtained: (i) Chan-Galatius-Payne have recently shown that the homology of Kontsevich's commutative graph complex injects into the homology of the mapping class groups of surfaces, producing an enormous amount of highly unstable homology classes. We show that this homomorphism factors through the homology of the corresponding handlebody mapping class groups. (ii) The handlebody mapping class group is a virtual duality group in the sense of Bieri-Eckmann, with dualizing module given by a certain complex of nonsimple disk systems; the analogous result for mapping class groups of surfaces is a theorem of Harer. (Joint with Louis Hainaut and with Ric Wade.)

Abstract: Graph complexes, as originally defined by Kontsevich, are chain complexes over the rational numbers, defined explicitly by generators and relations. They appear in surprisingly many areas of topology. I will briefly survey their connection to operads, and discuss how parts of the theory may be upgraded from rational chain complexes to integral and even space-level objects.

Abstract: Fix a field $k$ of positive characteristic. We consider the stabilization of Toën's category of affine higher stacks over $k$, referred to as ''unipotent spectra.'' This construction is motivated by stable homotopy-theoretic ideas and gives rise to unipotent stable homotopy groups and unipotent homology, which are new invariants for $k$-schemes valued in unipotent group schemes.
We discuss how the formal groups associated to schemes of Artin--Mazur may be recovered from unipotent homology. Further, we consider a variant obtained by restricting to sites of perfect schemes. We show that syntomic cohomology and Milne's arithmetic duality admit natural refinements to this setting.
This is joint work with Shubhodip Mondal and Tasos Moulinos.

Abstract: Tambara functors are a generalization of commutative rings which model multiplicative structures that arise in equivariant homotopy theory. There is a robust analogy between the algebraic theory of Tambara functors and classical commutative algebra, giving us intuition for how constructions in the equivariant setting should be carried out. In this talk, I will discuss how some recent results on the theory of radical and prime ideals in Tambara functors. In both cases, parts of the analogy with commutative rings work perfectly, while others parts are more subtle.

Abstract: String topology studies algebraic structures arising from the interactions of loops and paths on a geometric space. The subject originated in 1999 with work of Chas and Sullivan who discovered that classical Poincaré duality and intersection theory has a rich manifestation on the homology of the free loop space, recovering structures also appearing in mathematical physics such as BV-algebras, TQFTs, and Lie bialgebras. This insight emerged from the broader question: what characterizes the algebraic topology of manifolds? Since then, string topology has developed into a vibrant area, revealing a wealth of new operations describing string interactions, with deep connections to knot theory, symplectic geometry, homotopy theory, homological algebra, and mathematical physics. In this talk, I will survey some developments from the past decade regarding new understanding of the structure and computations in string topology as well as their significance in other fields of mathematics, with focus on operations that capture geometric information beyond the oriented homotopy type of the underlying manifold.
This talk will be followed by a more technical second talk at the CUNY Graduate Center on Friday (details here).

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract: