Spring 2025 Algebraic topology seminar

Abstract: Floer homotopy theory is the programme that seeks to upgrade various existing Morse/Floer theories from (co)chain complexes to module spectra over some E_\infty ring spectrum. In the first half of the talk, we review the general construction proposed in 1995 by Cohen, Jones, and Segal, which takes as input the data of all the higher dimensional moduli spaces of broken flow lines between critical points, together with suitable stable framings thereof. Then, we briefly illustrate how in the case of classical Morse-Bott theory on a closed smooth manifold M, the construction recovers the Thom spectra M^E for all reduced KO-theory classes [E] on M. In the second half, we show that one can still obtain weaker but non-trivial invariants (in the form of modules over Postnikov truncations of bordism spectra) if moduli spaces are smooth and compact only up to a certain dimension (as is often the case due to bubbling, e.g. in Instanton and Lagrangian Floer theories.) We end by illustrating how our construction can provide further constraints on the topology of the intersection between a monotone Lagrangian submanifold L of (M, $\omega$), and a Hamiltonian perturbation of L, and explore the concrete example of RP^n inside CP^n.

Abstract: Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour beyond the smooth case remains less well understood. Building upon recent advancements by Bachmann, Elmanto, Morrow, and Bouis, we establish its A^1-homotopty invariance for a broader class of "smooth" schemes. This is part of ongoing work in collaboration with Tess Bouis.

Abstract: Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic 0 are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic p admit canonical (and algebraisable) lifts to characteristic 0, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.

Abstract: How much of a closed smooth manifold M is captured by the homotopy type of its framed configuration spaces? This talk serves to make this question precise and to explain recent results in this direction in the case where M is an exotic sphere, obtained as part of joint work in progress with A. Kupers and F. Mezher.

Abstract: Multiple polylogarithms appear to be central to many seemingly unrelated areas of mathematics, including the volumes of hyperbolic polytopes, scissors congruence, algebraic K-theory, and special values of zeta functions. Despite this broad network of connections, the most fundamental properties of these functions, as predicted by the Goncharov program, remain conjectural. I will discuss recent progress in the Goncharov program, which is based on the connection between multiple polylogarithms and the Steinberg module. The talks are based on joint work with Steven Charlton and Danylo Radchenko, as well as on ongoing work with Alexander Kupers and Ismael Sierra.

Abstract: Work of Hill-Hopkins-Ravanel and Nardin pioneered a delicate relationship between (homotopy coherent) equivariant algebraic structures and genuine-equivariant notions of semiadditivity. The former is controlled by algebras over the commutative G-operad $\mathrm{Comm}_G$, and the latter by Wirthmüller isomorphisms in equivariant higher category theory. Motivated by equivariant chromatic homotopy theory, $\mathrm{Comm}_G$ was generalized by Blumberg-Hill to $\mathcal{N}_{\infty}$ operads, a family of sub-terminal objects of $\mathrm{Op}_G$ classified by subcategories $I \subset \mathbb{F}_G$ called indexing categories. Guided by discrete algebra, they conjectured an equivalence $\mathcal{N}_{I \infty} \otimes \mathcal{N}_{J \infty} \simeq \mathcal{N}_{I \vee J \infty}$. In this talk, I will establish that $\mathcal{N}_{I \infty}$ is universal among $G$-operads whose monoids possess $I$-indexed Wirthmüller isomorphisms and affirm Blumberg-Hill's conjecture as a corollary.

Abstract: The surface cobordism category $\text{Cob}_2^{SO}$ is the topologically enriched category with objects finite disjoint union of circles and spaces of morphisms are disjoint union of moduli spaces of surfaces. For any space X, we can define a category $\text{Cob}_2^{SO}(X)$ with objects 1-dimensional closed manifold equipped with a map to X and morphisms are surfaces equipped with a map to X compatible with the data on the boundary. In this talk, I will define a category $\text{Cob}_2^{SG}(X)$ with the same objects as $\text{Cob}_2^{SO}(X)$, but where morphisms are surfaces up to self-homotopy equivalences instead of diffeomorphisms. I will then describe a formula for the geometric realisation of the nerve $\text{BCob}_2^{SG}(X)$, as well as for the first polynomial approximation of the functor $\text{BCob}_2^{SG}(-)$.