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Maths Lecture Theatre

School of Mathematical Sciences

Queen Mary University of London

School of Mathematical Sciences

Queen Mary University of London

Contact: b.noohi@qmul.ac.uk

**Jack Davies:**An introduction to algebra in stable homotopy theory, Part 1Abstract. The objects in the stable homotopy category are called spectra, which are a homotopy-theoretic concept representing the generalised cohomology theories of Eilenberg--Steenrod. Various examples and constructions of spectra will be given, as well as a survey of common tools and techniques used by modern homotopy theorists. There will be a focus on the analogy to the derived category of an abelian category and also the so-called chromatic perspective. In this mini-series, we will not assume any background in stable homotopy theory.

**Noah Porcelli:**Floer homology and homotopy typesAbstract. Floer homology is the main tool currently used in symplectic geometry, and supports a rich family of algebraic structures, which in many cases mirrors structures in algebraic geometry. Recently progress has been made in finding stable-homotopical enrichments. We'll outline the construction of Morse and Hamiltonian Floer homologies, and how in some cases they can be realised as the homology of a spectrum. No symplectic background will be assumed.

**Jack Davies:**An introduction to algebra in stable homotopy theory, Part 2Abstract. The objects in the stable homotopy category are called spectra, which are a homotopy-theoretic concept representing the generalised cohomology theories of Eilenberg--Steenrod. Various examples and constructions of spectra will be given, as well as a survey of common tools and techniques used by modern homotopy theorists. There will be a focus on the analogy to the derived category of an abelian category and also the so-called chromatic perspective. In this mini-series, we will not assume any background in stable homotopy theory.

- Coffee 🍵
**Drew Heard:**The spectrum of Goodwillie calculusAbstract. We will explain a recent computation of the Balmer spectrum of d-excisive functors from finite spectra to spectra. We will spend the majority of the time trying to explain exactly what this means. This is joint work with Arone, Barthel, and Sanders.

- Lunch
**Neil Strickland:**Recent progress in chromatic homotopy theoryAbstract. Chromatic homotopy theory is a branch of algebraic topology which has seen an explosion of new activity in recent years. In particular, in the summer of 2023 Burklund, Hahn, Levy and Schlank announced the negative resolution of Ravenel's Telescope Conjecture, which had been open since 1984. That conjecture said that two categories are the same; now we know that they are different, and we expect to find a rich and intricate structure in the gap between them. In this talk I will attempt to survey, for a general pure mathematical audience, the main ideas of chromatic homotopy theory and some highlights of recent progress.

- Coffee 🍵
**Constanze Roitzheim:**Homotopy theory of finite total orders, trees and chicken feetAbstract. A transfer system is a graph on a lattice satisfying certain restriction and composition properties. They were first studied on the lattice of subgroups of a finite group in order to examine equivariant homotopy commutativity, which then unlocked a wealth of links to combinatorial methods and other homotopy theory. On a finite total order [n], transfer systems can be used to classify different model category structures on [n]. The talk will involve plenty of examples and not assume any background knowledge.

**Jack Davies:**Topological modular forms and stable Hecke operatorsAbstract. The cohomology theory of topological modular forms is a derived algebro-geometric refinement of the usual ring of modular forms. Inspired by the classical constructions, we present an algebra-geometric perspective to construct Hecke operators on topological modular forms. We would like to highlight some of the immediate applications of these operations, some of their structural properties, and some future potential directions.

- Coffee 🍵
**Noah Porcelli:**Floer homotopy theory and smooth structures on LagrangiansAbstract. The Fukaya category is an A_\infty category over \Z associated to certain symplectic manifolds which is important in the study of Lagrangian submanifolds, particularly detecting homological/homotopical information. We use Floer homotopy theory to prove that in some cases the Fukaya category sees more: it can detect information about smooth structures on certain Lagrangian submanifolds. Based on joint work-in-progress with Ivan Smith.

- Lunch
**Özgür Bayındır:**Algebraic K-theory of the two-periodic first Morava K-theoryAbstract. Using a root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of the algebraic K-theory of the complex K-theory spectrum. Furthermore, our computational methods also provide the algebraic K-theory of the two-periodic Morava K-theory spectrum of height 1.

- Coffee 🍵
**Clark Barwick:**World-objects and quantum field theories in arithmeticAbstract. Work of Kim and Ben-Zvi−Sakellaridis−Venkatesh suggests strongly the possibility and promise of quantum field theories on objects in arithmetic geometry. But it's not at all clear how one might go about making such notions precise. In this talk, we will describe our first stab at doing so by categorically extending the theory of factorization algebras to these contexts. Along the way, we uncover a strange piece of additional structure that generally remains implicit in discussions of factorization algebras in other contexts.

- Behrang Noohi (Queen Mary University of London)

- Behrang Noohi (Queen Mary University of London)

- Clark Barwick (The University of Edinburgh)
- Özgür Bayındır (City, University of London)
- Jack Davies (Universität Bonn)
- Drew Heard (Trondheim University)
- Noah Porcelli (Imperial College London)
- Constanze Roitzheim (University of Kent)
- Neil Strickland (The University of Sheffield)

We are, unfortunately, unable to provide accommodation for participants. If you would like to book accommodation, the following are some of the options among the huge variety of other possibilities.

The theme of this meeting is (loosely) around arithmetic and Galois theoretic aspects of stable homotopy theory, with a little bit of geometry (Floer homotopy theory). We hope it will be attractive to researchers in Number Theory and Geometry who are keen to explore possible connections to homotopy thoery. The research talks will take place on Tuesday-Wednesday, January 16-17. In addition, there will be introductory talks by Jack Davies (on basics of Stable Homotopy Theory) and Noah Porcelli (on Symplectic Geometry and Floer theory) on Monday, January 15.

**Registration** is now open: registration form. Please register by midnight Sunday, 14 of January. You do not need to register to attend, but registration helps us estimate how much tea/coffee and biscuits will need to be provided by the catering team. (So if you show up without registering bring your own tea/coffee and biscuits! 🍵)