Isabel Longbottom

Isabel Longbottom

Mathematics graduate student at Harvard

Talks and presentations

Intro to equivariant homotopy theory

September 2024, Harvard Babytop Seminar

In Fall 2024, the babytop seminar was on Hill-Hopkins-Ravenel’s resolution of the Kervaire invariant one problem. In this talk, I gave a brief introduction to equivariant homotopy theory. In the non-equivariant setting, homotopy theory is concerned with topological spaces up to weak equivalence. Before we can do equivariant homotopy theory, we need an equivariant notion of weak equivalence. Through a selection of examples, we try to motivate the relevant definitions. We then discuss Elmendorf’s Theorem and how it gives us a very nice, concrete model for the \(\infty\)-category of G-spaces as presheaves on the orbit category. We conclude by saying a few words about equivariance in families. This talk was based on Chapter 1 of Blumberg’s Burnside Category, and there are notes I prepared.

Cyclotomic Frobenius and the Segal conjecture

August 2024, MIT Talbot Workshop

This talk was given as an attendee at the 2024 Talbot workshop, which was about THH of ring spectra. I gave a proof of the Segal conjecture for \(BP\langle n \rangle\) by first proving it for graded polynomial \(\mathbb{E}_2\)-\(\mathbb{F}_p\)-algebras, and then using (the décalage of) the Adams filtration on \(BP\langle n \rangle\) to reduce to the \(\mathbb{F}_p\)-algebra case. The talk was based on Section 4 of “Redshift and multiplication for truncated Brown-Peterson spectra” by Jeremy Hahn and Dylan Wilson. Notes from all the talks given at Talbot 2024 will eventually be available on the workshop website.

Rational homotopy of the K(n)-local sphere

April 2024, MIT Babytop Seminar

In Spring 2024, the babytop seminar was about the two (p-adic) towers and the paper “On the rationalization of the K(n)-local sphere” by Barthel, Schlank, Stapleton, and Weinstein. I gave the last talk of the semester, explaining how to finish the proof of Theorem B by computing the proétale cohomology of the Drinfeld symmetric space and the rigid analytic open unit disc. My talk covered Sections 6 and 3.9 of the paper.

The Hopf invariant one problem

April 2024, Harvard Zygotop Seminar

An expository talk given for zygotop in Spring 2024, a learning seminar for young graduate students interested in homotopy theory and related areas. I gave an introduction to the Hopf invariant one problem and its solution, including the classical results relating real division algebra structures on \(\mathbb{R}^n\), parallelizability of \(S^{n-1}\), H-space structures on \(S^{n-1}\), and elements of Hopf invariant one in \(\pi_{2n-1}(S^n)\). I defined the Hopf invariant in terms of complex topological K-theory, and gave Adams-Atiyah’s proof (using the splitting principle and Adams operations on topological K-theory) that elements of Hopf invariant \(\pm 1\) exist only for \(n = 1, 2, 4, 8\).

\(K\)-theory of the \(K(1)\)-local sphere via \(TC\)

February 2024, Harvard Thursday Seminar

An expository talk given for the Harvard Thursday Seminar in Spring 2024, which was on the topic of the recent paper “K-theoretic counterexamples to Ravenel’s telescope conjecture” by Burklund, Hahn, Levy, and Schlank. In this talk, I stated and gave proofs of the key theorems from “The algebraic \(K\)-theory of the \(K(1)\)-local sphere via \(TC\)” by Levy, including computation of the \(K\)-theory of the \(K(1)\)-local sphere in terms of \(TC\) of the Adams summand, and an analogue to the Dundas-Goodwillie-McCarthy theorem for certain \((-1)\)-connective rings. I also explained the Land-Tamme $\odot$ construction, which, given the input of a pullback square of \(\mathbb{E}_1\)-rings, allows comparison with a closely related commuting square which becomes a pullback under \(K\)-theory or another localizing invariant. The relevance of this talk to the seminar is that the counterexample to the telescope conjecture at height 2 is \(K(L_{K(1)}\mathbb{S})\). Notes are available.

Sylow Theorems for Infinity Groups

November 2023, Harvard Trivial Notions Seminar

Taking the fundamental group gives an equivalence of categories between pointed spaces whose homotopy groups \(\pi_n\) for \(n \neq 1\) all vanish, and groups. From this perspective, group theory can be viewed as the truncation of a more general theory of pointed connected spaces. Then the natural question is: to what extent can we do group theory in this new homotopical setting (where we don’t require the higher homotopy groups to vanish)? In this talk, we translate the Sylow theorems for finite groups to the context of finite $\infty$-groups, and use this to get a group-theoretic classification of finite nilpotent spaces, by analogy to the classification of finite nilpotent groups. We also discuss a failure of normality for $\infty$-groups, as something of a cautionary tale. Notes are available, and the primary reference is “Sylow theorems for \(\infty\)-groups” by Prasma and Schlank.

\(K\)-Theory from the Ground Up

October 2023, Harvard Zygotop Seminar

A 2-hour expository talk given for zygotop in Fall 2023, a learning seminar for young graduate students interested in homotopy theory and related areas. I gave an introduction to algebraic \(K\)-theory, starting from the classical perspective and then spending some time on the universal constructions of connective and non-connective \(K\)-theory (in terms of the universal additive and localising invariant, respectively). I also discussed Waldhausen’s \(wS_\bullet\) construction, and we concluded with a summary of Quillen’s computation of the \(K\)-theory of finite fields via the \(BGL(R)^+\)-construction. Notes are available.

On the \(K\)-theory of pullbacks and pushouts

May 2023, Harvard Zygotop Seminar

A 2.5-hour expository talk given for zygotop in Spring 2023, a learning seminar for young graduate students interested in homotopy theory and related areas. I gave an overview of the Land-Tamme paper “On the \(K\)-theory of pushouts” shortly after it was uploaded to the arXiv. I stated and proved their main results, namely that we can measure the failure of excision for localising invariants (think \(K\)-theory) via the \(\odot\)-construction, and that this \(\odot\)-construction can be expressed as a pushout in \(\text{Alg}(k)\) in good circumstances. This makes it computable in many situations of interest. The talk finished with a tour of some applications.

Abouzaid-Blumberg Flow Categories

March 2023, MIT Juvitop Seminar

A 1-hour talk given for Juvitop in Spring 2023, which was about Floer Homotopy Theory. After the work of Abouzaid-Blumberg, I defined (framed) flow categories and flow bimodules, and constructed the infinity category of flow categories. I gave the proof that in the framed case, this is a stable infinity category and in fact equivalent to the category of finite spectra. Auxiliary material included some background on stability and how to recognise the category of finite spectra.

A Proof of the Thom-Pontryagin Theorem

November 2022, MIT Kan Seminar

A 50-minute talk given for the MIT Kan seminar in Fall 2022. I gave a proof of the Thom-Pontryagin theorem, which states that as graded rings, the cobordism ring with a given universal tangential structure is isomorphic to the stable homotopy groups of the corresponding Thom spectrum. I focused first on the case of framed cobordism giving rise to the stable homotopy groups of spheres, and then explained how to generalise.

On the Cobordism Ring and a Complex Analogue, after Milnor

October 2022, MIT Kan Seminar

A 50-minute talk given for the MIT Kan seminar in Fall 2022. I read and presented Milnor’s “On the Cobordism Ring $\Omega$* and a Complex Analogue,” giving a modern treatment of the material in terms of spectra.

Reflections of Quiver Representations

November 2021, ANU, Mathematical Sciences Institute

A 15-minute talk giving an introduction to derived reflection functors. We start with basic definitions, introduce the bounded derived category of quiver representations, and explain how derived reflection functors are induced from reflection functors. The upshot is that for an acyclic graph, any choice of edge orientations gives rise to the same derived category, with explicit isomorphisms via derived reflections. The video is available on Youtube, and here are the slides as a PDF.