## Spring 2023

This semester Juvitop will be about Floer Homotopy Theory.

We meet at 3:59 on Wednesday in 2-151 unless otherwise noted.

• Feb 082023
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### Natalie Stewart

This talk will be a bird's eye view of the foundations of stable homotopy theory; without delving into models, we will sample some classical homotopy theory in the language of infinity categories. We will define stable infinity categories and t structures, emphasizing the derived category as a guiding example. We will conclude by defining the infinity category of spectra and listing some properties, including its relationship to spaces and the derived category.

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• Feb 152023
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### References:

Review Morse functions, Morse lemma, Morse charts, (pseudo)gradient flows, stable and unstable manifolds. Define the Morse complex/Morse homology. Maybe discuss how the moduli of gradient flows can be compactified to the moduli of broken trajectories as a teaser trailer for next time.

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• Feb 222023
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### Cameron Krulewski

Given a finite-dimensional, compact Riemannian manifold X and a Morse-Bott function f from X to R satisfying the Smale transversality condition, we can construct the flow category. From this category, we would like to be able to recover the stable homotopy type of X. I will discuss the structure carried by moduli spaces of gradient flow lines and how this provides the information to recover the suspension spectrum of X. This talk is a warm-up for later, when we generalize to the infinite-dimensional case.

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• Mar 012023
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• Mar 082023
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### Natalia Pacheco-Tallaj

Describe how the boundary strata of the Morse moduli space give a Maurer Cartan element in a graded spectrum.

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• Mar 152023
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### References:

Define the moduli stack of broken lines and factorizable spaces over it. Relate the Abouzaid-Blumberg and Lurie-Tanaka pictures: an object in Abouzaid-Blumberg's category gives a factorizable space over Broken with framing.

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• Mar 222023
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### Clair Xinle Dai

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• Mar 292023
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### None

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• Apr 052023
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### References:

For spin manifolds with almost complex structure, the Floer homotopy type determined by the smooth, compact flow category defined in CJS is given by the suspension spectrum of the free loop space.

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• Apr 122023
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### References:

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• Apr 192023
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### Daniel Alvarez-Gavela

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• Apr 262023
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### Zihong Chen

Define Fukaya categories and show they are A_∞.

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• May 032023
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### Mary Stelow

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• May 102023
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### TBD

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This seminar is organized by Cameron Krulewski, Ishan Levy, Natalia Pacheco-Tallaj, and Mary Stelow.