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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  Crash Course on Spectra and Infinity Categories

  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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  Morse Theory

  Daniel Alvarez-Gavela

  References:

  • Audin-Damian, Morse Theory and Floer Homology

  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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  Morse Theory over Spectra

  Cameron Krulewski

  References:

  • Cohen-Jones-Segal, Morse Theory and Classifying Spaces
  • Cohen-Jones-Segal, Floer's infinite dimensional Morse theory and homotopy theory, sections 3 and 5

  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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  Abouzaid Blumberg Reformulation of Flow Categories

  Isabel Longbottom

  References:

  • Abouzaid's Floer Homotopy talks at Séminaire de Mathématiques Supérieures 2022: Floer Homotopy Theory (see also Amanda Hirschi's texed notes)
  • Blumberg-Abouzaid MSRI lectures 1-2: talk 1 talk2

  Flow categories, flow modules and their bordisms, quasicategory of flow categories, the quasi-category of flow categories is stable.

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• • Mar 082023
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  Morse Moduli Spaces and a Geometric Maurer Cartan Equation

  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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  The Moduli Stack of Broken Lines

  Ishan Levy

  References:

  • Lurie-Tanaka, Associative Algebras and Broken Lines

  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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  Classical Floer Theory in the Symplectic Setting

  Clair Xinle Dai
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• • Mar 292023
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  No Talk: MIT Spring Break

  None
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• • Apr 052023
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  Free Loop Space of a Symplectic Manifold and Symplectic Floer Theory

  Sanath Devalapurkar

  References:

  • Cohen-Jones-Segal, Floer's infinite dimensional Morse theory and homotopy theory
  • Cohen, Floer homotopy theory, revisited

  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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  Complex cobordism, Hamiltonion loops, and global Kuranishi charts

  Charlotte Kirchhoff-Lukat

  References:

  • Abouzaid-McLean-Smith
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• • Apr 192023
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  Floer theory and algebraic K-theory

  Daniel Alvarez-Gavela
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• • Apr 262023
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  Fukaya Categories

  Zihong Chen

  Define Fukaya categories and show they are A_∞.

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• • May 032023
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  Bauer Furuta Invariants

  Mary Stelow
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• • May 102023
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  TBD

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