Juvitop Seminar
Spring 2023
This semester Juvitop will be about Floer Homotopy Theory.
We meet at 3:59 on Wednesday in 2-151 unless otherwise noted.
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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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Daniel Álvarez-Gavela
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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Cameron Krulewski
References:
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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- 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
Isabel Longbottom
References:
Flow categories, flow modules and their bordisms, quasicategory of flow categories, the quasi-category of flow categories is stable.
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Ishan Levy
I will discuss the work of Lurie-Tanaka on the stack of broken lines, which is a topological stack recording families of flow lines and their degenerations, such as those appearing in Morse theory. I will explain how to present this stack as a Lie groupoid with corners and explain why factorizable sheaves on it are equivalent to non-unital algebras.
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No Talk: Harvard Spring Break
None
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Clair Xinle Dai
In the mid 80s, Floer invented his theory to prove the Arnold conjectures on the number of fixed points of Hamiltonian diffeomorphisms. There are many versions of Floer homology that have been constructed over the last thirty years. In this talk, I'll first talk about the history of the Arnold conjecture and then introduce the construction of Hamiltonian Floer homology.
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No Talk: MIT Spring Break
None
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Sanath Devalapurkar
References:
We'll show that the spherical Floer chains of T*M can be identified with the suspension spectrum of the free loop space of M. This'll be a consequence of the existence of framings on the compactified Floer moduli spaces, and it can be viewed as a version of the Legendre transform between Lagrangians and Hamiltonians.
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Daniel Álvarez-Gavela
I will discuss some recent connections between Floer theory and algebraic K-theory.
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Complex cobordism, Hamiltonion loops, and global Kuranishi charts
Charlotte Kirchhoff-Lukat
References:
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Bauer Furuta Invariants
Mary Stelow
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Fukaya Categories
Zihong Chen
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Q & A
Mohammed Abouzaid
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This seminar is organized by Cameron Krulewski, Ishan Levy, Natalia Pacheco-Tallaj, and Mary Stelow.