Juvitop Seminar
Fall 2020
This semester Juvitop will be about The Coboridsm Hypothesis after HopkinsLurie.
We meet at 3:59 on Wednesday in zoom unless otherwise noted.
Discussion sections will be on Tuesdays at 3pm. Times are EST.

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On the Classification of Topological Field Theories by J. Lurie

Higherdimensional Algebra and Topological Quantum Field Theory by J. Baez and J. Dolan

The Cobordism Hypothesis by D. Freed

Topological Quantum Field Theory and the Cobordism Hypothesis by J. Lurie

Topological Field Theories by M. Hopkins
Araminta Amabel
Notes:
References:
Videos:
What does the cobordism hypothesis say? Why did anyone ever hypothesize it? This talk is based on pages 215 of Lurie's paper. We introduce Atiyah's definition of a topological field theory and examine what data a TFT provides in dimensions 1 and 2. Using these examples, we motivate Baez and Dolan's Cobordism Hypothesis.
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Miscellaneous tutorial on $\infty$categories by D. Wilson

Chapter 1, Section 4 of Factorization Algebras in Quantum Field Theory by K. Costello and O. Gwilliam

Chapter 3 of Lectures on Factorization Homology, $\infty$Categories, and Topological Field Theories by A. Amabel, A. Kalmykov, L. Müller, and H. Tanaka

2Dimensional Topological Field Theories by N. Wahl

TQFTs and Higher Categories by C. Scheimbauer

A Factorization View on States/Observables in Topological Field Theories by C. Scheimbauer

Topological Field Theories in Homotopy Theory by U. Tillmann
Organizers
References:
Videos:
We'll discuss examples that came up in last week's talk and some new material from pages 1524 of Lurie's paper. This includes comparing models of field theories, defining strict ncategories, and some ∞category basics.
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Adrian Clough
Videos:
Pages 2434 of Lurie's paper
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Factorization Homology as a Fully Extended Topological Field Theory by C. Scheimbauer

A Note on the $(\infty,n)$Category of Bordisms by D. Calaque and C. Scheimbauer
Organizers
References:
Pages 3438 of Lurie's paper
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Jackson Van Dyke
Guided by what we know about finitedimensional vector spaces, we will first define the notion of a dualizable object in any monoidal category. Then, guided by what we know about adjoint functors, we will define a notion of duality for 1morphisms in any 2category. With these two definitions in hand, we will define what it means for an object of a monoidal (∞,n)category to be kdualizable.
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Dualizability in LowDimensional Higher Category Theory by C. SchommerPries

On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis by P. Pstragowski

Duality Notes by D. Culver and M. Faulk
Discussion Section
Organizers
References:
Videos:
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Kiran Luecke
Pages 4348 of Lurie's paper
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Mumford's Conjecture  A Topological Outlook by U. Tillmann

The Homotopy Type of the Cobordism Category by S. Galatius, I. Madsen, U. Tillmann, and M. Weiss
Natalia PachecoTallaj
References:
Pages 4851 of Lurie's paper
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Ishan Levy
Page 51 of Lurie's paper
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TBD
Pages 5257 of Lurie's paper
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Lucy Yang
Pages 5761 of Lurie's paper
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Micah Darrell
Pages 6170 of Lurie's paper
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Cameron Krulewski
Pages 7079 of Lurie's paper
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Dylan Wilson
Pages 7986 of Lurie's paper
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The Cobordism Hypothesis by D. Ayala and J. Francis

Section 5.5 of Higher Algebra by J. Lurie

Factorizationhomology and the Cobordism Hypothesis by J. Francis

$E_n$Algebras, Extended Topological Field Theories and Dualizability by C. Scheimbauer
Tashi Walde
References:
Videos:
Recall the definition of E_nalgebras and define topological chiral homology. Describe the class of TFTs that can be produced using topological chiral homology. Formulate a version of the cobordism hypothesis in terms of topological chiral homology (Theorem 4.1.24).
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Topological conformal field theories and CalabiYau categories by K. Costello

Section 5.5 of String Topology by M. Chas and D. Sullivan
Adela Zhang
References:
"Discuss some consequences of the cobordism hypothesis and related results in the case of manifolds of dimension 1 and 2. [...] Relate the contents of this paper to the work of Costello and to the ChasSullivan theory of string topology operations on the homology of loop spaces of manifolds"
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Will Stewart
"Describe a generalization of the cobordism hypothesis, which gives a geometric description of symmetric monoidal (∞, n)categories (again assumed to have duals) having more complicated presentations."
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Morgan Opie
Use the material from the previous talk "to sketch a proof of a version of the BaezDolan tangle hypothesis, which characterizes (∞, n)categories of embedded bordisms and can be regarded as an “unstable” version of the cobordism hypothesis."
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This seminar is organized by Araminta Amabel, Peter Haine, and Lucy Yang.