Juvitop Seminar
Fall 2020
In Fall 2020, Juvitop was about The Coboridsm Hypothesis after Hopkins-Lurie.Discussion sections will be on Tuesdays at 3:30pm. Times are EST.
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On the Classification of Topological Field Theories by J. Lurie
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Higher-dimensional Algebra and Topological Quantum Field Theory by J. Baez and J. Dolan
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The Cobordism Hypothesis by D. Freed
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Topological Quantum Field Theory and the Cobordism Hypothesis by J. Lurie
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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 2-15 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
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Chapter 1, Section 4 of Factorization Algebras in Quantum Field Theory by K. Costello and O. Gwilliam
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Chapter 3 of Lectures on Factorization Homology, $\infty$-Categories, and Topological Field Theories by A. Amabel, A. Kalmykov, L. Müller, and H. Tanaka
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2-Dimensional Topological Field Theories by N. Wahl
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TQFTs and Higher Categories by C. Scheimbauer
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A Factorization View on States/Observables in Topological Field Theories by C. Scheimbauer
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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 15-24 of Lurie's paper. This includes comparing models of field theories, defining strict n-categories, and some ∞-category basics.
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Adrian Clough
Videos:
Pages 24-34 of Lurie's paper
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Factorization Homology as a Fully Extended Topological Field Theory by C. Scheimbauer
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A Note on the $(\infty,n)$-Category of Bordisms by D. Calaque and C. Scheimbauer
Organizers
References:
Pages 34-38 of Lurie's paper
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Jackson Van Dyke
Guided by what we know about finite-dimensional 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 1-morphisms in any 2-category. With these two definitions in hand, we will define what it means for an object of a monoidal (∞,n)-category to be k-dualizable.
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Dualizability in Low-Dimensional Higher Category Theory by C. Schommer-Pries
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On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis by P. Pstragowski
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Duality Notes by D. Culver and M. Faulk
Discussion Section
Organizers
References:
Videos:
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Kiran Luecke
In this talk I will present some versions of the Cobordism Hypothesis. Together we will ponder what they are really saying.
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Mumford's Conjecture - A Topological Outlook by U. Tillmann
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The Homotopy Type of the Cobordism Category by S. Galatius, I. Madsen, U. Tillmann, and M. Weiss
Nat Pacheco-Tallaj
References:
Pages 48-51 of Lurie's paper
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Ishan Levy
I will sketch the main ingredients of each step of the proof of the cobordism hypothesis.
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Organizers
Pages 52-57 of Lurie's paper
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Lucy Yang
I will discuss the strategy for reducing the cobordism hypothesis for manifolds with (X,zeta)-structure to the cobordism hypothesis for Bord_n (i.e. with structure group O(n)).
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Discussion Section
Organizers
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Micah Darrell
Pages 61-70 of Lurie's paper
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Discussion Section
Organizers
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The space of framed functions by K. Igusa
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The space of framed functions is contractible by Y. Eliashberg and N. Mishachev
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Three applications of delooping to $h$-principles by A. Kupers
Dylan Wilson
References:
Pages 70-79 of Lurie's paper
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NO Discussion Section (go vote instead)
To be determined
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Peter Haine
Pages 79-86 of Lurie's paper
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Discussion Section
Organizers
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The Cobordism Hypothesis by D. Ayala and J. Francis
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Section 5.5 of Higher Algebra by J. Lurie
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Factorizationhomology and the Cobordism Hypothesis by J. Francis
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$E_n$-Algebras, Extended Topological Field Theories and Dualizability by C. Scheimbauer
Tashi Walde
References:
Videos:
Recall the definition of E_n-algebras 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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Factorization homology of topological manifolds by D. Ayala and J. Francis
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Geometry of iterated loop spaces by J.P. May
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Factorization Homology Miniature Seminar organized by A. Amabel and P. Haine
Discussion Section: factorization homology
Organizers
References:
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Topological conformal field theories and Calabi-Yau categories by K. Costello
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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 Chas-Sullivan 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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Local structures on stratified spaces by D. Ayala, J. Francis, and H. Tanaka
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Section A.5 of Higher Algebra by J. Lurie
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On the homotopy theory of stratified spaces by P. Haine
Discussion Section
Organizers
References:
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The Tangle Hypothesis by C. Schommer-Pries
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The 1-dimensional tangle hypothesis by D. Ayala
Morgan Opie
Videos:
Use the material from the previous talk "to sketch a proof of a version of the Baez-Dolan 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 was organized by Araminta Amabel, Peter Haine, and Lucy Yang.