Fall 2019

In Fall 2019, Juvitop was about Differential Cohomology.

Collected notes from this seminar, by Amabel, Debray, and Haine.

• • Sep 042019
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  Introduction

  Peter Haine

  References:

  • Dan Freed's Talk on Differential Cohomology

  Notes

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• • Sep 112019
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  No Talk

   
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• • Sep 182019
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  Background: Chern–Weil Theory and Equivariant deRham Theory

  Greg Parker

  References:

  • An introduction to equivariant cohomology and the equivariant first Chern class

  • Chapters 2–4 of Supersymmetry and Equivariant de Rham Theory

  Notes

  The first part (maybe two thirds?) of my talk will give an introduction to Chern–Weil theory. First, we'll review the notions of connections and curvature, then we will develop the main ideas of the Chern–Weil description of characteristic classes (the Weil homomorphism in particular) and give some examples. The second part of the talk will give an introduction to equivariant de Rham theory. There, we will give the construction, discuss the equivariant de Rham theorem and the Cartan model.

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• • Sep 252019
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  Definitions of Differential Cohomology

  Peter Haine

  References:

  • Quadratic functions in geometry and topology, and M-theory

  • Structured bundles define differential K-theory

  • Differential cohomology theories as sheaves of spectra

  Notes

  In this talk we introduce the "abstract homotopy theory" perspective on differential cohomology theories: sheaves of spectra on the category Man of manifolds. From this perspective, spectra can be recovered as the homotopy-invariant sheaves on Man (equivalently, the constant sheaves). Complementary to the homotopy-invariant sheaves are the pure sheaves: sheaves with trivial global sections. After explaining some of the basic features of sheaves on Man, we’ll explain the "fracture square" that says that the ∞-category of sheaves of spectra on Man is built from the subcategories of homotopy-invariant and pure sheaves. This fracture square will then allow us to produce a "differential cohomology diagram" associated to every sheaf of spectra on Man. Next week’s talk will be dedicated to showing how ordinary differential cohomology fits into this framework as well as providing a number of other examples that arise from K-theory and bundles with connection.

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• • Oct 022019
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  Examples of Differential Cohomology Theories

  Araminta Amabel

  References:

  • Section 4 of Quadratic functions in geometry and topology, and M-theory

  • Structured bundles define differential K-theory

  • Notes on differential cohomology and gerbes

  • Section 4 of Differential cohomology theories as sheaves of spectra

  Notes

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• • Oct 092019
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  Chern-Simons forms and applications to (the configuration spaces of) Lens spaces

  Adela Zhang

  Referencse:

  • On the deleted squares of lens spaces

  • Chern-Simons invariants of 3-manifolds and representations of and representation spaces of knot groups

  • Classic Chern-Simons theory, part 2

  Notes

  In this talk, we introduce the Chern-Simons forms and explore their relation to the Cheeger-Simons differential characters. We will present a construction of the Chern-Simons invariants of topological 3-manifolds. As an application, we will compute these invariants for Lens spaces and see how they can help us understand the homotopy types of the configuration spaces of Lens spaces.

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• • Oct 162019
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  Differential Characteristic Classes and Conformal Immersions

  Charlie Reid

  References:

  • Characteristic forms and geometric invariants

  • Differential characters and geometric invariants

  Notes

  In this talk we are going to focus on differential Characteristic classes of the frame bundle of a Riemannian manifold. In particular we will show that the differential Pontrjagin classes of the Levi-Civita connection of a Riemannian manifold M are conformal invariants. We will use these differential Pontrjagin classes to construct obstructions to M conformally immersing in R^n. As an example, non-vanishing of differential p_1 shows that SO(3) does not conformally immerse in R^4. If M is three dimensional, differential p_1 gives us a map from conformal structures on M to R/Z. Investigating the differential of this function, we find that its critical points are conformally flat metrics.

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• • Oct 232019
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  Work of Freed–Hopkins

  Dexter Chua

  References:

  • Chern–Weil forms and abstract homotopy theory

  • On the Chern-Weil Homomorphism and the Continuous Cohomology of Lie-Groups

  Notes

  I will prove that the only natural way to obtain a differential form from a principal G-bundle is via Chern–Weil theory, following Freed–Hopkins. If time permits, I will discuss the relation to equivariant de Rham cohomology.

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• • Oct 302019
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  Differential Cohomology in Field Theory

  Dan Freed

  Arun's Notes

  I know two general applications of differential cohomology in field theory, namely Dirac charge quantization and topological terms in actions. I will illustrate both of these using particular examples.

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• • Nov 062019
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  Characteristic Classes in Differential Cohomology

  Mike Hopkins

  Science Center Hall E

  I will discuss the construction of characteristic classes in differential cohomology, and how they can be used to realize central extensions of the orientation preserving diffeomorphism group of the circle.

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• • Nov 132019
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  Deligne Cup Product and Differential Fiber Integration

  Araminta Amabel

  Notes

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• • Nov 202019
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  The Virasoro Group and Differential Cohomology

  Arun Debray
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• • Nov 272019
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  No Talk

   
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• • Dec 042019
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  Intertwining of Positive Energy Representations after Pressley-Segal

  Sanath Devalapurkar

  Reference: Theorem 13.4.3 of Loop Groups

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• • Dec 112019
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  Segal–Sugawara Construction

  Peter Haine

  Reference: Geometric Realizations of the Segal-Sugawara Construction

  Notes

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This seminar was organized by Araminta Amabel and Peter Haine.