Juvitop Seminar
Fall 2019
In Fall 2019, Juvitop was about Differential Cohomology.Collected notes from this seminar, by Amabel, Debray, and Haine.

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Introduction
Peter Haine
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No Talk
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Greg Parker
References:
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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Peter Haine
References:
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 homotopyinvariant sheaves on Man (equivalently, the constant sheaves). Complementary to the homotopyinvariant 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 homotopyinvariant 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 Ktheory and bundles with connection.
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Examples of Differential Cohomology Theories
Araminta Amabel
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Adela Zhang
Referencse:
In this talk, we introduce the ChernSimons forms and explore their relation to the CheegerSimons differential characters. We will present a construction of the ChernSimons invariants of topological 3manifolds. 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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Charlie Reid
References:
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 LeviCivita 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, nonvanishing 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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Dexter Chua
I will prove that the only natural way to obtain a differential form from a principal Gbundle 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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Dan Freed
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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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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Deligne Cup Product and Differential Fiber Integration
Araminta Amabel
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The Virasoro Group and Differential Cohomology
Arun Debray
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No Talk
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Intertwining of Positive Energy Representations after PressleySegal
Sanath Devalapurkar
Reference: Theorem 13.4.3 of Loop Groups
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Segal–Sugawara Construction
Peter Haine
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This seminar was organized by Araminta Amabel and Peter Haine.