## math-ph seminar

This seminar is an informal, discussion-style seminar for students in mathematical physics. Any and all are welcome to speak and/or come and discuss. You can even request a topic you'd like someone to talk about here.

schedule: MWF 3 - 4pm @ Lunt 102

October 25, 2017 — Nilay — differential characters
• Differential cohomology is a geometric refinement of ordinary cohomology that naturally lives in the smooth category. The Chern-Weil homomorphism, for instance, factors naturally through differential cohomology, yielding differential refinements of topological characteristic classes. Moreover one finds that (higher) circle bundles with connection are classified by differential cohomology. I will outline the original model of differential characters, as constructed by Cheeger and Simons, and work through some basic properties.

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October 27, 2017 — Sean — the variational bicomplex

October 30, 2017 — Pax — heat kernel asymptotics
• We will discuss the small-time asymptotics of the heat kernel of the laplacian (on functions) on a compact oriented manifold. In particular, we will explore the general form of the coefficients appearing in this expansion.

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November 6, 2017 — Nilay — characteristic forms and simplicial sheaves I
• Chern-Weil allows us to construct concrete differential form representatives of characteristic classes, sometimes called characteristic forms. Freed and Hopkins ask the question: are the Chern-Weil characteristic forms the only characteristic forms? We will follow their precise formulation of this question in terms of simplicial sheaves (necessary due to the presence of principal bundle automorphisms) and sketch their (affirmative) answer.

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November 8, 2017 — Nilay — characteristic forms and simplicial sheaves II
• We will use the setup of simplicial presheaves discussed last time to prove that the de Rham complex of $$E_\nabla G$$ is precisely the Weil algebra $$(\text{Kos}^\bullet \mathfrak{g}^*, d_K)$$. We will take for granted some results from invariant/representation theory as well as a bit of model category theory.

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November 10, 2017 — Pax — the Witt and Virasoro algebras
• We will discuss the basics of the Witt and Virasoro algebras.

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• Francesco, Mathieu, Senechal — Conformal field theory

November 13, 2017 — Sean — particle mechanics on jet space

November 17, 2017 — Sean — variational approach to bv I
• I'll start by recalling the geometric building blocks for classical BV in the case of zero-dimensional field theory. Then I will say how these can be generalized to positive dimensions by building on the variational techniques introduced previously.

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November 20, 2017 — Mike — the local Weyl law
• The classical Weyl law is a theorem about the growth of eigenvalues of an elliptic operator. In this talk I'll specialize to the Laplacian and present Hormander's proof of a suped up version called the local Weyl law, the spiciest ingredient in the proof of quantum ergodicity. [qual practice talk hosted by the analysis seminar]

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November 27, 2017 — Sean — variational approach to bv II

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January 10, 2018 — Nilay — smooth Deligne cohomology
• We will introduce smooth Deligne cohomology as a model for differential cohomology. We will compute these groups explicitly in low degrees with an eye towards gerbes as well as outline the multiplicative structure. As natural examples of degree one and two classes we will sketch the construction of the eta invariant and the determinant line bundle associated to families of Dirac operators in odd and even dimensions, respectively.

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January 12, 2018 — Pax — modular tensor categories
• Fusion, modular, and unitary modular tensor categories have played an increasingly large role in modeling exotic symmetries in physics. In fact, it is widely believed that unitary modular tensor categories (with an additional rational number parameter $$c$$) are in one to one corrospondence with topological phases of matter. In this talk, we aim to define such objects and discuss basic examples and properties of such categories, with an eye toward exploring physical applications in the future.

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January 17, 2018 — Nilay — more Deligne cohomology
• Last time we rushed the definition of the multiplicative structure on Deligne cohomology so we will review the definition as well as motivate its seemingly-asymmetric construction and sketch its homotopy coherence. We will then move to the definition of a bundle gerbe and give some examples arising from geometry.

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January 22, 2018 — Pax — review of Gaussian processes
• We review the basics of random variables and Gaussian processes with an eye towards spin glasses, which we will discuss next time.

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January 24, 2018 — Pax — spin glasses
• TBA

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January 26, 2018 — Nilay — introduction to operads
• Our discussions from last week on homotopy coherence and multiplications on the singular and Deligne complexes motivated me to learn a bit of formalism to make some of our statements precise. We will cover the very basics of operads: definitions and most importantly, simple examples. We will sketch an $$A_\infty$$-structure on the based loop space $$\Omega X$$ and an $$E_\infty$$-structure on (normalized) singular cochains.

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February 7, 2018 — Sean — bv: the spinning particle I
• TBA

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February 12, 2018 — Sean — bv: the spinning particle II
• TBA

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February 21, 2018 — Sean — bv: the spinning particle III
• TBA

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February 25, 2018 — Nilay — differential forms and Hochschild homology
• We'll discuss how the Hochschild complex of the dga of differential forms $$\Omega^*X$$ is quasi-isomorphic to $$\Omega^*\text{Maps}(S^1, X)$$ if $$X$$ is simply-connected. We'll sketch how one might extend this to compact manifolds that are not the circle and how this relates to higher Hochschild homology or factorization homology.

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March 12, 2018 — Nilay — integration as an $$A_\infty$$-morphism
• We'll follow Gugenheim's 1976 paper that shows that the integration map $$\Omega^*X \to C^*(X,\mathbb{R})$$, which is not a map of algebras, can be extended to an $$A_\infty$$-morphism.

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March 14, 2018 — Piotr — homology theories and Hopf algebroids
• If $$E$$ is a reasonable homology theory, then $$E_{*}E$$ acquires a structure of a so-called Hopf algebroid, and for any spectrum $$X$$, the homology $$E_{*}X$$ is canonically a comodule over $$E_{*}E$$. We will discuss how this basic algebraic invariant is the lens through which we understand stable homotopy theory.

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