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Consider a polynomial differential operator in one variable, depending on a small parameter (Planck constant). Under appropriate conditions, the low-energy spectrum admits an asymptotic expansion in hbar.
I will present a way to calculate such a series via a purely "commutative problem", a mixture of variations of Hodge structures and of the Stirling formula. This result came from discussions with A.Soibelman. It seems that we obtain an explanation of an old observation by J.Zinn-Justin of the 

 universal appearance of Bernoulli numbers.

In recent years, there has been a great deal of progress on ideas related to twisted supergravity, building on the definition given by Costello and Li. Much of what is explicitly known about these theories comes from the topological B-model, whose string field theory conjecturally produces the holomorphic twist of type IIB supergravity. Progress on eleven-dimensional supergravity has been hindered, in part, by the lack of such a worldsheet approach. I will discuss a rigorous computation of the twist of the free eleven-dimensional supergravity multiplet, as well as an interacting BV theory with this field content that passes a large  number of consistency checks. Surprisingly, the resulting holomorphic theory on flat space is closely related to the infinite-dimensional exceptional simple Lie superalgebra E(5,10). This is joint work with Surya Raghavendran and Brian Williams.

Zoom Link: https://pitp.zoom.us/j/99622967785?pwd=YmlQWW1sNW1qS1FhQkV4NXFlY0Nsdz09

I will connect approaches to classical integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations. In particular, I will consider holomorphic Chern-Simons theory on twistor space, defined using a range of meromorphic (3,0)-forms. On shell these are, in most cases, found to agree with actions for anti-self-dual Yang-Mills theory on space-time. Under symmetry reduction, these space-time actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to 4d Chern-Simons theory.

Zoom Link: https://pitp.zoom.us/j/99193672959?pwd=RUJ3N3h2V3RFK3ZNVVVCK1E3bXJ2Zz09

Factorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will discuss how to compute categorical factorization homology on oriented surfaces with principal D-bundles, for D a finite group, in terms of categories of modules over algebras defined in purely combinatorial terms. This is an extension of the work of Ben-Zvi, Brochier and Jordan to D-decorated surfaces. The main example for us comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups associated to a reductive group G. We will see that in this case factorization homology gives rise to a quantization of character varieties which are twisted by the group of outer automorphisms of G.

This talk is based on joint work with L. Müller.

Zoom Link: https://pitp.zoom.us/j/93950433494?pwd=WXI2VE9IdnRweEh5RmZsZ21BV1BQQT09

I will describe a concrete, computable example of holographic geometry emerging purely from the combinatorics of large matrices. Correlation functions of determinant operators in a matrix-valued free chiral algebra will be matched to holomorphic curves in SL(2,C).

Zoom Link: https://pitp.zoom.us/j/99341570607?pwd=TVVmSkF3d1haa0hBOWlPeDhGNi9aQT09

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a
genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Finally, we shall look into the appearance of branes within the moduli space of generalized hyperpolygons as well as of Higgs bundles, and consider mirror symmetry for such branes. Time permitting, we will mention some other recent results in various areas of science. 

Zoom Link: https://pitp.zoom.us/j/91592778202?pwd=WnM2VS9pS2c0QVIxVWFOdGhFMTdEdz09

The program of Ben-Zvi--Sakellaridis--Venkatesh connects the construction of L-functions in number theory with S-duality of boundary conditions in 4d. In particular this predicts certain equivalences of categories between equivariant D-modules on the formal loop space of a smooth variety X and equivariant quasi-coherent sheaves on a Hamiltonian manifold. I discuss an extension of this conjecture to certain singular varieties X and the possibility of quantizing the equivalence. I will explain joint work with Yiannis Sakellaridis on computing a certain factorization algebra which plays a role in the story. 

Zoom Link: https://pitp.zoom.us/j/95543248994?pwd=bmZIRnEyLzZnNmlEWW5oNTEwaEhNUT09

Recently, an intriguing connection between the exceptional Jordan algebra h_3(O) and the standard model of particle physics was noticed by Dubois-Violette and Todorov (with further interpretation by Baez). How do the standard model fermions fit into this story? I will explain how they may be neatly incorporated by complexifying h_3(O) or, relatedly, by passing from RxO to CxO in the so-called "magic square" of normed division algebras. This, in turn, suggests that the standard model, with gauge group SU(3)xSU(2)xU(1), is embedded in a left/right-symmetric theory, with gauge group SU(3)xSU(2)xSU(2)xU(1). This theory is not only experimentally viable, but offers some explanatory advantages over the standard model (including an elegant solution to the standard model's "strong CP problem"). Ramond's formulation of the magic square, based on triality, provides further insights, and possible hints about where to go next.

This talk will be about two applications of Jordan algebras. The first, to quantum mechanics, follows on from the talk of John Baez. I will explain how time dependence makes use of the associator, and how this is related to the commutator in the standard density matrix formulation. The associator of a Jordan algebra also determines the curvature of a Riemannian metric on its positive cone, invariant under the symmetry group of the norm (mentioned in the talk of John Baez); the cone is foliated by hypersurfaces of constant norm. This geometry is relevant to a class of N=2 5D supergravity theories (from the early 1980s) which arise (in some cases, at least) from Calabi-Yau compactification of 11D supergravity. The 5D interactions are determined by the structure constants of a euclidean Jordan algebra with cubic norm. The exceptional JA of 3x3 octonionic matrices yields an ``exceptional’’ 5D supergravity which yields, on reduction to 4D, an ``exceptional’’ N=2 supergravity with many similarities to N=8 supergravity, such as a non-compact global E7 symmetry. However, it has a compact `composite’ E6 gauge invariance (in contrast to the SU(8) of N=8 supergravity). An old speculation is that non-perturbative effects break the N=2 supersymmetry and cause the E6 gauge potentials to become the dynamical fields of an E6 GUT. Potentially (albeit improbably) this provides a connection between M-theory, the exceptional Jordan algebra, and the Standard Model.

I will take a look at the SO(10) grand unified theories from the perspective of the typical phenomenology constraints imposed on their structure. The current status of the minimal potentially realistic models will be briefly commented upon.