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.
Human contact patterns are highly heterogeneous in terms of the both the number and nature of interactions. To incorporate these heterogeneities into infectious disease models one naturally represents a population as a weighted network. While there is a large literature on the spread of diseases on networks, most techniques are highly computational in nature. In this talk I will talk about an analytical framework for modeling infectious diseases as a percolation process on weighted networks based on probability generating functions.
We introduce the notions of (G,q)-opers and Miura (G,q)-opers, where G is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q)-opers of a certain kind and the set of nondegenerate solutions of a system of XXZ Bethe Ansatz equations. This can be viewed as a generalization of the so-called quantum/classical duality which I studied with D. Gaiotto several years ago. q-Opers generalize classical side, while on the quantum side we have more general XXZ Bethe Ansatz equations.
The subject of equivariant elliptic cohomology finds itself at the interface of topology, string theory, affine representation theory, singularity theory and integrable systems. These connections were already known to the founders of the discipline, Grojnowski, Segal, Hopkins, Devoto, but have come into sharper focus in recent years with a number of remarkable developments happening simultaneously: First, there are the geometric constructions, due to Kitchloo, Rezk and Spong, Berwick-Evans and Tripathy, building on the older program of Stolz-Teichner, and of course Segal.
I will present recent work (to appear soon) on the homological mirror symmetry about the universal centralizers $J_G$, for any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category of $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center).
Borcherds Kac-Moody (BKM) algebras are a generalization of familiar Kac-Moody algebras with imaginary simple roots. On the one hand, they were invented by Borcherds in his proof of the monstrous moonshine conjectures and have many interesting connections to new moonshines, number theory and the theory of automorphic forms. On the other hand, there is an old conjecture of Harvey and Moore that BPS states in string theory form an algebra that is in some cases a BKM algebra and which is based on certain signatures of BKMs observed in 4d threshold corrections and black hole physics.
To any vertex algebra one can attach in a canonical way a certain Poisson variety, called the associated variety. Nilpotent Slodowy slices appear as associated varieties of admissible (simple) W-algebras. They also appear as Higgs branches of the Argyres-Douglas theories in 4d N=2 SCFT’s. These two facts are linked by the so-called Higgs branch conjecture. In this talk I will explain how to exploit the geometry of nilpotent Slodowy slices to study some properties of W-algebras whose motivation stems from physics.
Every toric variety is a GIT quotient of an affine space by an algebraic torus. In this talk, I will discuss a way to understand and compute the symplectic mirrors of toric varieties from this universal perspective using the concept of window subcategories. The talk is based on results from a work of myself and a joint work in progress with Peng Zhou.
The theory of quasimaps to Nakajima quiver varieties X has recently been used very effectively by Aganagic, Okounkov and others to study symplectic duality. For certain X, namely Hilbert schemes of ADE surfaces, it turns out quasimap theory is equivalent to a particular flavor of Donaldson-Thomas theory on a related threefold Y. I will explain this equivalence and how it intertwines concepts and tools from the two sides. For example, symplectic duality has something to say about the crepant resolution conjecture for Y.
A classical result of R. Courant gives an upper bound for the count of nodal domains (connected components of the complement of where a function vanishes) for Dirichlet eigenfunctions on compact planar domains. This can be generalized to Laplace-Beltrami eigenfunctions on compact surfaces without boundary. When considering random linear combinations of eigenfunctions, one can make this count more precise and pose statistical questions on the geometries appearing amongst the nodal domains: what percentage have one hole? ten holes?