ERC SYNERGY GRANT

The talk is an overview of resurgence results in generic linear or nonlinear meromorphic ODEs and classes of PDEs with emphasis on the
time-periodic Schrödinger equation and its non-perturbative analysis. I
will also explain the method of trans-asymptotic matching. Then I will
present a number of applications ranging from the proof of the Dubrovin
conjecture, the calculation of connection constants in closed form for
integrable models, and talk about two recent results, on the optimal
reconstruction of functions from truncated Maclaurin series and the
solution of a 60 year old problem concerning the domain of convergence
of spherical harmonic expansions

The talk is an overview of resurgence results in generic linear or nonlinear meromorphic ODEs and classes of PDEs with emphasis on the
time-periodic Schrödinger equation and its non-perturbative analysis. I
will also explain the method of trans-asymptotic matching. Then I will
present a number of applications ranging from the proof of the Dubrovin
conjecture, the calculation of connection constants in closed form for
integrable models, and talk about two recent results, on the optimal
reconstruction of functions from truncated Maclaurin series and the
solution of a 60 year old problem concerning the domain of convergence
of spherical harmonic expansions

Virasoro/W-constraint has been playing a central role in the context of enumerative geometry, which provides algebraic characterization of the partition function (generating function; tau function) and the correlation functions. Instanton counting is one of such enumerative problems originally motivated by 4d supersymmetric gauge theory, which shares a lot of concepts with other enumerative problems, e.g., spectral curve and its quantization, WKB expansion, integrability, etc. In this talk, I will show that a discrete analog of the Virasoro/W-constraint, which is in fact the q-deformation of the ordinary Virasoro/W-constraint, would emerge in the context of instanton counting through double quantization of the gauge theory moduli space. I would in particular discuss a possible interplay between geometric representation theory and enumerative geometry.

There is a connection between N=2 supersymmetric field theory in four dimensions and the theory of linear differential and difference equations with meromorphic coefficients. This connection has been studied by many different authors from different perspectives. One way of understanding it is as an instance of the general phenomenon that turning on Omega-background naturally quantizes algebraic structures in the operator algebras of supersymmetric QFT. I will recall this connection and describe some of its uses, along with some recent and ongoing extensions of the story.

Some recent work in the quantum gravity literature has considered what happens when the amplitudes of a TQFT are summed over the bordisms between fixed in-going and out-going boundaries. We will comment on these constructions. The total amplitude, that takes into account all in-going and out-going boundaries can be presented in a curious factorized form.

I will discuss a curious relation satisfied by many superconformal indices and index-like quantities. The relation takes the form of a large-N, large charge asymptotic expansion, but is actually exact.

Conventional Floer theory studies real symplectic manifolds by means of algebraic and categorical structures associated with their Lagrangian submanifolds. The Floer complex of a pair of Lagrangian submanifolds is an example. Its upgrade to the Fukaya category of a symplectic manifold is another example. In 2014 jointly with Maxim Kontsevich we started the project "Holomorphic Floer theory" (HFT) in which complex symplectic manifolds were the main object of study. This brings methods of symplectic topology into the realm of complex geometry without passing to the mirror dual manifold. In this talk I plan to give an introduction to the project based mainly on our version of the Riemann-Hilbert correspondence. The latter relates deformation quantization and the Fukaya category of the same complex symplectic manifold. In particular it gives a tool for study some of the questions of "brane quantization" story in physics like e.g. the meaning of coisotropic branes, or how to associate a module over quantum algebra to a complex Lagrangian submanifold (maybe singular). Another story in which HFT point of view is useful is the one of exponential integrals. HFT explains the geometric meaning of the Stokes phenomenon, wall-crossing formulas and resurgence in the framework of exponential integrals (maybe infinite-dimensional).

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