It has long been known that many elliptic partial differential equations can be reformulated as Fredholm integral equations of the second kind on the boundaries of their domains. The kernels of the resulting integral equations are weakly singular, which has historically made their numerical solution somewhat onerous, requiring the construction of detailed and typically sub-optimal quadrature formulas. Recently, a numerical algorithm for constructing generalized Gaussian quadratures was discovered which, given 2n essentially arbitrary functions, constructs a unique n-point quadrature that integrates them to machine precision, solving the longstanding problem posed by singular kernels.

When the domains have corners, the solutions themselves are also singular. In fact, they are known to be representable, to order n, by a linear combination (expansion) of n known singular functions. In order to solve the integral equation accurately, it is necessary to construct a discretization such that the mapping (in the L^2-sense) from the values at the discretization points to the corresponding n expansion coefficients is well-conditioned. In this talk, we present exactly such an algorithm, which is optimal in the sense that, given n essentially arbitrary functions, it produces n discretization points, and for which the resulting interpolation formulas have condition numbers extremely close to one.

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One of the standard methods for the solution of elliptic boundary value problems calls for reformulating them as systems of integral equations.  The integral operators that arise in this fashion typically have singular kernels, and, in many cases of interest, the solutions of these equations are themselves singular.  This makes the accurate discretization of the systems of integral equations arising from elliptic boundary value problems challenging.

Over the last decade, Generalized Gaussian quadrature rules, which are n-point quadrature rules that are exact for a collection of 2n functions, have emerged as one of the most effective tools for discretizing singular integral equations. Among other things, they have been used to accelerate the discretization of singular integral operators on curves, to enable the accurate discretization of singular integral operators on complex surfaces and to greatly reduce the cost of representing the (singular) solutions of integral equations given on planar domains with corners.

We will first briefly outline a standard method for the discretization of integral operators given on curves which is highly amenable to acceleration through generalized Gaussian quadratures. We will then describe a numerical procedure for the construction of Generalized Gaussian quadrature rules.

Much of this is joint work with Zydrunas Gimbutas (NIST Boulder) and Vladimir Rokhlin (Yale University).

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This is a report on joint work with Martijn Kool. 

Recently, Marian-Oprea-Pandharipande established a generalization of Lehn’s conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. 

Using Mochizuki’s formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to  verify our conjectures in examples. 

We study the minimum Wasserstein distance from the empirical measure to a space of probability measures satisfying linear constraints. This statistic can naturally be used in a wide range of applications, for example, optimally choosing uncertainty sizes in distributionally robust optimization, optimal regularization, testing fairness, martingality, among many other statistical properties. We will discuss duality results which recover the celebrated Kantorovich-Rubinstein duality when the manifold is sufficiently rich and associated test statistics as the sample size increases. We illustrate how this relaxation can beat the statistical curse of dimensionality often associated to empirical Wasserstein distances.

The talk builds on joint work with S. Ghosh, Y. Kang, K. Murthy, M. Squillante, and N. Si.

Let G be a reductive group, E an elliptic curve, and Bun_G the moduli stack of principal G-bundles on E. In this talk, I will attempt to explain why Bun_G is a very interesting object from the perspectives of both singularity theory on the one hand, and shifted symplectic geometry and representation theory on the other. In the first part of the talk, I will explain how to construct slices of Bun_G through points corresponding to unstable bundles, and how these are linked to certain singular algebraic surfaces and their deformations in the case of a "subregular" bundle. In the second (probably much shorter) part, I will discuss the shifted symplectic geometry of Bun_G and its slices. If time permits, I will sketch how (conjectural) quantisations of these structures should be related to some well known algebras of an "elliptic" flavour, such as Sklyanin and Feigin-Odesskii algebras, and elliptic quantum groups.

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points and a question of McDuff-Salamon on Hamiltonian diffeomorphisms of finite order.

In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. 

One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration.

We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.