Vafa-Witten theory is a topologically twisted version of 4d N=4 super Yang-Mills theory. In my talk I will tell how to derive a holomorphic anomaly equation for its partition function on a Kaehler 4-manifold with b_2^+=1 and b_1=0 from the path integral of the effective theory on the Coulomb branch. I will also briefly mention an alternative and somewhat similar computation of the same holomorphic anomaly in the effective 2d theory obtained by compactification of the corresponding 6d (2,0) theory on the 4-manifold.
 

Abstract: Gradient descent is known to converge quickly for convex objective functions, but it can be trapped at local minimums. On the other hand, Langevin dynamic can explore the state space and find global minimums, but in order to give accurate estimates, it needs to run with small discretization step size and weak stochastic force, which in general slows down its convergence. This work shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if Langevin dynamic yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighbourhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. This is joint work with Xin Tong at National University of Singapore.

 I will report on my recent work on dimension, definable groups, and definable fields in vector spaces over algebraically closed [real closed] fields equipped with a non-degenerate alternating bilinear form or a non-degenerate [positive-definite] symmetric bilinear form. After a brief overview of the background, I will discuss a notion of dimension and some other ingredients of the proof of the main result, which states that, in the above context, every definable group is (algebraic-by-abelian)-by-algebraic [(semialgebraic-by-abelian)-by-semialgebraic]. It follows from this result that every definable field is definable in the field of scalars, hence either finite or definably isomorphic to it [finite or algebraically closed or real closed].
 

Let ?={??}?∈ℤ be zero-mean stationary Gaussian sequence of random variables with covariance function ρ satisfying ρ(0)=1. Let φ:R→R be a function such that ?[?(?_0)2]<∞ and assume that φ has Hermite rank d≥1. The celebrated Breuer–Major theorem asserts that, if ∑|?(?)|^?<∞ then
the finite dimensional distributions of the normalized sum of ?(??) converge to those of ?? where W is
a standard Brownian motion and σ is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the
space ?([0,1]) of càdlàg functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornstein–Uhlenbeck semigroup,
we show that tightness holds under the sufficient (and almost necessary) natural condition that E[|φ(X0)|p]<∞ for some p>2.

Joint work with D Nualart
 

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Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to globally synchronize, starting from random initial phases? One expects that dense networks have a strong tendency to synchronize and the basin of attraction for the synchronous state to be the whole phase space. But, how dense is dense enough? In this (hopefully) entertaining Zoom talk, we use techniques from numerical linear algebra and computational Algebraic geometry to derive the densest known networks that do not synchronize and the sparsest networks that do. This is joint work with Steven Strogatz and Mike Stillman.

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The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20^th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

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