Abstract: The affine GAGA theorem of Peterzil and Starchenko says that a closed analytic subset of complex n-space which is definable in an o-minimal expansion of the ordered field R is actually algebraic. It is a one of the main tools in recent work on Hodge theory by Bakker, Brunebarbe and Tsimerman. In my talk I'll show how to give a very short proof of Peterzil--Starchenko using a much older affine GAGA theorem of Stoll along with a volume estimate for definable sets in an o-minimal structure (due to Kurdyka and Raby in a slightly different form and to and Nguyen and Valette in the o-minimal setting). The main point of my talk is really to advertise the o-minimal methods. So I'll give the (relatively easy) proof the volume estimate. But I'll also try to say something about the way Peterzil--Starchenko has been used in Hodge theory.
Abstract: In this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far-field data and a known function. It can be shown that the imaging functional is strictly positive in the scatterer and decays as the sampling point moves away from the scatterer. The analysis uses the factorization of the far-field operator and the Funke-Hecke formula. This method can also be shown to be stable with respect to perturbations in the scattering data. We will discuss the inverse scattering problem for acoustic waves. This is joint work with A. Kleefeld.
Abstract: This talk is concerned with a generalization of the classical gradient flow problem to the case where the time derivative is replaced by the fractional Caputo derivative. We first define the energy solution for this problem and prove its existence and uniqueness, under suitable assumptions, via a generalized minimizing movements scheme. We then introduce and discuss a semi-discrete numerical scheme where we only discretize in time. We then present a posteriori error estimates, which allow to adaptively choose time steps. We also give convergence rates. Several numerical experiments are also presented in the end to help the understanding of the problem and the numerical scheme.
Abstract: We consider the approximation of singular integrals on closed smooth contours and surfaces. Such integrals frequently arise in the solution of Boundary Integral Equations. We introduce a new quadrature method for these integrals that attains high-order accuracy, numerical stability, ease of implementation, and compatibility with the "fast" algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Our quadrature method exploits important connections between the punctured trapezoidal rule and the Riemann zeta function, leading to remarkably simple construction procedures.
Abstract: Numerous results in Analysis concern rigidity of homomorphisms between topological groups or Banach algebras. In this talk, I will concentrate on topological rigidity of homomorphisms between topological groups, namely automatic continuity. Classical results of Banach, Sierpinski, Steinhaus and Weil established that every Haar measurable homomorphism between locally compact second countable groups is continuous and a similar result was found for Baire category by Pettis. In the late 1960s, Christensen introduced an appropriate notion of Haar null sets in Polish topological groups, which later found use in the theory of dynamical systems by influential work of Hunt, Sauer and Yorke. In connection with this work, Christensen asked whether every universally measurable homomorphism between Polish groups is continuous and gave positive answers for special cases. We will present the general solution to Christensen's problem and also show how the associated techniques have consequences in mathematical logic.
Abstract: It was shown by Tim Austin that if an orbit equivalence between probability-measure-preserving actions of finitely generated amenable groups is integrable then it preserves entropy. I will discuss some joint work with Hanfeng Li in which we show that the same conclusion holds for the maximal sofic entropy when the acting groups are countable and sofic and contain an amenable w-normal subgroup which is not locally virtually cyclic, and that it is moreover enough to assume that the Shannon entropy of the cocycle partitions is finite (what we call Shannon orbit equivalence). It follows that two Bernoulli actions of a group in the above class are Shannon orbit equivalent if and only if they are conjugate. I will also describe a topological version of our measure entropy invariance result, along with an application to the construction of simple C*-simple groups whose von Neumann algebras have property Gamma.
Abstract: The self-dual U(1)-Yang-Mills-Higgs functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly in the 2-dimensional and Kaehler settings) are objects of long-standing interest in low-dimensional gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.
Abstract: I introduce trace definability, a weak notion of interpretability. This notion arose out of attempts to extend the Peterzil-Starchenko trichotomy to non o-minimal settings. Two theories are trace equivalent if each trace defines the other. Trace equivalence seems to be an interesting weak equivalence notion of theories, in particular for NIP theories. NIP-theoretic properties are preserved under trace definibility and there are non-trivial trace equivalences between NIP structures of interest. For example the theory of real closed fields is trace equivalent to the theory of real closed valued fields. It may also be possible, and interesting, to classify homogeneous structures up to trace definability, and this classification problem turns out to be closely related to indiscernible collapse.
Abstract: The self-dual U(1)-Yang-Mills-Higgs functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly in the 2-dimensional and Kaehler settings) are objects of long-standing interest in low-dimensional gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.
Abstract: In recent years, Multi-Agent Deep Reinforcement Learning (MADRL) has been successfully applied to various complex scenarios such as playing computer games and coordinating robot swarms. In this talk, we investigate the impact of âimplementation tricksâ for SOTA cooperative MADRL algorithms, such as QMIX, and provide some suggestions for tuning. In investigating implementation settings and how they affect fairness in MADRL experiments, we found some conclusions contrary to the previous work; we discuss how QMIXâs monotonicity condition is critical for cooperative tasks. Finally, we propose the new policy-based algorithm RIIT that achieves SOTA among policy-based algorithms.