dean baskin | The Feynman propagator in some model singular settings
In this talk I will describe the existence and asymptotic properties of the Feynman propagator in three model singular settings: the scalar wave equation on cones, the scalar wave equation on Minkowski space with an inverse square potential, and the massless Dirac equation in 3 dimensions coupled to a Coulomb potential. The proof combines techniques of Gell-Redman–Haber–Vasy as well as prior work with Booth, Gell-Redman, Marzuola, Vasy, and Wunsch. One novelty of the proof is that it does not rely on Wick rotation (though a shadow of it survives in some special function analysis at infinity).
charles epstein | Static currents in type I superconductors: Numerical methods and the λL → 0 limit
jesse gell-redman | Fredholm approach to the Schrodinger equation
We discuss a new approach, inspired by work of Hintz and Vasy, to solving the Schrodinger equation (i∂t − Δ)u = f using the Fredholm method. Specifically, we use ’parabolic’ pseudodifferential operators (reflecting the parabolic nature of the symbol of P = i∂t − Δ) to obtain families of function spaces X, Y for which P : X → Y is an isomorphism. The spaces further allow us to read off precise regularity and decay information about u directly from that of f. We discuss applications to the nonlinear Schrodinger equation, and extensions of this method to equations with compact spatial perturbations, such as smooth decaying potential functions, using the N-body calculus of Vasy. This includes joint work with Dean Baskin, Sean Gomes, and Andrew Hassell.
robin graham | X-Ray Transform on Asymptotically Hyperbolic Manifolds
This talk will describe recent developments concerning the geodesic X-ray transform on asymptotically hyperbolic manifolds. Joint work with Guillarmou, Stefanov and Uhlmann will be discussed which provides a natural formulation of the transform in this setting and proves injectivity results. A stability estimate of Eptaminitakis will be described which relies on a parametrix construction in the
0-calculus for a 0-pseudodifferential operator. Finally, the talk will conclude with a discussion of a version of the celebrated local injectivity result of Uhlmann-Vasy in the asymptotically hyperbolic setting, which is joint work with Eptaminitakis.
colin guillarmou | Scattering theory in conformal field theory, from Liouville to Toda
We will explain how scattering theory in an infinite dimensional setting can allow to solve a two dimensional conformal field theory (CFT), i.e. to compute the correlation functions explicitly by a method introduced in physics called the conformal bootstrap. This is applied to the Liouville CFT, which has certain similarities with b-geometries, but as we will point out, it could also be used for Toda CFT, for which the Hamiltonian has similarities with Laplacians on higher rank symmetric spaces (a topic on which Mazzeo and Vasy have developed a method to analyse scattering theory).
jim isenberg | Convergence-Stability for Ricci Flow and Other Geometric Heat Flows on Bounded Geometries*
In previous work with Eric Bahuaud and Christine Guenther, we introduced the notion of convergence-stability for Ricci flow on compact manifolds. Depending on stability results and on continuous dependence for the flow, we were able to use this idea to show that the Ricci flow solutions which begin in an open set of geometries must all converge to a flat metric. A key tool for proving these results was the demonstration that the geometric operator governing the flow is sectorial. In this talk, after illustrating the notion of convergence-stability, I discuss the recent work with Bahuaud and Guenther together with Rafe in which we prove that for a much wider class of geometric operators defined on bounded geometries, sectoriality holds. The proof involves the use of tools from microlocal analysis. This result should lead to verifying convergence-stability for geometric heat flows on bounded geometries, and consequently to showing that many flows beginning with nonsymmetric initial data converge to fixed points for those flows.
svitlana mayboroda | PDEs vs. Geometry: analytic characterizations of geometric properties of sets
In this talk we will discuss connections between the geometric and analytic/PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions d < n, a d-dimensional set in Rn is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by
affine functions (distance to the hyperplanes).
richard melrose | Twisted index and smoothing operators
I will discuss recent work with Mathai Varghese on the twisted index theorem corresponding to small gerbes, i.e. smooth finite dimensional representations of integral cohomology 3-classes. This will be approached through semiclassical, twisted, smoothing operators.
andy neitzke | On Higgs bundles and the WKB method
Hitchin discovered the marvelous fact that moduli spaces of Higgs bundles carry natural hyperkahler metrics. Relatively little was known explicitly about these hyperkahler metrics until the work of Mazzeo-Swoboda-Weiss-Witt, which provided concrete formulas for their leading asymptotics near infinity. To go beyond the leading asymptotics there is a more detailed conjectural picture available, which was introduced in my joint work with Gaiotto-Moore some time ago. One way to understand this picture is as an application of the exact WKB method for studying one-parameter families of differential equations. I will explain how this picture works, and describe the current status of the program.
frank pacard | Geodesics on surfaces and the Allen-Cahn equation*
There is strong parallel between the theory of minimal hypersurfaces and the solutions of the double-well phase transition Allen-Cahn equation when the phase transition parameter tends to 0. In this talk, I will report some recent progress on the relations between geodesics on surfaces and solutions of the Allen-Cahn equation whose Morse indices are controlled. This is a joint work with Juncheng Wei and Yong Liu.
paolo piazza | K-Theory of pseudodifferential operators and secondary invariants of Dirac operators*
Let M be a connected compact manifold without boundary and let Γ be its fundamental group. The Higson-Roe analytic surgery sequence is a long exact sequence in K-theory that encodes in an elegant way primary and secondary K-theory invariants of Γ-equivariant Dirac operators on the universal cover of M. Among them, the index class of a Dirac operator and the rho class of an invertible Dirac operator. In this talk I will begin by explaining that this sequence can be realized solely in termsof algebras of pseudodifferential operators. Next I will explain how to extract numeric invariants out of the K-Theory classes defined by Dirac operators, in particular out of the rho class. This is achieved,
under additional assumptions on Γ, by defining a pairing between the cyclic cohomology of the group algebra of Γ and the relevant K-Theory groups appearing in the Higson-Roe sequence. We will define in this way the higher rho numbers associated to the rho class of an invertible Dirac operator and I willend my talk explaining how these numeric invariants can be used in order to study the moduli space
of metrics of positive scalar curvature (when the latter is non-empty). This is joint work with Thomas Schick and Vito Felice Zenobi.
julie rowlett | The strength of diversity
Diversity can be beneficial in many contexts. In business, research, and education diversity can increase creativity by providing a wider variety of perspectives and increasing impact through alarger network. In biology, the health of an ecosystem is often measured in terms of its biodiversity.A diverse ecosystem has a greater capacity for adaptive responses to new challenges. In finance, a cornerstone of modern portfolio theory is diversification of investments. I will discuss joint work withCJ Karlsson and M Nursultanov. Our main result is ‘the diversity theorem’ which shows that a teamof diverse individuals is strong in competition with other teams of individuals. The proof is basedon a game theoretic model that interpolates between individual-level interactions and collective-level ramifications.
yanir rubinstein | On Nazarov’s proof of the Bourgain-Milman inequality
Over 85 years ago, Mahler conjectured several sharp inequalities concerning the volume of convex bodies in two cases: symmetric bodies and general bodies with neither implying the other. In the 80’s Bourgain-Milman proved a breakthrough sub-optimal version of these inequalities and since then many researchers have gradually improved the best known constant in these inequalities with each
new proof revealing beautiful new aspects of the problems or connections to other fields/problems. All proofs so far have derived the general case from the symmetric case via the standard symmetrization trick. A decade ago Nazarov devised a beautiful Bergman kernel proof of the Bourgain-Milman in thesymmetric case and suggested that this approach should yield a direct, though sub-optimal, proof of
the general case, without using symmetrization. In joint work with V. Mastrantonis, we follow many of Nazarov’s ideas, some ideas of Berndtsson and Hultgren, as well as some new ingredients, to confirm Nazarov’s prediction.
mariel sáez | Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
In this talk I will discuss joint work with M.M Gonzalez. Our work studies bounds for the first eigenvalue of the Paneitz operator P and its associated third-order boundary operator B3 on fourmanifolds. We restrict to orientable, simply connected, locally conformally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered manifolds are confomally equivalent to canonical models. This equivalence is proved by showing the injectivity of suitable developing maps. Then the bounds on the eigenvalues are obtained through explicit computations on the canonical models and its connections
with the classes of manifolds that we are considering. In particular, we give an explicit bound for a 4-dimensional annulus with a radially symmetric metric. The fact that P and B3 are conformal in four dimensions is key in the proof.
peter sarnak | Prescribing the spectra of locally uniform geometries
After reviewing recent developments (conformal bootstrap and random covers) concerning the Laplace spectra of hyperbolic manifolds and of large regular graphs, we focus on rigidity features to creating spectral gaps.
david sher | Bessel functions and a conjecture of Pólya
We consider the classical problem of finding asymptotics for Bessel functions as the order and
the argument both approach infinity. We use blow-up analysis to find complete asymptotic descriptions
of the modulus and the phase of the Bessel functions. These descriptions are valid in any regime and
thus unify, and slightly extend, previously known results. In this talk I will first explain these results and
then discuss how insights obtained in this analysis led to the proof (with M. Levitin and I. Polterovich)
of a conjecture of Pálya concerning the Dirichlet and Neumann eigenvalues of the unit disk.
gunther uhlmann | The fractional Calderón Inverse Problems
We describe some of the recent results on the fractional analog of Calderón’s inverse problem.