Microlocal analysis

Spyros Alexakis : The lens rigidity problem in 2 dimensions
abstract: We will discuss aspects the geometric inverse problem of finding a Riemannian metric on a manifold-with boundary, from knowledge of the lens map: Assume we know the exit point and exit time for any geodesic that is shot into a manifold from its boundary in any direction, can we reconstruct the metric? Can local metric reconstruction be performed from local lens data? If the whole metric cannot be reconstructed, how much of it can? Is the reconstruction stable? We present recent work on these questions, for 2-dimensional metrics. Some brief discussion the relation of this work with wave inverse problems will also be provided. Joint work with Matti Lassas.

Gabriele Benedetti : Rigidity and flexibility of periodic Hamiltonian flows
abstract: An old problem in classical mechanics asks for the existence of periodic flows within specific classes of Hamiltonian systems such as central forces and geodesic flows. While Bertrand showed that only trivial examples of periodic flows among central systems exist (the gravitational and elastic force), Zoll and, later, Guillemin proved that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin’s microlocal approach, the goal of this talk is to construct magnetic flows (a generalization of geodesic flows in which the particle is subject to a Lorentz force) on the two-torus which are periodic for just one single value of the energy. This is joint work with Luca Asselle and Massimiliano Berti.

Yannick Bonthonneau : Tunneling for the magnetic Laplacian
abstract: For the magnetic Laplacian in the plane, we are able to prove an asymptotic tunneling formula in the presence of symmetry and magnetic wells. This is the second formula of its kind! (Joint work with Fournais, Morin and Raymond)

Nguyen Viet Dang : Wick rotating the φ⁴₃ measure on 3 dimensional manifolds
abstract: In this talk, I will describe joint work with Bailleul-Ferdinand and To where we construct a certain model of Euclidean quantum field theory (the so called φ⁴₃ model) on every compact 3 dimensional Riemannian manifold. as invariant measure under the dynamics described by some stochastic partial differential equation. In the second part of the talk I will describe the challenges that we must overcome in order to Wick rotate the theory from the three sphere to the three dimensional de Sitter space, which is joint work with Bonthonneau-Ferdinand-Lin and To.

Andreas Debrouwere : Boundary values of zero solutions of hypoelliptic PDO
abstract: In this talk, I will discuss boundary values of zero solutions of hypoelliptic partial differential operators in various spaces of generalized functions (distributions, ultradistributions, hyperfunctions). I will explain how this framework unifies and extends classical results about boundary values of holomorphic functions, harmonic functions, and zero solutions to the heat equation. This is joint work with Thomas Kalmes.

Christian Gaß : Operator-theoretic approaches to QFT on curved spacetimes
abstract: I will describe two distinct approaches to Quantum Field Theory of a scalar field on curved spacetimes based on operator-theory: the on-shell approach based on the Krein space of solutions of the Klein-Gordon equation and the off-shell approach based on the space of square integrable functions on the spacetime. The off-shell approach is often used implicitly to define the Feynman propagator via the path integral. In well-behaved situations, the so-defined propagator corresponds to a time-ordered two-point function between two Hadamard states.
After an introduction to the framework, I will apply these methods to various patches of de Sitter space, where we encounter a particularly rich structure. Due to the infinite expansion of de Sitter space, this example is "not well-behaved": the path integral in the off-shell approach does not yield a propagator with Hadamard type singularities, i.e., it does not belong to one of the four parametrix classes of Duistermaat and Hörmander.

András Vasy : Spectral theory for Dirac type operators on asymptotically Minkowski spaces and second microlocalization
abstract: I will discuss a second microlocal approach to spectral theory on asymptotically Minkowski spaces which has interesting implications for both the proof of the selfadjointness of scalar wave operators and also for the spectral theory of Dirac type operators. This is joint work with Nguyen Viet Dang and Michal Wrochna.

Jian Wang : Topographic scattering of internal waves
abstract: Internal waves are a central topic in oceanography and the theory of rotating fluids. They are gravity waves in density-stratified fluids. In a two-dimensional aquarium, the velocity of linear internal waves can concentrate on certain attractors. Locations of internal wave attractors are related to periodic orbits of homeomorphisms of the circle, given by a nonlinear "chess billiard" dynamical system. This relation provides a surprising "quantum--classical correspondence" in fluid dynamics. In this talk, I will explain connections between homeomorphisms of circles, spectral theory, and internal wave dynamics.

Alden Waters : Scattering Theory for Maxwell's equations
abstract: This talk starts with a gentle introduction to scattering theory for Maxwell's equations. In specific it will lead into recent acheivements in the understanding of geometric scattering theory. Different aspects of evolution operators for topological scatterers will be discussed as well as a fun mathematical history of the topic. Future applications to dispersive estimates will be discussed.