The APDE seminar on Monday, 03/09 will be given by Mihai Tohaneanu in Evans 939 from 4:10 to 5pm.

Title: Local energy estimates on black hole backgrounds.

Abstract: Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is joint work with Lindblad, Marzuola, Metcalfe, and Tataru.

The APDE seminar on Monday, 02/24 will be given by Mihaela Ifrim in Evans 939 from 4:10 to 5pm.

Title: Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions.

Abstract: We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We systematically investigate all the possible quadratic null form type quasilinear strong coupling nonlinearities, and provide a new, robust approach for the proof.

The APDE seminar on Monday, 03/16 will be given by Johannes Sjöstrand in Evans 939 from 4:10 to 5pm.

Title: Resonances over a potential well in an island.

Abstract: Recent work with M. Zerzeri. Let V : R^n → R be a sufficiently analytic potential which tends to 0 at infinity. Assume that for an E > 0 we have V^{-1}(]- ∞ ,E[)=U(E) ⊔ S(E), where U(E) ∩ S(E) = ∅ , with U(E) connected and bounded (the well) and S(E) connected (the sea). The distribution of the resonances of -h^2 Δ + V near E has been thoroughly studied since more than 30 years. If we increase E a natural scenario is that the decomposition persists until the closures of U(E) and S(E) intersect at a critical energy E = E_0. Under some natural assumptions we show that near E_0 most of the resonances are close to the real axis and obey a Weyl law. In one dimension there are more detailed results (Fujiie-Ramond ’98).

The APDE seminar on Monday, 01/13 Monday, 01/27 will be given by Alexander Volberg in Evans 939 from 4:10 to 5pm.

Title: Box condition versus Chang–Fefferman condition for weighted multi-parameter paraproducts.

Abstract: Paraproducts are building blocks of many singular integral operators and the main instrument in proving “Leibniz rule” for fractional derivatives (Kato–Ponce). Also multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgues spaces with respect to a measure in the polydisc. The latter problem (without loss of information) can be often reduced to  boundedness of weighted dyadic multi-parameter paraproducts. We  find the necessary and sufficient  condition for this boundedness in n-parameter case, when n is 1, 2, or 3.  The answer is quite unexpected and seemingly goes against the well known difference between box and Chang–Fefferman condition that was given by Carleson quilts example of 1974.

The APDE seminar on Monday, 11/25 will be given by Charles Hadfield in Evans 939 from 4:10 to 5pm.

Title: Dynamical zeta functions at zero on surfaces with boundary

Abstract: The Ruelle zeta function counts closed geodesics on a Riemannian manifold of negative curvature. Its zeroes are related to Pollicott-Ruelle resonances which have been heavily studied in the setting of Anosov dynamical systems. In 2016, Dyatlov-Zworski proved an unexpected result relating the structure of the zeta function near the origin to the topology of the manifold. This extended a formula previously only known to hold in the constant curvature setting.

This talk will consider the situation where the manifold has boundary. A similar story can be told and the ultimate result extends the constant curvature setting (understood in 2001) to the variable curvature setting.

The microlocal tools required to consider this problem had been well developed in earlier papers (principally Dyatlov-Guillarmou 2016) and it remained to manipulate correctly relative cohomology (in this case à la Bott-Tu) in order to understand the space of 1-form Pollicott-Ruelle resonances.

The APDE seminar on Monday, 11/18 will be given by Dean Baskin in Evans 939 from 4:10 to 5pm.

Title: Asymptotics of the radiation field on cones

Abstract:
Radiation fields are rescaled limits of solutions of wave equations near “null infinity” and capture the radiation pattern seen by a distant observer.  They are intimately connected with the Fourier and Radon transforms and with scattering theory.  We consider the wave equation on a product cone and show that the associated radiation field has an asymptotic expansion; the exponents seen in this expansion are the resonances of the hyperbolic cone with the same link.  This talk is based on joint work with Jeremy Marzuola (building on prior work with Andras Vasy and Jared Wunsch).

The APDE seminar on Monday, 11/04 will be given by Jared Wunsch in Evans 939 from 4:10 to 5pm.

Title: A tale of two resolvent estimates

Abstract:
I will discuss two new results concerning the best of resolvent estimates and the worst of resolvent estimates.  In the former, case, that of nontrapping obstacles or metrics, we have obtained (in joint work with Galkowski and Spence) optimal, dynamically determined, constants in the standard non-trapping estimate for the (chopped off) resolvent.  In the latter case, that of obstacles or metrics that may have very strong trapping, I will discuss joint work with Lafontaine and Spence that shows the estimates to be a far, far better thing than you might have expected, provided you omit a small set of frequencies from consideration.

The APDE seminar on Monday, 10/28 will be given by Melissa Tacy in Evans 939 from 4:10 to 5pm.

Title: Adapting analysis/synthesis pairs to pseudodifferential operators

Abstract:
Many problems in harmonic analysis are resolved by producing
an analysis/synthesis of function spaces. For example the Fourier or
wavelet decompositions. In this talk I will discuss how to use Fourier
integral operators to adapt analysis/synthesis pairs (developed for the
constant coefficient PDE case) to the pseudodifferential setting. I will
demonstrate how adapting a wavelet decomposition can be used to prove
$L^{p}$ bounds for joint eigenfunctions.

The APDE seminar on Monday, 10/21 will be given by Benjamin Küster in Evans 939 from 4:10 to 5pm.

Title: Pollicott-Ruelle resonances and Betti numbers

Abstract:
In joint work with Tobias Weich, we study the multiplicity of
the Pollicott-Ruelle resonance 0 of the Lie derivative along the
geodesic vector field on the cosphere bundle of a closed negatively
curved Riemannian manifold, acting on flow-transversal one-forms. We
prove that if the manifold admits a metric of constant negative
curvature and the Riemannian metric is close to such a constant
curvature metric, then the considered resonance multiplicity agrees with
the first Betti number of the manifold, provided the latter does not
have dimension 3. In dimension 3 and for constant curvature, it turns
out that the resonance multiplicity is twice the first Betti number.

The APDE seminar on Monday, 10/07 will be given by Deng Zhang in Evans 939 from 4:10 to 5pm.

Title – The stochastic nonlinear Schrödinger equations: defocusing mass and energy critical cases

Abstract – In this talk we will present our recent results on stochastic nonlinear Schrödinger equations with linear multiplicative noise, particularly, in the defocusing mass-critical and energy-critical cases. More precisely, for general initial data, we obtain the global existence and uniqueness of solutions in both mass-critical and energy-critical case. When the quadratic variation of noise is globally bounded, we also prove the rescaled scattering behavior of stochastic solutions in the spaces L2, H1 as well as the pseudo-conformal space. Furthermore, the Stroock-Varadhan type theorem is derived for the topological support of solutions to stochastic nonlinear Schrödinger equations in the Strichartz and local smoothing spaces.