UToronto Graduate Analysis Seminar and Joaquin's personal website

Adam presented on a 2011 paper of J. Kato and Pusateri concerning an alternative proof of the long-time behaviour of small solutions to the cubic nonlinear Schroedinger equation. This proof utilises the method of space-time resonances, an elegant and easily generalized framework for understanding the long-time asymptotics of nonlinear dispersive waves.

Mitchell spoke on his recent work (joint with Ben Pineau, Mihaela Ifrim, and Daniel Tataru) on the non-existence of solitary waves in the pure capillary 2D water waves equations. The proof of this non-existence result used the holomorphic-coordinate formulation of the water waves problem and a careful understanding of the "Tilbert transform" and its symbol.

During this meeting we featured three short talks from some Master's students in our department. Cameron Davis spoke on aggregation equations, Kevin Dembski spoke on blow-up of solutions to ideal fluid models, and Cameron Martin spoke about geodesics in discrete last passage percolation. Click the file icon above to download a zip archive of all three speakers' lecture notes.

Sebastian gave us an introduction to perturbation theory in the context of quantum dynamics with diferenced scales. This talk was a continuation of his introduction to the Born-Oppenheimer theory. He focused in adiabatic perturbation theory.

Fabian spoke about filtering theory in stochastic analysis. after introducing us to the basics of Lévy processes, we saw how the main problem of filtering theory could be studied using tools from stochastic DEs. The Girsanov change-of-measure formula and the Zakai equation were described.

Joaquín continued the theme of his January 29th presentation, describing a program for studying rough spacetimes in general relativity through optimal transport theory. This time, the main focus was on a stochastic DE-type approach to the problem, rather than Wasserstein geometry.

Tomás gave us an introduction to the mathematics of the Sherrington-Kirkpatrick model of spin glasses. Much attention was paid to showing how the problem can be understood by adapting objects from thermodynamics, such as Gibbs measures. The main result of the presentation was the statement and partial proof of the Parisi formula.

David spoke on harmonic measure, extremal lengths, and the Carleson-Makarov theorem. Connections between Brownian motion and potential theory were critical to understanding many of the main ideas: we saw lots of way to view constructions from elliptic PDE theory as probabilistic objects and vice versa.

Adam spoke on some classical results regarding the scattering of small solutions to the generalized Benjamin-Bona-Mahony equation. We saw that establishing an upper bound for the time decay of linear solutions was critical to proving scattering. Duhamel's principle and the bootstrap method could be used to handle nonlinear terms in the model PDE perturbatively.

Tristan spoke on the connection between Wasserstein geometry (arising from optimal transport theory) and generative modelling. We learned about the special structure of the Wasserstein 1-distance, and saw how one could use neural networks to approximately compute Wasserstein distances and solve optimal transport problems.

Sebastian introduced us to the Born-Oppenheimer approximation in quantum molecular dynamics. This approximation was used to motivated an investigation of perturbation theory (a la Kato-Rellich) and the Feshbach-Schur projection method for estimating the eigenvalues of perturbed Schroedinger operators.

Saeyon spoke to us about wave equations in black hole spacetimes. We learned about how the Aretakis charge contributes to the asymptotic behaviour of solutions to such equations.

Carrie spoke about concentration compactness and its applications to minimizing the energy of certain nonlocal interactions. We learned that concentration compactness is a useful tool for dividing candidate minimizing sequences into particular classes: specifically, the main focus of the talk was related to so-called "dichotomous" sequences.

Gianmarco (from the University of Birmingham) gave us a talk on extremizing Strichartz estimates with applications to Schroedinger-type equations. Along the way, we saw some results on the relationship between Fourier restriction bounds and the geometry of dispersion relations.

Cintia (from Aalto University, Finland) spoke on the existence of parabolic minimizers to the total variation flow on metric measure spaces. We learned that, with the right definitions in place, one can adapt techniques from variational calculus to solve problems in the very general setting of metric measure spaces.

Joaquín presented a curvature-dimension condition in terms of optimal transport theory. The goal is to understand results in general relativity (including Hawking's singularity theorem) which are re-written using optimal transport machinery.

In his talk, Adam described how to sensibly formulate the generalized Benjamin-Bona-Mahony-Burgers equation on networks modelling a model a subset of the human arterial tree. He also explained the results of some numerical simulations of this model, which sometimes predict nonlinear wave reflection off of network vertices.