Research Seminars: Applied and Computational Mathematics

Spring 2024

Time & Location: Typically talks will be in Gibson Hall 126 at 3:00 PM on a Friday.
Organizers: Punshon-Smith, Samuel and Buvoli, Tommaso

Archives

 

February 2
Title: The planar Coulomb gas on a Jordan curve

 Klara Courteaut - NYU Courant

Abstract:  The eigenvalues of a uniformly distributed unitary matrix (CUE) have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve, at any positive temperature. In a paper with Johansson, we obtained the asymptotic partition function and the Laplace transform of linear statistics at any positive temperature. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator.

Time: 3:00pm
Location:  Gibson Hall 126

 

February 16
Title: Self-Similar Blow up Profiles for Fluids via Physics-Informed Neural Network

Javier Gomez Serrano - Brown University

Abstract:  In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution (or asymptotically self-similar solution) for different equations in fluid dynamics, such as Euler or Boussinesq. The new numerical framework is shown to be both robust and readily adaptable to several situations. Joint work with Tristan Buckmaster, Gonzalo Cao-Labora, Ching-Yao Lai and Yongji Wang.

Time: 3:00
Location:  Gibson Hall 126

 

March 1
Title: On Maxwell-Bloch systems with inhomogeneous broadening and one-sided nonzero background

Barbara Prinari - University at Buffalo

Abstract:  We present the inverse scattering transform to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The inverse problem is formulated in terms of a suitable matrix Riemann-Hilbert problem, and the formulation of the direct scattering problem combines features of the methods with decaying as well as non-decaying fields. We also discuss the asymptotic state of the medium and of the optical pulse.

Time: 3:00
Location:  Gibson Hall 126

 

March 8
Title: A tractable algorithm, based on optimal transport, for computing adversarial training lower bounds.

Nicolas Garcia-Trillos - University of Wisconsin Madison

Abstract:  Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this talk, I will leverage the MOT connection to discuss new computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk. The key insight in the AT setting is that one can harmlessly truncate high order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. I’ll present a rigorous complexity analysis of the proposed algorithm and validate our theoretical results experimentally on the MNIST and CIFAR-10 datasets, demonstrating the tractability of our approach. This is joint work with Matt Jacobs (UCSB), Jakwang Kim (UBC), and Matt Werenski (Tufts).

Time: 3:00
Location:  Gibson Hall 126

 

March 15
Title: Response theory for dissipative SPDEs.

Giulia Carigi

Abstract:  A framework suitable to establish response theory for a class of nonlinear stochastic partial differential equations is presented. With response theory we mean in this context the following: one considers a dynamical system whose dynamical law depends on a parameter (here given by an SPDE where the parameter is in the forcing) and we say that one has a response theory if one can show a regularity in the dependence of the invariant measure on the parameter (here differentiability or Hölder continuity in weak topology). The results are applied to the 2D stochastic Navier-Stokes equation and the stochastic two-layer quasi-geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. This work is jointly with Jochen Bröcker (University of Reading) and Tobias Kuna (University of L’Aquila).

Time: 3:00
Location:  Gibson Hall 126

 

March 22
Title: The restriction of the Laplacian operator on manifolds

Padi Fuster Aguilera - University of Colorado Boulder

Abstract:  On a Riemannian manifold, there is no canonical Laplace operator for vectors fields or forms, and it is not clear what is the “correct” Laplacian to use when formulating fluid dynamics equations. In this talk, we will walk through different approaches for obtaining a viscosity operator when considering a Riemannian submanifold in the Euclidean space, as well as present some concrete examples.

Time: 3:00
Location:  Gibson Hall 126

 

April 5
Title: TBA

Grady Wright - Boise State

Abstract:  TBA

Time: 3:30
Location:  Gibson Hall 126

 

April 19
Title: TBA

Cole Graham - Brown University

Abstract:  TBA

Time: 3:00
Location:  Gibson Hall 126