On a nonlinearly coupled stochastic fluid-structure interaction model.

The HADES seminar on Tuesday, September 26th will be at 3:30pm in Room 740.

Speaker: Krutika Tawri

Abstract: In this talk, we will present a constructive approach to investigate
the existence of martingale solutions to a benchmark fluid-structure interaction
problem that involves an incompressible, viscous fluid interacting with a linearly
elastic membrane subjected to a multiplicative stochastic force. The fluid flow
is described by the Navier-Stokes equations while the elastodynamics of the
thin structure is modeled by the Koiter shell equations. We will discuss the
challenges arising due to the random motion of the time-dependent fluid domain
and present our recent findings. This is joint work with Sunčica Čanić.

Low regularity well-posedness for the surface quasi-geostrophic front equation

The HADES seminar on Tuesday, September 19th will be at 3:30pm in Room 740.

Speaker: Ovidiu-Neculai Avadanei

Abstract: We consider the well-posedness of the generalized surface quasi-geostrophic (gSQG) front equation. In the present paper, by making use of the null structure of the equation, we carry out a paradifferential normal form analysis in order to obtain balanced energy estimates, which allows us to prove the low regularity local well-posedness of the g-SQG front equation in the non-periodic case at a low level of regularity (in the SQG case, it is only one half of a derivative above scaling). In addition, we establish global well-posedness theory for small and localized rough initial data, as well as modified scattering, by using the testing by wave packet approach of Ifrim-Tataru.

This is joint work with Albert Ai.

Convergence of Lindblad Dynamics towards Fokker-Planck Equations beyond Ehrenfest time

The HADES seminar on Tuesday, September 12th will be at 3:30pm in Room 740.

Speaker: Zhen Huang

Abstract: The goal of this talk is to introduce the topic of semi-classical analysis of open quantum systems to the audience.
Semi-classical analysis of closed quantum systems is a very well-established topic (for example, see Zworski’s book). However, rigorous analytical studies of open quantum systems in the semi-classical regimes are rarely done so far. This is partly because open quantum dynamics often do not have properties as nice as Schrodinger equations. The lack of analytic results also hinders the design and analysis of numerical algorithms.
Quantum-classical correspondence in the Schrodinger equation is well known to hold for O(log(1/h)) time scale (h is the non-dimensionalized Planck constant). We will discuss a very recent work that addresses the quantum-classical correspondence for the simplest open quantum system model (which is still complicated), i.e. Lindblad dynamics. We present a rigorous proof that a classical description is valid for O(1/sqrt(h)) time, which is much longer than the Ehrenfest timescale. We will also discuss several open questions along this line, and possible generalizations to more complicated open quantum systems.

Illposedness for dispersive equations: Degenerate dispersion and Takeuchi-Mizohata condition

The HADES seminar on Tuesday, September 5th will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, my aim is to provide a unified viewpoint on two illposedness mechanisms for dispersive equations, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. For a linear dispersive equation, degenerate dispersion is a property of the principal term in the presence of degenerating coefficients, and the Takeuchi-Mizohata condition concerns the effect of the subprincipal term. First, I will demonstrate how these two effects manifest in the context of wave packet construction. Then, I will exhibit a simple energy and duality argument (similar to testing by wave packets of Ifrim-Tataru) that allows one to extend this illposedness phenomenon to a variety of quasilinear(!) degenerate dispersive PDEs, including singular generalized SQG, surface growth model, Rosenau-Hyman model, etc. This talk is mostly based on joint projects with In-Jee Jeong and Dongho Chae.

Three Things About Polynomials

The HADES seminar on Tuesday, August 29th will be at 3:30 pm in Room 740.

Speaker: Ruixiang Zhang

Abstract: I will talk about three interesting ingredients that go into the results on Hörmander type operators I presented at APDE seminar (joint with Shaoming Guo and Hong Wang). They are all related to algebraic or geometric properties of multivariate polynomials.

Nonlinear Coupled Systems of PDEs for Modeling of Multi-Lane Traffic Flow Problems

The HADES seminar on Tuesday, May 9th will be at 3:30 pm in Room 740.

Speaker: Nadim Saad

Abstract: In this talk, first, we start with the traditional Lighthill-Whitham-Richards (LWR) model for unidirectional traffic on a single road and present a novel traffic model which incorporates realistic driver behaviors through a non-linear velocity function. We develop a particle-based traffic model to inform the choice of velocity functions for the PDE model. We incorporate various driver behaviors in the particle-based model to generate realistic velocity functions. We explore various impacts of numerous driving behaviors on different traffic situations using both the PDE model and the particle-based model, and compare the traffic distributions and throughput of cars on the road obtained by both models. Second, we extend the one-lane model to a multi-lane traffic model and incorporate source functions representing lanes exchanges. We derive desirable mathematical conditions for source functions to ensure $L^1$ contractivity for the system of PDEs. We build a multi-lane particle-based model to inform the choice of source functions for the PDE model. We study various driver behaviors in the particle-based model to develop realistic source functions. We explore various impacts of different driving scenarios using both models.

Quantitative stratification for harmonic maps

The HADES seminar on Tuesday, May 2nd will be at 3:30 pm in Room 740.

Speaker: Jason Zhao

Abstract: It is well-known that stationary harmonic maps are singular on a set of at least codimension $2$. We will exposit the work of Cheeger and Naber which improves the result by establishing effective volume estimates of tubular neighborhoods of the singular set. The primary purpose of the talk is to highlight the two key ingredients in the proof,

  • quantitative differentiation; functions in a given class cannot be far away from the infinitesimal behavior except at finitely many scales,
  • cone-splitting; lesser symmetries can be combined to form a greater symmetry,

which have proven extremely robust in the fields of geometric PDE and metric geometry. Combined with $\epsilon$-regularity theorems, one can pass to a priori estimates, e.g. for minimizing harmonic maps in $W^{1, p} \cap W^{2, p/2}$ in the sub-critical regime $p < 3$.

Asymptotics of non-linear and linear waves on asymptotically flat spacetimes in three space dimensions

The HADES seminar on Tuesday, April 25th will be at 3:30 pm in Room 740.

Speaker: Shi-Zhuo Looi

Abstract: In this talk, we start with basic examples of wave decay and then delve into the investigation of asymptotic expansions for both non-linear and linear wave propagation in asymptotically flat spacetimes, allowing for non-stationary spacetimes without spherical symmetry assumptions. The analysis encompasses Schwarzschild spacetime and Kerr spacetimes within the full subextremal range. We present an exposition of a novel approach combining either integrated local energy decay or the limiting absorption principle, the r^p method, and, from a spectral perspective, resolvent expansions near zero energy. Potential applications of this research include scenarios involving waves interacting with spatially-localized objects, such as solitons.

Magic Angles in Randomly Perturbed Twisted Bilayer Graphene

The HADES seminar on Tuesday, April 18th will be at 3:30 pm in Room 740.

Speaker: Izak Oltman

Abstract: One way to predict magic angles in twisted bilayer graphene (TBG) is to look for flat bands of the Bloch-Floquet transformed Hamiltonian modeling the chiral limit of the continuum model for TBG. In this talk, I will address the question: What happens to the spectrum when this Hamiltonian is randomly perturbed?

To answer this, I will provide an overview of multiscale analysis describing localization and delocalization for random self-adjoint operators and show how it applies to the TBG setting.

This is based on joint work with Hermann-Weyl lecturer Dr. Simon Becker.

Kahler-Einstein Metrics and the Complex Monge-Ampere Equation

The HADES seminar on Tuesday, April 11th will be at 3:30 pm in Room 740.

Speaker: Garrett Brown

Abstract: A central question in geometric analysis is as follows: given a smooth manifold, can one find a Riemannian metric with “special” curvature properties? A classic example of this is the uniformization theorem, which states that any smooth 2-manifold has a metric of constant curvature, and the Gauss-Bonnet theorem relates the sign of the curvature to the genus of the surface.

In complex geometry, one can consider the possible higher dimensional generalizations of the uniformization theorem. One candidate is the following: given a complex manifold, does it have a metric which is Kahler-Einstein, that is, the complex structure is parallel with respect to the metric, and the metric is proportional to the Ricci curvature? This question was answered in the affirmative by Aubin and Yau in the negative first Chern class case, and by Yau in the zero first Chern class case via the more general Calabi conjecture in the late 70s (the positive case was resolved in 2015, requiring a deeper analysis). The crucial step is establishing a priori estimates for a fully nonlinear elliptic equation.

I will do my best to explain the ideas from geometry that are beyond a basic acquaintance with Riemannian geometry.