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

Speaker: Sung-Jin Oh

Abstract: The electron magnetohydrodynamics equation (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, (E-MHD) is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. In this talk, I’ll discuss a recent proof, joint with In-Jee Jeong, of the local wellposedness of the Cauchy problems for (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity.

More specifically, my goal is to explain the main new ideas introduced in this work and the related work of Pineau-Taylor on quasilinear ultrahyperbolic Schrodinger equations, which also have nondegenerate but nonelliptic principal terms. Both works significantly improve upon the classical work of Kenig-Ponce-Rolvung-Vega on such PDEs, in the sense that the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola-Metcalfe-Tataru in the case of an elliptic principal term.

The HADES seminar on Tuesday, March 4th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: Solving an underdetermined PDE such as a divergence equation plays a central role in problems like general relativistic gluing. Starting from divergence equations on Euclidean spaces, I will introduce a method of constructing integral solution operators for a wide class of underdetermined differential operators with prescribed support properties. By duality, this will also produce integral representation formulas for overdetermined differential operators. The method extends various ideas from Bogovskii, Oh-Tataru, and Reshetnyak. The construction is based on an assumption called the recovery on curves condition (RC) imposed on the operators. I will also give an algebraic sufficient condition of RC that is easier to verify, which is called the finite-codimensional cokernel condition (FC). At the end, I will show some examples that satisfy FC on space forms and derive their integral formulas in the flat case. This is joint work with Philip Isett, Sung-Jin Oh, and Zhongkai Tao.

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

Speaker: Jason Zhao

Abstract: It has been observed both experimentally and mathematically that solutions to linear dispersive equations, such as the Schrodinger and Airy equations, posed on the torus exhibit dramatically different behaviors between rational and irrational times. For example, the evolution of piecewise constant data remains so at rational times, while it becomes continuous and fractalised at irrational times. A natural question to ask is whether this Talbot effect, as it broadly known, persists under non-linear dispersive flows. Focusing on the KdV equation, we will present two perspectives which follow in the spirit of the seminal works of Bourgain (1993) and Babin-Ilyin-Titi (2011): the first is the non-linear smoothing effect observed by Erdogan-Tzirakis (2013), and the second is the numerical work of Hofmanova-Schratz (2017) and Rousset-Schratz (2022}.

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

Speaker: Zhongkai Tao

Abstract: The Strichartz estimate is an important estimate in proving the well-posedness of dispersive PDEs. People believed that the lossless Strichartz estimate could not hold on manifolds with trapping (for example, the local smoothing estimate always comes with a logarithmic loss in the presence of trapping). Surprisingly, in 2009, Burq, Guillarmou, and Hassell proved a lossless Strichartz estimate for manifolds with trapping under the “pressure condition”. I will talk about their result as well as our recent work with Xiaoqi Huang, Christopher Sogge, and Zhexing Zhang, which goes beyond the pressure condition using the fractal uncertainty principle and a logarithmic short-time Strichartz estimate.

The HADES seminar on Tuesday, February 4th, will be at 3:00pm in Room 740 (UNUSUAL TIME).

Speaker: Mengxuan Yang

Abstract: Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles, the resulting material is superconducting. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer–MacDonald Hamiltonian, including the generic existence of Dirac cones and the mathematical characterization of magic angles. I will also discuss topological aspects of this model, as well as some new mathematical discoveries in twisted multilayer graphene.

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.

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 $ϵ$-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 $p<3$.

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.

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.

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.