Author Archives: zhongkaitao

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.

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.

Fractal uncertainty principle via Dolgopyat’s method in higher dimensions

The HADES seminar on Tuesday, March 21st will be at 3:30 pm in Room 740.

Speaker: Zhongkai Tao

Abstract: The fractal uncertainty principle (FUP) was introduced by Dyatlov–Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in proving spectral gaps and quantum chaos. Bourgain–Dyatlov proved the fractal uncertainty principle in dimension $1$, which leads to an essential spectral gap, and was applied by Dyatlov–Jin and Dyatlov–Jin–Nonnenmacher to show quantum limits on closed negatively curved surfaces have full support. The higher dimensional version of the fractal uncertainty principle for large fractal sets is widely open, and there is a recent work by Alex Cohen who addressed the case of $2$ dimensional arithmetic Cantor sets.

I will talk about the history of the fractal uncertainty principle and explain its applications via examples. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov–Jin using Dolgopyat’s method. As an application, we get effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups. The new ingredients include a “non-orthogonality condition”, an explicit construction of Christ cubes and a statistical argument.

The Tensorial X-ray Transform on Asymptotically Conic Manifolds

The HADES seminar on Tuesday, March 14th will be at 3:30 pm in Room 740.

Speaker: Qiuye Jia

Abstract: In this talk I will discuss the invertibility of the geodesic X-ray
transform on one forms and 2-tensors on asymptotically conic
manifolds, up to the natural obstruction, allowing existence of
certain kinds of conjugate points. We use the 1-cusp
pseudodifferential operator algebra and its semiclassical foliation
version introduced and used by Vasy and Zachos, who showed the same type
invertibility on functions.

The complication of the invertibility of the tensorial X-ray
transform, compared with X-ray transform on functions, is caused by
the natural kernel of the transform consisting of `potential
tensors’. We overcome this by arranging a modified solenoidal gauge condition,
under which we have the invertibility of the X-ray transform.

Well-posedness for the Surface Quasi-Geostrophic front equation

The HADES seminar on Tuesday, March 7th will be at 3:30 pm in Room 740.

Speaker: Ovidiu-Neculai Avadanei

Abstract: We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation in the non-periodic case, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim-Tataru.

This is joint work with Albert Ai.

Curve tangencies and maximal functions

The HADES seminar on Tuesday, February 28th will be at 3:30 pm in Room 740.

Speaker: Joshua Zahl

Abstract: I will discuss a class of maximal operators that arise from averaging functions over thin neighborhoods of curves in the plane. Examples of such operators are the Kakeya maximal function and the Wolff and Bourgain circular maximal functions. To understand the behavior of these operators, we need to study the possible intersection patterns for collections of curves in the plane: how often can these curves intersect, how often can they be tangent, and how often can they be tangent to higher order?

Stable phase retrieval in function spaces

The HADES seminar on Tuesday, February 14th will be at 3:30 pm in Room 740.

Speaker: Mitchell A. Taylor

Abstract: Let $(\Omega,\Sigma,\mu)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(\mu)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have     \begin{equation}       \inf_{|\lambda|=1} \|f-\lambda g\|\leq C\||f|-|g|\|.    \end{equation}    In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics.

In this talk, I will present some elementary examples of subspaces of $L_p(\mu)$ which do stable phase retrieval, and discuss the structure of this class of subspaces. This is based on a joint work with M. Christ and B. Pineau, as well as a joint work with D. Freeman, B. Pineau and T. Oikhberg.

Almost-sure scattering below scaling regularity for the nonlinear Schrodinger equation in high dimensions

The HADES seminar on Tuesday, January 24th will be at 3:30 pm in Room 740.

Speaker: Marsden Katie Sabrina Catherine Rosie

Abstract: In this talk we will discuss the Cauchy problem for the energy-critical nonlinear Schrodinger equation in high dimensions. It is well-known that this problem is well-posed for data in Sobolev spaces with regularity $s>1$. The critical case $s=1$ was also shown to be globally well-posed with scattering by Ryckman-Vişan in the mid-2000s. In this talk we will show that even for some super-critical regularities, $s<1$, the equation is “almost-surely” globally well-posed with respect to a certain randomisation of the initial data and exhibits scattering.