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.

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 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.

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.

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?

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.

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.

The HADES seminar on Tuesday, November 29th will be at 3:30 pm on Zoom.

Speaker: Martin Fraas

Abstract: Quantum trajectory models time evolution of a quantum system including a particular measurement strategy. Quantum trajectories were introduced in the 1970s and, in the last decade, became a standard experimental tool to monitor and control quantum systems with few degrees of freedom. In this talk, I will introduce the theory of quantum trajectories, and discuss a model example of a particle whose position is repeatedly measured.

The HADES seminar on Tuesday, November 8th will be at 3:30 pm in Room 740.

Speaker: Jianhui (Franky) Li

Abstract: We will discuss some $L^2$ restriction estimates for smooth compact surfaces in $\mathbb{R}^3$ with affine surface measure and certain powers thereof. The primary tool is a decoupling theorem for these surfaces. The results are also uniform for polynomial surfaces of bounded degrees and coefficients. Some of the results we will discuss are joint with Tongou Yang.