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 $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ be a measure space, and $1\le p\le \mathrm{\infty }$. A subspace $E\subseteq L_p(\mu )$ is said to do stable phase retrieval (SPR) if there exists a constant $C\ge 1$ such that for any $f,g\in E$ we have     $$\underset{|\lambda |=1}{inf}\Vert f-\lambda g\Vert \le C\Vert |f|-|g|\Vert .$$    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.

The HADES seminar on Tuesday, November 1st will be at 3:30 pm in Room 740.

Speaker: Peng Zhou

Abstract: Let $(M,\omega )$ be a smooth compact Kahler manifold and $(L,h)$ a positive hermitian line bundle on $M$. Given a smooth real valued function $H$ on $M$, we may consider the Toeplitz quantization $T_{H,k}$ acting on $H^0(M,L^k)$. Let $[a,b]$ be an interval, the partial Bergman kernel is the orthogonal projection from $H^0(M,L^k)$ to sum of eigenspaces of $T_{H,k}$ with eigenvalue within $[a,b]$. We study the behavior of the projection kernel near the “boundary”. This was based on joint work with Steve Zelditch.

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

Speaker: Lingfu Zhang

Abstract: I will talk about the Anderson model (i.e., the random Schordinger operator of Laplacian plus i.i.d. potential on the lattice). It is widely used to understand the conductivity of materials in condensed matter physics. An interesting phenomenon is Anderson localization, where eigenfunctions have exponential decay, and the spectrum of this random operator is pure-point (in some intervals). This phenomenon was first rigorously established in the 1980s, while one main remaining question is on the case of Bernoulli potential. A continuous space analog of this problem was proved in a seminal paper by Bourgain and Kenig, and the 2D lattice setting was proved by Ding and Smart. Following their framework, we prove 3D lattice Anderson-Bernoulli localization near the edges of the spectrum. Our main contribution is proving a 3D discrete unique continuation principle, using combinatorial and polynomial arguments. This is joint work with Linjun Li.