Category Archives: Uncategorized

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.

Quantum trajectories and the appearance of particle tracks in detectors

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.

Affine restriction estimates for surfaces in $\mathbb{R}^3$ via decoupling

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.

Interface and partial Bergman kernel

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.

3D lattice Anderson–Bernoulli localization

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.

A fractal uncertainty principle for discrete 2D Cantor sets

The HADES seminar on Tuesday, October 18th will be at 3:30 pm over zoom. Zoom link:

Speaker: Alex Cohen

Abstract: A fractal uncertainty principle (FUP) states that a function $f$ and its Fourier transform cannot both be large on a fractal set. These were recently introduced by Semyon Dyatlov and collaborators in order to prove new results in quantum chaos. So far FUPs are only understood for fractal sets in $\mathbb{R}$, and fractal sets in $\mathbb{R}^2$ remain elusive. In this talk, we prove a sharp fractal uncertainty principle for Cantor sets in $\mathbb{Z}/N\mathbb{Z} \times \mathbb{Z}/N\mathbb{Z}$, a discrete model for $\mathbb{R}^2$. The main tool is a quantitative form of Lang’s conjecture from number theory due to Beukers and Smyth.

A Proto Inverse Szemerédi–Trotter Theorem

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

Speaker: Olivine Silier

Abstract: A point-line incidence is a point-line pair such that the point is on the line. The Szemerédi–Trotter theorem says the number of point-line incidences for $n$ (distinct) points and lines in $\mathbb{R}^2$ is tightly upperbounded by $O(n^{4/3})$. We advance the inverse problem: we geometrically characterize `sharp’ examples which saturate the bound using the cell decomposition and crossing lemma proofs of Szemerédi–Trotter. This result is also an important step towards obtaining an $\epsilon$ improvement in the unit-distance problem. (Ongoing work with Nets Katz)

No background required, all welcome!

Quantitative Convergence of Semiclassical Particle Trajectories

The HADES seminar on Tuesday, September 27th will be at 3:30 pm in Room 740.

Speaker: Yonah Borns-Weil

Abstract: We study the trajectories of a quantum particle in a detector under repeated indirect measurement, in the semiclassical regime. We extend the results of Benoist, Fraas, and Fröhlich to discrete-time quantum maps on the quantized torus, and provide the first numerics illustrating the results. In addition, we derive quantitative bounds on the convergence to a classical trajectory based on classical dynamical measures of chaos of the system. No prior knowledge of quantum mechanics will be assumed. This is joint work with Izak Oltman.

Heat Trace Asymptotics for the Generalized Harmonic Oscillator on Scattering Manifolds

The HADES seminar on Tuesday, September 20th will be at 3:30 pm in Room 740.

Speaker: Moritz Doll

Abstract: On a scattering manifold, we consider a Schrödinger operator of the form
$H = -\Delta + V(x)$, where the potential satisfies a growth condition that
generalizes quadratic growth for Euclidean space. These types of
operators were first investigated by Wunsch, who proved a relationship
between singularities of the wave trace and a Hamiltonian flow. On the
other hand, it is easy to see that the heat trace is smooth away from
$t=0$ and our goal is to calculate the asymptotic expansion of the heat
trace as $t \to 0$. We follow the approach of Melrose by constructing a
suitable space on which the integral kernel of the heat operator is
smooth and then using the push-forward theorem to calculate the heat
trace asymptotics. This is based on ongoing joint work with Daniel Grieser.

Solitary waves for infinite depth gravity water waves with constant vorticity

The HADES seminar on Tuesday, September 13th will be at 3:30 pm in Room 740.

Speaker: James Rowan

Abstract:The existence of solitary waves has been a key question for mathematical models of water waves since the 1830s. The model I will discuss is the infinite depth, gravity, zero surface tension case in the presence of nonzero constant vorticity, a model that applies in settings with countercurrents. Because the infinite depth gravity water waves equations with constant vorticity are well-approximated (on a suitable timescale) by the Benjamin-Ono equation, which has solitary waves, one might expect a solitary wave to exist. We show that this is indeed the case, and that this wave is close to the solitary wave for the Benjamin-Ono soliton. This work is joint with Lizhe Wan.