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.

The HADES seminar on Tuesday, October 18th will be at 3:30 pm over zoom. Zoom link:https://berkeley.zoom.us/j/96232331895.

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.

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!

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.

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.

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.

The HADES seminar on Tuesday, September 6th will be at 3:30 pm in Room 740.

Speaker: Zhongkai Tao

Abstract: The Schur complement formula is a very simple formula in linear algebra. Yet it is very useful in spectral theory. I will introduce the Schur complement formula and talk about how to use it to prove a strong convergence of kinetic Brownian motion to the Laplace operator on locally symmetric spaces. This is joint work with Qiuyu Ren.

The HADES seminar on Tuesday, May 10 will be at 3:30 pm in Room 1015 (Notice the room change).

Speaker: Benjamin Pineau

Abstract: Let $(X, \mathcal A, \mu)$ be a measure space. Let $V$ be a closed subspace of the (real or complex) Hilbert space $L^2 = L^2 (\mu)$. We say that $V$ does Holder-stable phase retrieval if there exists a constant $C < \infty$ and $\gamma \in (0, 1]$ such that \begin{equation}\label{eq} \min_{|z|=1} \|f − zg\|_{L^2} \leq C\||f| − |g|\|_{L^2}^\gamma (\|f\|_{L^2} + \|g\|_{L^2} )^{1−γ}\,\forall f, g \in V,(*)\end{equation}

Recently, Calderbank, Daubechies, Freeman, and Freeman have studied real subspaces of real-valued $L^2$ for which (*) holds with $\gamma = 1$ and constructed the first examples of such infinite-dimensional subspaces. In this situation, if $|f|$ is known then $f$ is uniquely determined almost everywhere up to an unavoidably arbitrary global phase factor of $\pm 1$. Moreover, if $|f|$ is known within a small tolerance in norm then up to such a global phase factor, f is determined within a correspondingly small tolerance. This issue arises for instance in crystallography, where one seeks to recover an unknown function $F \in L^2 (\mathbb R)$ from the absolute value of its Fourier transform $\hat F$.

In this talk, I will discuss a set of simple sufficient conditions for constructing infinite-dimensional (real and complex) subspaces $V \subset L^2 (\mu)$ which satisfy (*) and show how to construct some natural examples in which (*) holds. These examples include certain variants of Rademacher series and lacunary Fourier series. This is a joint work with Michael Christ and Mitchell Taylor.

The HADES seminar on Tuesday, May 3rd will be at 3:30 pm in Room 740.

Speaker: Michael Christ

Abstract: An archetypal (bilinear) oscillatory integral inequality states that $$ \Big| \iint_{\mathbb{R}^d\times\mathbb{R}^d} f(x)\,g(y)\,e^{i\lambda\phi(x,y)}\,\eta(x,y)\,dx\,dy\Big|\le C|\lambda|^{-\gamma} \|{f}\|_{L^2}\|{g}\|_{L^2}$$ where $\lambda\in\mathbb{R}$ is a large parameter, $\phi$ is a smooth real-valued phase function which is nondegenerate in a suitable sense, $f,g$ are arbitrary $L^2$ functions, $\eta$ is a  smooth compactly supported cutoff function, and $\gamma>0$ and $C<\infty$ depend on $\phi$ but not on $f,g,\lambda$. Its main features are the decaying factor $|\lambda|^{-\gamma}$, the absence of any smoothness hypothesis on the measurable factors $f,g$, and the interplay between the structure of $\phi$ and the product structure of $f(x)\,g(y)$. If $\phi$ is nonconstant then $e^{i\lambda\phi}$ oscillates rapidly, creating cancellation that potentially results in smallness of the integral.

Implicitly oscillatory integrals involve no overtly oscillatory factor $e^{i\lambda\phi}$; instead, the measurable factors $f_j$ are themselves assumed to be oscillatory, but in a less structured way. A multilinear integral of this type takes the form \[ \int_{\mathbb{R}^2} \prod_{j=1}^N (f_j\circ\varphi_j)(x)\,\eta(x)\,dx\] where $\varphi_j:\mathbb{R}^2\to\mathbb{R}^1$ are smooth submersions, and the functions $f_j$ are merely measurable. The desired upper bound is expressed in terms of strictly negative order Sobolev norms of these functions. Thus the functions $f_j$ are rapidly oscillatory in the sense that they consist mainly of high frequency components.

I will give an introduction to recent (and ongoing) work on this topic, and on associatedsublevel set inequalities.