Fall 2023

Date	Speaker	Topic
Sept. 7	Organizational Meeting	Organizational Meeting .
Sept. 14	TBA	TBA Abstract: TBA .
Sept. 21	TBA	TBA Abstract: TBA .
Sept. 28	Kasso Okoudjou Tufts University	Optimal point distributions on the d-dimensional unit sphere Abstract: Finding optimal configurations of point masses under the action of energy functionals defined on $d-$ dimensional Euclidian space appears in several fields such as numerical integration, coding theory, and chemistry. The problem often involves minimizing a functional of the form $\iint_{S^{d-1}\times S^{d-1}}f(x, y)d\mu(x)d\mu(y)$ over all probability measures $\mu$ defined on the unit sphere $S^{d-1}$ , or its discrete version, minimizing $\sum_{k\neq \ell}f(\varphi_k, \varphi_\ell)$ over all sets of $N$ vectors $\Phi=\{\varphi_k\}_{k=1}^N\subset S^{d-1}$ for some function $f$ . In this talk, we will consider the case where $f(x, y)=\|\langle x, y\rangle\|^p$ for $p\in [0, \infty]$ which is referred to as the $p^{th}$ frame potentials. We will motivate this special case with applications from frame theory, spherical designs, and the Zauner conjecture in quantum information theory. After a survey of recent results concerning the minimizers of the $p^{th}$ frame potentials, we will focus on the interplay between the continuous and the discrete problems, especially when the dimension $d$ is small.
Oct. 5	Effie Papageorgiou Universität Paderborn & Tufts University	Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces Abstract: The Central Limit Theorem of probability represented in the PDE setting can be described as follows: consider the heat equation on $\mathbb{R}^n$ , $\begin{align} \partial_{t}u(t,x)\,=\,\Delta_{x}u(t,x),\quad u(0,x)\,=\,f(x), \end{align}$ with initial data $f \in L^1(\mathbb{R}^n)$ . Denote by $M = \int_{\mathbb{R}^n}f(x)\, dx$ the mass and by $h_t(x) = (4\pi t)^{-\frac{n}{2}}e^{-\frac{\\|x\\|^2}{4t}}$ the heat kernel. Then the following asymptotics are known to hold in $L^p(\mathbb{R}^n)$ , for all $1 \leq p \leq \infty$ : $\lim_{t\rightarrow +\infty} t^{\frac{n}{2p'}}\\|u(t,\cdot)-M\,h_t\\|_{L^p(\mathbb{R}^n)}=0.$ Analogous heat asymptotics may or may not hold on Riemannian manifolds. Our aim is to discuss noncompact symmetric spaces $G/K$ of arbitrary rank, generalizing earlier results of J.L. V{\’a}zquez on real hyperbolic spaces. More precisely, we discuss the heat equation related to the Laplace-Beltrami operator and to the distinguished Laplacian. In the first case, if the data is bi- $K$ -invariant, the convergence is true, but unlike the Euclidean case, if this symmetry on initial data is removed, the convergence may fail. In the case of the distinguished Laplacian, we observe phenomena resembling to the Euclidean setting. Joint work with J.-Ph. Anker (Université d’ Orléans, France) and H.-W. Zhang (Ghent University, Belgium). .
Oct. 12	Fulton Gonzalez Tufts University	The Snapshot Problem for the Wave Equation Abstract: By definition, a \emph{wave} is a $C^\infty$ solution $u(x,t)$ of the wave equation on $\mathbb R^n$ , and a \emph{snapshot} of the wave $u$ at time $t$ is the function $u_t$ on $\mathbb R^n$ given by $u_t(x)=u(x,t)$ . We show that there are infinitely many waves with given $C^\infty$ snapshots $f_0$ and $f_1$ at times $t=0$ and $t=1$ respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave $u$ to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to shifted wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the shifted wave equation on the $n$ -sphere. .
Oct. 19	Dominique Maldague MIT	Small cap decoupling for the moment curve in R^3 Abstract: In decoupling theory, we study $L^p$ estimates for exponential sums whose frequencies live in restricted sets. It is easier for a function whose frequencies lie in a line to stay large than for a function whose frequencies lie in a curve. Thus allows us to prove strong $L^p$ bounds, called decoupling estimates, when the restricted sets satisfy some curvature assumptions. I will focus on small cap decoupling, which is a version of decoupling with applications in number theory. I will explain how the high-low method for decoupling, established for the parabola by Guth, Maldague, and Wang, may be generalized to the moment curve in $R^3$ . This leads to the full solution of the 3-dimensional case of Conjecture 2.5 from the original small cap decoupling paper of Demeter, Guth, and Wang. This is based on joint work with Larry Guth. .
Oct. 26	Marjorie Drake MIT	Finiteness Principles for Smooth Convex Functions Abstract: Let $E \subset \mathbb{R}^n$ be a compact set and the function $f:E \to \mathbb{R}$ . Let $C_c^{1,1}(\mathbb{R}^n)$ be the space of convex, differentiable functions with Lipschitz continuous gradient. How can we tell if there exists a convex function $F \in C_c^{1,1}(\mathbb{R}^n)$ extending $f$ (satisfying $F\|_E=f\|_E$ )? I will present recent work of mine proving that if a function satisfies a finiteness hypothesis for strongly convex functions in $C^{1,1}_c(\mathbb{R}^n)$ , then there exists a strongly convex function in $C^{1,1}_c(\mathbb{R}^n)$ extending the given function. Despite obstacles to their direct application, this theorem brings techniques developed by P. Shvartsman, C. Fefferman, A. Israel, and K. Luli for smooth extension and selection to smooth, convex extension. A key component of this result is D. Azagra and C. Mudarra’s theorem on extension of 1-jets of functions in $C^{1,1}_c(\mathbb{R}^n)$ . We will finish with a discussion of challenges in adapting my more general 1d-theorem ( $E \subset \mathbb{R}$ ) to higher dimensions. .
Nov. 2	Masaharu Kobayashi Hokkaido University	Rendered by QuickLaTeX.com Abstract: Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $\\| f \\|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2} ( \|{\mathcal F}[f](\xi)\| ( 1+ \|\xi\| )^s )^q d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$ . We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ . This is joint work with Prof. Sato (Yamagata University). .
Nov. 9	Joris Roos UMass Lowell	Poincare and isoperimetric inequalities on the Hamming cube Abstract: Classical isoperimetric inequalities can be viewed as providing a lower bound on the size of the boundary of a set in terms of its area or volume. There are some well-known versions of such inequalities for subsets of the Hamming cube and we are particularly interested in sharp inequalities near and at the critical exponent $1/2$ which are still open. These are closely related to $L^1$ Poincare inequalities for functions on the Hamming cube. We will see that at the heart of this problem lies a certain two-point inequality for an unknown envelope function. We will then discuss a computational approach to approximate the envelope and some improved bounds. This is ongoing joint work with Polona Durcik and Paata Ivanisvili. .
Nov. 16	Todd Quinto Tufts University	Microlocal properties of novel Ellipsoidal and hyperbolic Radon transforms Abstract: This talk with include deep functional analysis and cool pictures to describe the properties of a new integral transform (Radon transform) that integrates over generalized ellipsoids and hyperboloids. We discuss applications to Ultrasound Reflection Tomography (URT). In URT, one is interested in the structure of the body, and we will describe what body features are visible using this data and which are not visible. Artifacts, which are added streaks or other “features” that appear in the reconstruction but are not in the object. can occur in reconstructions We will use microlocal analysis, functional anlaysi related to the Fourier transform, to justify this. We will also describe artifacts We will occur that are and In this case, backprojection type reconstruction operators such as the normal operator $R^* R$ do not add artifacts to the reconstruction. We apply our results to a cylindrical geometry that could be used in URT. We investigate the visible singularities in this modality. In addition, we present reconstructions of image phantoms in two dimensions that illustrate our microlocal theory. .
Nov. 23	Thanksgiving Break
Nov. 30	Effie Papageorgiou Universität Paderborn	Large sets containing no copies of a given infinite sequence Abstract: An analog of the Erdős similarity problem “in the large” can be stated as follows: consider a discrete, unbounded, infinite set $A$ in the real line. Given $0\leq p<1$ , can we find a “large” -in the sense that it intersects every interval of unit length in a set of measure at least $p-$ measurable set $E$ , which does not contain any affine copy of $A$ ? A recent result by Bradford, Kohut and Mooroogen shows that this is indeed possible, by giving an explicit construction of such a set $E$ when $A$ is an infinite arithmetic progression. We generalize this to sequences that do not grow too fast (e.g. “subexponentially”), for some “non-linear” copies of the sequence, as well as to higher dimensions. Our method is probabilistic. Joint work with M. Kolountzakis (University of Crete) .
Dec. 7	TBA	TBA Abstract: TBA .