Clemson Analysis and PDE Seminar

Abstract: A recent breakthrough in learning low-degree Boolean functions by Eskenazis and Ivanisvili employs a family of dimension-free polynomial inequalities named after Bohnenblust and Hille, originating from Littlewood's 1930 work. In this talk, I will review some recent progress that extends these results from discrete hypercubes to qubit systems. Further extensions to more general discrete quantum systems require new Fourier analysis inequalities on cyclic groups. Along the way, dimension-free discretization inequalities were obtained as unexpected byproducts. This is based on joint work with Lars Becker, Ohad Klein, Alexander Volberg and Joseph Slote.

Abstract: We show that the best constant in the weak-type (1,1) bound for the vector Riesz transform on R^n is at most 2, independent of the dimension n. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

Abstract:  I will present a proof of the discrete nonlinear Hausdorff-Young inequality with constant one for the \(\mathrm{SU}(1,1)\) nonlinear Fourier transform. A discrete-to-continuous limiting argument then yields

$$

\|(\log |a_f|^2)^{1/2}\|_{L^{p'}(\mathbb R)}

\le \|f\|_{L^p(\mathbb R)},\quad 1\le p<2,

$$

where $a_f$ is the transmission coefficient and $p'$ is the Hölder conjugate of $p$. This resolves the Muscalu-Tao-Thiele uniformity problem. This talk is based on arXiv:2608.15895.

Abstract: Many dissipative partial differential equations have solutions that become analytic, or more generally Gevrey regular, for positive time. Analyticity and Gevrey regularity provide strong control of high frequencies, while quantitative restrictions on oscillation and zero sets also depend essentially on the underlying parabolic or elliptic structure of the equation. Together, these properties yield bounds on the number and size of zeros and show how information on a small region can control the solution elsewhere.

Starting from this perspective, the talk will provide an overview of several broader themes in the qualitative and quantitative study of partial differential equations. I will begin with the Fourier approach to analyticity for the Navier-Stokes equations and then discuss how regularity, equation structure, and unique continuation enter questions concerning the complexity of solutions, nodal sets, and observability. I will conclude with recent joint work with Qi Xu on the temporal decay of solutions to $u_t-\Delta u=vu$ on $\mathbb R^n$.

Abstract: In Werner's Quantum harmonic analysis (QHA) we develop notions in classical harmonic analysis, such as convolutions and the Fourier transform for operators in place of functions.

In this work, we set up QHA on the weighted Bergman space of a bounded symmetric domain,  where Toeplitz operators and Berezin transforms arise naturally as convolutions. Using the $\alpha$-Berezin transform, we obtain a characterization of Toeplitz operators and approximation results by Toeplitz operators, including a description of the radial Toeplitz algebra. We then introduce invariant differential operators that act on operators on the Bergman space. A key role is played by an explicit invariant differential operator that shifts the $\alpha$-Berezin transform, which leads to a discrete heat-type equation and a further characterization of the radial Toeplitz algebra through its domain.

This is joint work with Matthew Dawson and Gestur Ólafsson.

Abstract: In 1996, C. Heil, J. Ramanathan, and P. Topiwala conjectured that any finite collection of distinct time-frequency shifts — translations and modulations — of a nonzero square-integrable function is linearly independent. In August 2026, Faulhuber, Petersen, van Velthoven, and Voigtlaender constructed a counterexample consisting of twelve time-frequency shifts of a Schwartz function. Shortly afterward, Oussa and, independently, Dai, Deng, Shi, Wu, and Yang constructed counterexamples involving four-point configurations and Schwartz functions. Since any three distinct time-frequency shifts of a nonzero square-integrable function are linearly independent, these constructions settle the HRT conjecture in the negative with counterexamples of the smallest possible cardinality. All three constructions were developed with the assistance of large language models (LLMs).

Nevertheless, many questions remain open. For a fixed nonzero square-integrable function and an integer n≥4, can we characterize the configurations of n distinct points in the time-frequency plane that yield linear dependence? How large is the set of such configurations?

In this talk, I will give an overview of the conjecture and discuss these questions in the context of the known four-point counterexamples. Finally, I will present recent joint work with Kasso Okoudjou on computing the Lebesgue and Hausdorff measures of the set of four-point configurations that yield linear dependence for a fixed nonzero square-integrable function.

Abstract: TBA