12 problems

Let $g$ be a Schwartz function and let ${\mathcal G}(g,\alpha{\mathbb Z}\times\beta{\mathbb Z})$ be a Gabor frame, where $\alpha$ and $\beta$ are lattice constants. Janssen's conje…

In the real Jacquet–Rallis relative trace formula comparison, transfer refers to the correspondence between Schwartz functions on the general linear side and the compact unitary gr…

For $n\in\mathbb{Z}_+$, let … with $\mathcal{S}_{0}(\mathbb{R})=\mathcal{S}(\mathbb{R})$. Let $\mathcal{F}_{r}^{\lambda}$ denote the generalized Fourier transform appearing above.…

Viazovska's interpolation conjecture. The magic function is uniquely determined by its required roots; more generally, a radial Schwartz function on $\mathbb{R}^8$ is uniquely dete…

Let $\mathcal{S}_{even}(\mathbb{R})$ denote the space of real-valued even Schwartz functions on $\mathbb{R}$, and let $\widehat{f}$ be the Fourier transform of $f$. For a sequence…

Let $\mathcal{S}_{even}(\mathbb{R})$ denote the space of real-valued even Schwartz functions on $\mathbb{R}$, and let $\widehat{f}$ be the Fourier transform of $f$. For a sequence…

Let $\phi\in\mathcal{S}(\mathbb{R})\setminus\{0\}$, and let a finite wavelet system generated by $\phi$ be a finite collection of wavelet dilates and translates of $\phi$. Finite-w…

Let $A_{+}(28)$ denote the optimal constant in the uncertainty principle of Bourgain, Clozel, and Kahane in dimension $28$. An optimal function is a function attaining this optimal…

Demeter's Schwartz-class conjecture. The time-frequency translates $\{M_aT_bf:(a,b)\in\mathbb{R}^2\}$ are linearly independent.

Let $(u_0,0,T)$ be a homogeneous Schwartz set of data. Global regularity for homogeneous Schwartz data. There exists a smooth finite energy solution $(u,p,u_0,0,T)$ with the indica…

Let $(N,K)$ be a nilpotent Gelfand pair, let $\mathcal D=(D_1,\dots,D_d)$ be a $d$-tuple of self-adjoint generators of the algebra of left-invariant, $K$-invariant differential ope…

Let $N$) be a connected, simply-connected, two-step nilpotent Lie group, let $K$ be a compact group acting by automorphisms on $N$, and assume that $(K\ltimes N,K)$ is a Gelfand pa…