The hot spots ratio of a domain measures the degree of failure of Rauch's hot spots conjecture. We identify the largest possible value of this ratio over all connected Lipschitz domains in any dimension d. As d tends to infinity, we show that this maximal ratio converges to sqrt(e), asymptotically matching the previous best known upper bound...
The hot spots conjecture asserts that for any convex bounded domain $\Omega$ in $\mathbb{R}^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\Omega$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of...
We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping ρ↦Tr(OE[ρ]) to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill...
The periodic tiling conjecture asserts that if a region Σ⊂Rd tiles Rd by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in R, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for...
Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1)...
In this paper we prove a uniform Fourier restriction estimate over the class of simple curves where the last coordinate function can be extended to a holomorphic function of bounded frequency in a sufficiently large disc. The proof is based on a decomposition scheme for this class of functions.
We give new proofs of the description convex hulls of space curves $\gamma : [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(\gamma', \gamma'', \ldots, \gamma^{(d)})$ are positive. In particular, we...
We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube ${0,1}^d$, the $k-$higher energy of $A$ (i.e., the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is...
In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian...
We consider decoupling for a fractal subset of the parabola. We reduce studying l2Lp decoupling for a fractal subset on the parabola {(t,t2):0≤t≤1} to studying l2Lp/3 decoupling for the projection of this subset to the interval [0,1]. This generalizes the decoupling theorem of Bourgain-Demeter in...
The analysis of neural network training beyond their linearization regime remains an outstanding open question, even in the simplest setup of a single hidden-layer. The limit of infinitely wide networks provides an appealing route forward through the mean-field perspective, but a key challenge is...
The aim of this paper is to prove a uniform Fourier restriction estimate for certain 2−dimensional surfaces in R2n. These surfaces are the image of complex polynomial curves, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynomial counterpart to a...
Urban spatial networks are complex systems with interdependent roles of neighborhoods and methods of transportation between them. In this paper, we classify docking stations in bicycle-sharing networks to gain insight into the spatial delineations of three major United States cities from human...