arXiv subjects

Anshul Adve

Publications and source records attributed to Anshul Adve.

10 recordsLinked to original sources

arXiv2025

A converse theorem for hyperbolic surface spectra and the conformal bootstrap

The conformal bootstrap in physics has recently been adapted to prove remarkably sharp estimates on Laplace eigenvalues and triple correlations of automorphic forms on compact hyperbolic surfaces. These estimates derive from an infinite family of algebraic equations satisfied by this spectral data. The equations encode $G$-equivariance and associativity of multiplication on $\Gamma \backslash G$, for $\Gamma$ a cocompact lattice in $G = \text{PSL}_2(\mathbf{R})$. The effectiveness of the conformal bootstrap suggests that the equations characterize hyperbolic surface spectra, i.e., that every solution to the equations comes from a compact hyperbolic surface. This paper proves this rigorously with no analytic assumptions on the solution except discreteness of the spectrum. The key intermediate result is an axiomatic characterization of representations of $G$ of the form $L^2(\Gamma \backslash G)$.

math.SP↗

arXiv2025

Weyl bound for trilinear periods via conformal bootstrap

Let $f_1,f_2$ be holomorphic modular forms of the same weight for a cocompact lattice $\Gamma < \mathrm{PSL}_2(\mathbf{R})$. We estimate the rate of decay of the coefficients in the expansion of $f_1\overline{f_2}$ in a Laplace eigenbasis. By specializing our main theorem to the case where $\Gamma$ is arithmetic, we obtain new instances of the Weyl bound for triple product $L$-functions in the spectral aspect. Our method builds on the conformal bootstrap in physics.

math.NT↗

arXiv2025

A Spectral Gap for Spinors on Hyperbolic Surfaces

The purpose of this note is to construct a sequence of spin hyperbolic surfaces $\Sigma_n$ with genus going to infinity and with a uniform spectral gap for the Dirac operator. Our construction is completely explicit. In particular, the $\Sigma_n$ can be taken to be a tower of covers, with each $\Sigma_n$ an arithmetic hyperbolic surface.

math.NT↗

arXiv2024

Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms

We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap. This answers a question asked by Lin--Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same properties that geometrically converge to a tame limit manifold. Moreover, we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic rational homology 3-spheres with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have torsion homology that grows exponentially in volume.

math.GT↗

arXiv2023

Density criteria for Fourier uniqueness phenomena in $\mathbf{R}^d$

We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for a Schwartz function and its Fourier transform on $A$ (assuming those values are rapidly decreasing). More general results of the same nature hold for Fourier uniqueness pairs. This is an analog in all dimensions of the work of Kulikov, Nazarov, and Sodin in dimension $1$. Our methods are unrelated. As an application of our results, we produce Fourier uniqueness sets in higher dimensions which are optimally well-separated (up to constants). Our techniques also give the first purely analytic construction of discrete Fourier uniqueness pairs in higher dimensions. For a concrete example, consider \begin{align*} A = \{δ|n|^{t-1} n : n \in \mathbf{Z}^d\} \qquad \text{and} \qquad B = \{δ|n|^{u-1} n : n \in \mathbf{Z}^d\}, \end{align*} where $t,u,δ> 0$ and $t+u = 1$. We show that when $δ$ is sufficiently small, $(A,B)$ is a Fourier uniqueness pair, but when $δ$ is sufficiently large, there is an infinite-dimensional space of Schwartz functions $f$ with $f|_A = \hat{f}|_B = 0$.

math.CA↗

arXiv2021

An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete.

math.CO↗

arXiv2021

Computational complexity, Newton polytopes, and Schubert polynomials

The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class $NP\cap coNP$ of problems with "good characterizations". This suggests a new algebraic combinatorics viewpoint on complexity theory. This report discusses the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid, together with a theorem of A. Fink, K. Mészáros, and A. St. Dizier, which proved a conjecture of C. Monical, N. Tokcan, and the third author.

math.CO↗

arXiv2020

On nonexpansiveness of metric projection operators on Wasserstein spaces

In this paper we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p).$ When $d=1$, as $(\mathcal{P}_2(\mathbf{R}),W_2)$ is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When $d>1$, we show the failure of this property in two regimes: when $p>1$ is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces $(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $d\ge 2$ and $p\in(1,+\infty)$ are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion. As a corollary of the former, they have nonnegative upper Alexandrov curvature, in a precise sense that we define here. In our analysis a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.

math.FA↗

arXiv2017

Vanishing of Littlewood-Richardson polynomials is in P

J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming.

math.CO↗

arXiv2017

Symmetric group representations and Z

We discuss implications of the following statement about the representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation, and every nonnegative integer appears infinitely often as a Littlewood-Richardson coefficient and as a Kronecker coefficient.

math.CO↗