arXiv subjects

Danielle Wang

Publications and source records attributed to Danielle Wang.

7 recordsLinked to original sources

arXiv2025

Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture

We study certain new relative trace formulas on (non-reductive) period integrals involving Weil representations, in the context of the relative Langlands program. We study normal representatives using Galois theory, and establish geometric decompositions of relative trace formulas using normal representatives for good test functions. By comparing global representatives, local distributions and orbital integrals, we prove the twisted Gan--Gross--Prasad (GGP) conjecture on Asai L-functions, in any dimension under some local assumptions, allowing ramifications of number fields.

math.NT↗

arXiv2023

Twisted Gan-Gross-Prasad conjecture for unramified quadratic extensions

Using a relative trace formula approach, we prove the twisted global Gan-Gross-Prasad conjecture for $\operatorname{U}(V) \subseteq \operatorname{GL}(V)$, as well as its refinement, under some unramifiedness assumptions and local conditions on the quadratic extension and the automorphic representation.

math.RT↗

arXiv2021

The Convex Hull of Parking Functions of Length $n$

Let $\mathcal{P}_n$ be the convex hull in $\mathbb{R}^n$ of all parking functions of length $n$. Stanley found the number of vertices and the number of facets of $\mathcal{P}_n$. Building upon these results, we determine the number of faces of arbitrary dimension, the volume, and the number of integer points of $\mathcal{P}_n$.

math.CO↗

arXiv2019

The Eulerian distribution on involutions is indeed $γ$-positive

Let $\mathcal I_n$ and $\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\{1, \dots, n\}$, respectively, and let $\text{des}(π)$ denote the number of descents of the permutation $π$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 1$ and $J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents.

math.CO↗

arXiv2018

Uniform Bounds for the Number of Rational Points on Symmetric Squares of Curves with Low Mordell-Weil Rank

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, provided that the rank of the Jacobian is at most $g-4$. We also find rank-favorable uniform bounds (that is, bounds depending on the rank of the Jacobian) in the hyperelliptic case.

math.AG↗

arXiv2018

Modified Erdös--Ginzburg--Ziv Constants for $\mathbb Z/n\mathbb Z$ and $(\mathbb Z/n\mathbb Z)^2$

For an abelian group $G$ and an integer $t > 0$, the \emph{modified Erdös--Ginzburg--Ziv constant} $s_t'(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$. We compute $s_t'(G)$ for $G = \mathbb Z/n\mathbb Z$ and for $t = n$, $G = (\mathbb Z/n\mathbb Z)^2$.

math.CO↗

arXiv2018

On roots of Wiener polynomials of trees

The \emph{Wiener polynomial} of a connected graph $G$ is the polynomial $W(G;x) = \sum_{i=1}^{D(G)} d_i(G)x^i$ where $D(G)$ is the diameter of $G$, and $d_i(G)$ is the number of pairs of vertices at distance $i$ from each other. We examine the roots of Wiener polynomials of trees. We prove that the collection of real Wiener roots of trees is dense in $(-\infty, 0]$, and the collection of complex Wiener roots of trees is dense in $\mathbb C$. We also prove that the maximum modulus among all Wiener roots of trees of order $n \ge 31$ is between $2n-15$ and $2n-16$, and we determine the unique tree that achieves the maximum for $n \ge 31$. Finally, we find trees of arbitrarily large diameter whose Wiener roots are all real.

math.CO↗