Number Theory, Potential Theory, and Convex Optimization

该奖项的研究和更广泛的影响将有助于当前数论、计算机科学和数学物理的发展。计算机科学家和数学家对具有实根的积分多项式的分类和最优逼近感兴趣。PI和他的博士生（Bryce Orloski）将证明新的（最优的）结果，并在这个方向上引入新的策略。他们的方法和算法将应用于数学物理（相互作用粒子系统的基态）和信息论（纠错码）。PI热衷于向少数族裔教授数学。他将用他的基金来资助他的研究生。PI将在获得资助的第三个夏天在宾夕法尼亚州立大学组织一个关于数论和凸优化的研讨会。研讨会将介绍大约20名高级研究生和博士后进入一个活跃的研究领域，并使他们能够开始在该领域的工作，特别是与彼此或高级数学家合作。在更专业的层面上，理解具有实数根的积分多项式的根的分布有助于揭示有限域上阿贝尔变的zeta函数的根的分布、图的邻接矩阵的特征值的分布以及对称积分矩阵的特征值的分布。PI和他的博士生（Bryce Orloski）将对给定数域上代数整数共轭的可能渐近分布进行分类。主要目的是利用David Cantor和Robert Rumely所定义的广义超限直径，确定具有线性约束的代数整数的渐近数的首指数。此外，他们提出了自1984年Smyth工作以来的一种新方法，并通过形成势理论中的凸优化问题，推导出Schur-Siegel-Smyth迹问题的新界。他们将证明这个优化问题的唯一解析解的存在性，作为一些线性和积分方程的解。他们将开发并实现一个有效的算法来逼近最优解。此外，物理学家还利用线性规划（共形自举）方法来约束二维共形场理论的谱。PI关于六边形晶格的最优性和拉普拉斯算子的第一个非平凡特征值的极值的项目将为这些问题证明新的结果，并在数学物理方面引入新的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research and broader impacts of this award will contribute to current developments in number theory, computer science, and mathematical physics. Computer scientists and mathematicians are interested in the classification and optimal approximation of integral polynomials with real roots. The PI and his Ph.D. student (Bryce Orloski) will prove new (optimal) results and introduce new strategies in this direction. Their methods and algorithms will have applications in mathematical physics (ground states of interacting particle systems) and information theory (error-correcting codes). The PI is passionate and invested in teaching Mathematics to underrepresented minorities. He will use his funding to support his graduate students. The PI will organize a workshop on number theory and convex optimization at Penn State University in the third summer of the grant. The workshop will introduce around 20 advanced graduate students and beginning postdocs to an active area of research and enable them to start their work in this area, particularly in collaboration with each other or senior mathematicians. On a more technical level, understanding the distribution of roots of integral polynomials with real roots sheds light on the distribution of the roots of the zeta function of abelian varieties over finite fields, the distribution of eigenvalues of the adjacency matrices of graphs and the distribution of the eigenvalues of the symmetric integral matrices. The PI and his Ph.D. student (Bryce Orloski) will classify the possible asymptotic distributions of the conjugates of algebraic integers over a given number field. The main goal is to identify the leading exponent of the asymptotic number of algebraic integers with some adelic constraints with the generalized transfinite diameter defined by David Cantor and Robert Rumely. Moreover, they propose a new method since the work of Smyth in 1984 and derive new bounds for the Schur-Siegel-Smyth trace problem by formulating a convex optimization problem in potential theory. They will prove the existence of a unique analytic solution to this optimization problem as the solution to some linear and integral equations. They will develop and implement an efficient algorithm for approximating the optimal solution. Furthermore, physicists have used the linear programming (conformal bootstrap) method to constrain the spectrum of two-dimensional conformal field theories. The PI's project on the optimality of the hexagonal lattice and the extremal values of the first nontrivial eigenvalues of the Laplacian operator will prove new results for these problems and introduce new methods in this direction with applications in Mathematical Physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.