Cokernels of Random Matrices and the Geometry of Error-Correcting Codes

这个研究项目将调查几个关于组合学和数论中的对象如何在家庭中变化的问题。本文研究的对象在现代密码学和编码理论中起着重要的作用。 这些问题位于几个数学领域的边界，首席研究员将使用代数几何和数论的思想来回答组合问题，并增强我们对纠错码，格和沙堆群的理解。 在许多情况下，这些计数问题可以用随机矩阵族来理解。 一个主要的想法是从代数的角度研究纠错码，重点是代码之间的相互作用，有限域上的多项式，以及它们定义的代数簇。 首席研究员拥有丰富的经验，作为本科生和高中生的研究导师，并将领导这些领域的学生研究项目。 这些项目将得到研究生导师的支持，他们将获得宝贵的专业发展经验。 这里的研究项目联合收割机结合了来自纯数学和应用数学多个领域的思想，是促进跨学科合作的理想选择。连通图的沙堆群是阶等于图的生成树个数的有限阿贝尔群。 PI将研究这些组在某些图族中如何变化。 这些问题的动机是与数论中的科恩-伦斯特拉数学的联系。 PI将考虑有关随机格族的问题，重点是具有代数结构的格。 这里的一个重要工具来自群和环的zeta函数理论。 在某些情况下，这种额外的代数结构将导致非常不同的行为。 Reed-Muller码是Reed-Solomon码的多变量类似物，并且是编码理论中最基本的例子之一。 一个关键的想法是使用结果的理论插值问题的代数几何证明新的结果码。 首席研究员将结合联合收割机在代数和算术几何以及编码理论方面的专业知识，以进一步加深我们对这些重要物体的理解。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This research project will investigate several questions about how objects from combinatorics and number theory vary in families. The kinds of objects that are the focus of this research play important roles in modern cryptography and coding theory. These questions lie at the boundary of several mathematical areas, and the principal investigator will use ideas from algebraic geometry and number theory to answer combinatorial questions and enhance our understanding of error-correcting codes, lattices, and sandpile groups of graphs. In many situations, these counting problems can be understood in terms of families of random matrices. A major idea is to study error-correcting codes from an algebraic perspective, focusing on the interplay between codes, polynomials over finite fields, and the algebraic varieties that they define. The principal investigator has extensive experience as a mentor for research by undergraduates and high school students and will lead student research projects in these areas. These projects will be supported by graduate student mentors will be who will gain valuable professional development experience. The research projects here combine ideas from multiple areas of pure and applied mathematics and are ideal for promoting collaboration across disciplines.The sandpile group of a connected graph is a finite abelian group whose order is equal to the number of spanning trees of the graph. The PI will study how these groups vary within certain families of graphs. These questions are motivated by connections to the Cohen-Lenstra heuristics from number theory. The PI will consider problems about families of random lattices, focusing on lattices with algebraic structure. An important tool here comes from the theory of zeta functions of groups and rings. In certain cases this additional algebraic structure will lead to very different behavior. Reed-Muller codes are multivariable analogues of Reed-Solomon codes and are some of the most fundamental examples in coding theory. A key idea is to use results from the theory of interpolation problems in algebraic geometry to prove new results about codes. The principal investigator will combine expertise in algebraic and arithmetic geometry, as well as in coding theory, to further our understanding of these important objects.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.