Automorphic Forms on Higher Rank and Kac-Moody Groups

自守形式是现代数学中一个基本而复杂的结构。 虽然在历史上，它们主要由数论和表示理论家研究，但它们与其他科学分支有联系。 本研究探讨了一些连接数学和物理学不同领域的问题。 该项目包括与数学物理学家的合作，使用自守形式来描述弦理论中出现的广义相对论的修正。其他项目包括对某些爱森斯坦剩余的平方可积性的详细研究（它在酉对偶问题上有应用），以及对无限维Kac-Moody群上的爱森斯坦级数的研究（特别是它们常数项的亚纯延拓）。 另一个项目涉及Voronoi式求和公式在数论中的应用，例如自守L-函数的次凸性问题。最后，Miatello-Wallach猜想（自守形式理论中的适度增长条件在高阶群上实际上是多余的）也将是研究的重点。

Automorphic forms are a basic yet intricate structure in modern mathematics. Though historically they have been studied primarily by number theorists and representation theorists, they have connections to other branches of science. This research explores a number of questions connecting different areas of mathematics and physics. The project includes collaborative work with mathematical physicists that uses automorphic forms to describe corrections to general relativity that arise in string theory. Other projects include a detailed investigation of the square-integrability of certain Eisenstein residues (which has applications to the unitary dual problem), as well as a study of Eisenstein series on infinite-dimensional Kac-Moody group (in particular, the meromorphic continuation of their constant terms). Another project concerns applications of Voronoi-style summation formulas to number theory, such as to subconvexity problems for automorphic L-functions. Finally, the Miatello-Wallach conjecture (that the moderate growth condition in the theory of automorphic forms is in fact redundant on higher rank groups) will also be a focus of the research.