Distribution of the Hodge and the Tate locus

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Geometry and arithmetic were first studied by the Greeks, while algebra first emerged centuries later in the hands of Persian scholars as the art of solving equations. The interplay between these three disciplines has been at the center of mathematical research over the past century. Mathematicians have uncovered deep connections between them, leading to many spectacular results in mathematics (e.g., Fermat’s Last Theorem) and other fields, including applications in cryptography, quantum field theory, and string theory in physics. The main object of study at the intersection of these disciplines is a set of algebraic equations. While geometry helps understand the shape of the set of solutions with complex entries (also called algebraic varieties), the goal of arithmetic is to understand the set of solutions with integer entries. There are natural linear structures attached to algebraic varieties called Hodge structures, which in some cases capture faithfully the set of algebraic equations we started with. The study of Hodge structures, their symmetries, and their variations is the main object of investigation of this proposal. It is a topic at the crossroads of several areas of research such as complex algebraic geometry, number theory, and representation theory, with many long-standing conjectures. The PI will involve graduate students in this project and will organize a conference on recent advances in Hodge theory.This project aims to answer several questions regarding the distribution of the exceptional Hodge locus in the theory of variations of Hodge structures and their arithmetic counterpart, the Tate locus. These questions will be addressed using tools from Arakelov intersection theory, ergodic theory, Hodge theory, Ax-Schanuel theorem for Shimura varieties, and Diophantine geometry. The first goal is to study the atypical Hodge locus in some families of algebraic varieties. The second goal is to study the Tate locus and give a concrete application to exceptional algebraicity under specializations of Brauer classes on K3 surfaces. The third goal is to study the modularity behavior of the closure of special cycles in moduli spaces of K3 surfaces, or more generally in orthogonal Shimura varieties. These generating series exhibit a quasi-modularity behavior as well as a mixed mock modularity behavior, depending on the type of degeneration of the family of K3 surfaces.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.