This award supports the principal investigator's research on Number Theory and Representation Theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool, with diverse applications in areas such as data transmission and processing, communication systems, and internet security. Representation Theory emerges from people?s attempt to solve algebraic equations. The subject matured in the 20th century and is still fast developing. It studies the symmetries of objects and has applications to almost all branches of mathematics, physics, cosmology and material science. This project will explore some major conjectures in these two areas. Additionally, this award will support graduate students in the PI's institution. In the 1960s Langlands formulated a series of profound conjectures relating representation theory and number theory, and these conjectures have become guiding principles for number theorists and representation theorists since then. In the 1990s, Gross and Prasad formulated several conjectures on representation theory of the orthogonal groups, guided by the philosophy of Langlands. Together with Gan, they extended these conjectures to include all classical groups. These conjectures and their refinement and arithmetic counterparts stand at the crossroads of number theory, representation theory, and arithmetic geometry. The goal of the project is to study these conjectures. In more detail, we study the following two problems (1) the local Gan--Gross--Prasad conjecture for real orthogonal groups; (2) the archimedean counterpart of the arithmetic fundamental lemma arising from an arithmetic version of the Gan--Gross?Prasad conjecture. For the first problem we apply the method of theta correspondences, and a relative version of Shahidi?s local coefficients which is our main point of innovation. This method also gives an inductive construction of L-packets for orthogonal groups, which are themselves important objects of study in the conjectures of Langlands. For the second problem, we make use of an inductive structure of the relevant orbital integrals and spherical characters. It reduces the problem to explicit calculations using the theory of doubling zeta integrals. 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.