On Higher Turan inequalities for the Plane Partitions, Ellipsoidal T-Designs, and j-inversion

This thesis is about combinatorics and number theory. More precisely, this is based on the 3 papers I wrote during my time as a PhD student. The content of our thesis is on Higer Turan inequalities for plane partitions, introduction of a generalization of spherical t-designs, which we call ellipsoidal t-design, and providing inversion formulae for j-function around elliptic points, out of which the later work is a joint work with, my fellow PhD student at University of Virginia, Alejandro De Las Penas Castano.