We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.
The subject of counting positive lattice points in n-dimensional simplexes has interested
mathematicians for decades due to its applications in singularity theory and number the-
ory. Enumerating the lattice points in a right-angled simplex is equivalent to determining
the geometric genus of a singularity of a weighted homogeneous complex polynomial. It is
also a method to shed insight into large gaps in the sequence of prime numbers. Seeking
to contribute to these applications, this research project proves the Yau Geometric Con-
jecture in six dimensions, a sharp upper bound for the number of positive lattice points in
a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper
bounds for the number of points in 5-dimensional simplexes over the cross sections of the six-
dimensional simplex. This research project paves the way for the proof of a fully general sharp
upper bound for the number of lattice points in a simplex. It also moves the mathematical
community one step closer towards proving the Yau Geometric and Yau Number-Theoretic
Conjectures in full generality.