Lattices and Their Application: A Senior Thesis

Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick\u27s Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick\u27s Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick\u27s Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research