Tanis Nielsen

Thesis Advisor: Dr. Lenny Fukshansky (Claremont McKenna College - Department of Mathematics)

Second Reader: Dr. Michael Orrison (Harvey Mudd College - Department of Mathematics)

Contact: tnielsen@g.hmc.edu

Some Explorations into Well-Rounded Lattices

Lattices are discrete subgroups of Euclidean spaces. Analogously to vector spaces, they can be described as spans of collections of linearly independent vectors, but with integer (instead of real) coefficients. Lattices have many fascinating geometric properties and numerous applications, and lattice theory is a rich and active field of theoretical work. In this thesis, we present an introduction to the theory of Euclidean lattices, along with an overview of some major unsolved problems, such as sphere packing. We then describe several more specialized topics, including prior work on well-rounded ideal lattices and some preliminary results on the study of planar deep hole lattices.