This paper deals with the discrete subgroup problem, solvable through the study of higher Teichmüller theory. The discrete subgroup problem is as follows: given a Lie group (a group with continuity) such as R, how do you find discrete subgroups, such as Z, particularly those of a type similar to (isomorphic to) a certain group? This paper details how, using theorems pertaining to Higher Teichmüller theory graphing the eigenvalue gaps of a matrix group can be used to figure out if it is a discrete subgroup of a Lie Group. In it, I find two subsets of results: one is working towards an attempt to validate recently found results about a Lie group, and the other is a result about the method itself (specifically, projective convex bending) and how it can be used empirically.

(2013-05) Levenstein, Dustan; Ben-Zvi, David; Meth, John

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I have studied representation theory of finite groups, in particular of the symmetric group over fields of prime characteristics. Over C, there is a nice classification of the simple representations of symmetric groups. Here I give a description of how the standard representation behaves in prime characteristic, and I study the structure of the group algebras of small symmetric groups in more detail. The general subject of representation theory sits at the crossroads of a vast array of subjects in mathematics, including algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics, topology, fourier analysis, and harmonic analysis. Modular Representation theory, the study of representations of finite groups over a field of positive characteristic, has in particular been used in the classification of finite simple groups, and itself finds applications in a variety of areas of mathematics.