Lie Algebra Representation Theory:
Many problems in physics and mathematics may be reduced to finding all the ways that a group of symmetries can be realized as a collection of matrices. These realizations are known as representations of the group, and searching for such realizations is an area of great interest in the study of Lie groups. The structure of a Lie group is reflected in the structure of its Lie algebra (tangent space to the Lie group at the identity). Questions regarding the representations of a Lie group may be restated as questions regarding the representations of its associated Lie algebra. One of the most basic questions one can ask concerning the structure of a particular Lie algebra is the following: What are the dimensions of the generalized eigenspaces (the so- called root spaces) of the algebra? My research addresses this question. Root multiplicities for finite type Kac-Moody algebras are all known, and root multiplicities of indefinite type Kac-Moody algebras are well understood. However, there are many interesting, open problems regarding the multiplicities of roots of indefinite type Kac-Moody algebras. In my research, I view certain indefinite type Kac-Moody algebras as modules over (i.e. representations of ) appropriate Kac-Moody algebras of affine or finite type.Kang’s multiplicity formula reduces my problem to one of calculating weight multiplicities in certain highest weight modules over particular affine or finite Kac-Moody algebras. If the calculations are over finite algebras, I am able to apply well known formulas. If the calculations are over indefinite type Kac-Moody algebras, I calculate the weight multiplicities using crystal basis theory for affine algebras.
Symmetric Spaces for Lie Groups over Finite Fields:
Generalizations of symmetric spaces have recently been receiving wide attention. Real symmetric spaces were introduced by Élie Cartan as a special class of homogeneous Riemannian manifolds. Since then a rich and deep theory has been developed that plays a key role in many fields of active research such as Lie theory, differential geometry, and harmonic analysis. Symmetric spaces also appear in number theory, algebraic geometry, and representation theory. The theory of symmetric spaces has numerous generalizations, e.g., symmetric varieties, symmetric k-varieties, Vinberg's theta-groups, spherical varieties, Gelfand pairs, Bruhat- Tits buildings, Kac-Moody symmetric spaces, which provide a vast source of interesting open problems. I am part of a research group including Catherine Buell of Fitchburg State University, Ellen Ziliak of Benedictine College, Carmen Wright of Jackson State University, Jennifer Schaefer of Dickinson College, and Aloysius Helminck of University of Hawaii at Manoa. Our group is working to generalize several known results for Lie groups over algebraically closed fields, to Lie groups over finite fields. Because our base fields are finite, we often begin a new project with significant computational experimentation.
Algebraic Music Theory:
Many musical properties of the traditional 12-fold pitch system of octave division are related to symmetry and have been studied algebraically. For example, the circle of fifths can be viewed as the integers modulo twelve when the group is generated by seven. I have been working with students to study how some of the well-known algebraic properties of traditional 12-tone music translate to other microtonal systems.
Algebraic Voting Theroy:
I have been working with students to study connections between the way a voter's ranking of the candidates is scored, or weighted, and the outcome of an election. The question is important from a practical standpoint in considering the objectivity of voting procedures. Our results emphasize that the selection of a weight system may strongly influence the outcome of an election. Most recently I have begun thinking about how we can use mathematical tools to quantify gerrymandering of congressional districts (building from work by Jonathan Hodge et. all.), so look back for more information in this direction soon.