The department’s faculty specializing in algebra and topology are Eric Chesebro, Kelly McKinnie and Nikolaus Vonessen.

In a typical year, we offer 3-4 courses in algebra, topology, or geometry, with at least one at the undergraduate level.  Additionally, there is the weekly seminar on algebra/topology in which both faculty and graduate students give lectures on a variety of advanced topics. Besides introducing graduate students to many interesting subjects, this seminar provides valuable experience in exposition.

Faculty

Eric Chesebro

Professor Eric Chesebro received his Ph.D. from the University of Texas in 2006. He completed a postdoctoral appointment at Rice University in Houston, Texas before joining the Department of Mathematical Sciences at the University of Montana in Fall 2009. His interests lie in the study of the topology and geometry of three-dimensional manifolds. After W. Thurston's groundbreaking realization that many 3-manifolds admit homogeneous Riemannian metrics, geometry became an important and powerful tool in 3-dimensional topology. G. Perelman's recent proof of Thurston's geometrization conjecture is a strong confirmation of this philosophy. Amongst Thurston's eight model geometries, hyperbolic geometry plays a central role. Professor Chesebro is interested in the relationship between the topology of a hyperbolic 3-manifold and its geometry.

Kelly McKinnie

Professor Kelly McKinnie received her Ph.D. from the University of Texas in 2006. She then completed post-doctoral appointments at Emory University in Atlanta, GA and Rice University in Houston, Texas before joining the department of mathematical sciences at the University of Montana in Fall 2009. Her general research interests include finite dimensional division algebras, the Brauer group, valuation theory, and algebraic geometry. The index of a division algebra is the square root of the dimension of the algebra as a vector space over its center. One of the big open problems in the theory of division algebras is whether or not every division algebra of prime index can be described as a cyclic algebra, or equivalently, has a Galois maximal subfield. Prof. McKinnie has studied the existence of non-cyclic division algebras over a field with prime characteristic p, and index pn, n>1. She has also studied the existence of indecomposable division algebras with the same characteristics.

Recently Prof. McKinnie has worked with Profs. Eric Brussel of Emory University and Eduardo Tengan of ICMC, Brazil, to study the existence of non-cyclic and indecomposable division algebras over fields which are function fields of certain algebraic curves. She is very interested in using the techniques of algebraic geometry to solve problems in the theory of division algebras.

Nikolaus Vonessen

Professor Nikolaus Vonessen received his Ph.D. from MIT in 1988. His general research interests lie in ring theory, division algebras, and invariant theory. Much progress has been made in recent decades in understanding the structure of noncommutative rings and algebras. This makes it possible to study group actions and related invariant-theoretic questions in this setting. Prof. Vonessen's work in this direction is based on two different developments: first on the deep and well-understood commutative invariant theory, classical and geometric; and second, on the theory of finite group actions on noncommutative rings. In his research, he has been primarily concerned with actions of linear algebraic groups; one can call this area of research noncommutative invariant theory.