Many prominent trends in modern algebra represent mixtures of what was classically labeled algebra with geometry, combinatorics, and topology. The flux moves in both directions: Longstanding algebraic problems are solved with techniques adapted from neighboring fields, and conversely. Consequently, this area offers broad exposure to mathematical ideas.
Here are a few specific lines pursued by UCSB's algebra faculty:
- The representation theory of finite dimensional algebras focuses on `linearized snapshots' of certain nonlinear objects, such as finite groups. In the past decades, this branch of algebra has been strongly linked to combinatorics and geometry through seminal work of Auslander and Kac, among others, showing that much of the desired structural information on the nonlinear objects and their linear snapshots is encoded in directed graphs. Other aspects of a well-rounded structure theory can, for instance, be accessed by way of `derived categories' (due to Grothendieck and Verdier), powerful tools in studying maps among sets with various types of additional structure that one wishes to explore.
- Representation theory has proved equally important in the realm of infinite dimensional algebras, where it has long been utilized to `linearize' the study of objects such as infinite groups or differential operators. It has led, in particular, to the study of algebras with built-in `multiplicative twists', algebras which, in the past two decades, have played key roles in a rapidly developing new field labelled `quantum groups'. This new field originated in theoretical physics -- in quantum inverse scattering theory and the search for solutions to the `quantum Yang-Baxter equation', to be a little more precise -- but rapidly built connections with areas of mathematics as seemingly disparate as knot theory, algebras of operators on Hilbert spaces, and special functions. The algebraic side of this field includes the ongoing development of `noncommutative counterparts' to classical algebraic geometry, such as `quantized' versions of the algebras of polynomial functions on algebraic spaces.
- Algebraic Geometry is the study of the solutions of polynomial equations. At first sight this task would seem modest, but in fact this problem is so hard that algebraic geometry draws upon many areas of mathematics such as algebra, differential geometry, topology, number theory, analysis and differential equations to attack this problem. First one considers the zero set as a geometric object, a variety. The modern approach to the problem of classifying varieties involves classifying all possible embeddings into projective space. It turns out that this problem is intimately related with the study of the topology of the curves that lie on the variety, especially the combinatorial structure of an associated cone.
- Karel Casteels
- Kenneth Goodearl
- Birge Huisgen-Zimmermann
- Bill Jacob
- Jon McCammond
- David Morrison
- Xiaolei Zhao