Mathematics Algebra

Algebra

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.

	RESEARCH FIELDS
	Algebra
	Analysis
	Applied Math
	Geometry
	Number Theory
	PDE
	Topology

Faculty

	Larry Gerstein PhD: University of Notre Dame, 1967 Interests: Quadratic Forms, Number Theory Office: Room 6508 gerstein-A.T-math.ucsb.edu			Kenneth Goodearl PhD: University of Washington, 1971 Interests: Algebra, Functional Analysis Office: Room 6520 goodearl-A.T-math.ucsb.edu

	Birge Huisgen-Zimmermann PhD: University of Munich, 1979 Interests: Representation Theory of Algebras Office: Room 6516 birge-A.T-math.ucsb.edu			Bill Jacob PhD: Princeton University, 1979 Interests: Quadratic Forms, Division Algebra Office: Room 6719 jacob-A.T-math.ucsb.edu

	Jon McCammond PhD: Univesity of California, Berkeley, 1991 Interests: Geometric Group Theory Office: Room 6711 mccammon-A.T-math.ucsb.edu			James M^cKernan PhD: Harvard University, 1991 Interests: Algebraic Geometry Office: Room 6713 mckernan-A.T-math.ucsb.edu

	David Morrison PhD: Harvard University, 1980 Interests: Algebraic Geometry, String Theory Office: Room 6708 drm-A.T-math.ucsb.edu			Milen Yakimov PhD: University of California at Berkeley, 2001 Interests: Lie Theory, Poisson Geometry Office: Room 6703 yakimov-A.T-math.ucsb.edu

	Adil Yaqub PhD: University of California at Berkeley, 1955 Interests: Ring Theory, Universal Algebras Office: Room 6720 yaqub-A.T-math.ucsb.edu

Visiting Faculty

	Alex Dugas PhD: University of California at Berkeley, 2006 Interests: Representation Theory Office: Room 6510 asdugas-A.T-math.ucsb.edu			Pawel Gladki PhD: University of Saskatchewan, 2007 Interests: Algebra Office: Room 6721 gladki-A.T-math.ucsb.edu

Note: Please replace -A.T- with @ in the email addresses above.