What are Algebra and Discrete Mathematics?

Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as the integers.

Algebra (or abstract algebra) is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra when referring to abstract algebra.

Contemporary mathematics, engineering, and science make intensive use of abstract algebra and discrete mathematics; for example, theoretical physics draws on Lie algebras. Fields such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. At the same time, algebra and discrete mathematics are used in various applied areas such as cryptography, coding theory, and search algorithms for the internet.

Research in Algebra and Discrete Mathematics at Davis covers a wide range of topics, mainly (but not exclusively):

• polyhedral combinatorics, polytopes (De Loera)
• symmetric functions (Rains, Schilling, Tracy, Vazirani)
• quantum algebras and crystal bases (Kuperberg, Schilling, Vazirani)
• Hecke algebras (Rains, Vazirani)
• commutative algebra and symbolic computation (De Loera)
• representation theory (Schilling, Vazirani)
• alternating-sign matrices (Kuperberg)
• geometric and topological combinatorics (Babson, De Loera)