# Algebra

**Research in algebra at the University of Illinois has a long and distinguished history—in particular, the study of finite groups and representation theory and commutative algebra. The following is a short summary of this history by Derek Robinson and Phillip Griffith, professors emeriti.**

Group theory has a long history at Illinois, stretching back to G.A. Miller and H.R. Brahana in the period 1910-1930. Both worked on finite groups, the subject then being in its infancy. In 1938, Reinhold Baer arrived at Illinois from Germany. He worked mainly on infinite groups and did fundamental work on abelian groups and modules. W.W. Boone, by training a logician, came to Illinois in the early 1950s and was most famous for his work on the insolvability of the word problem for finitely presented groups. Around the same time, Irving Reiner and Michio Suzuki joined the department. Suzuki rapidly became world famous for his discovery of a new infinite family of finite simple groups, as well as a new simple group of sporadic type: both of these now bear his name. Reiner was prominent in the creation of integral representation theory of groups. Subsequently, John Walter arrived to work on finite simple groups, as did Everett Dade, who made major contributions to representation theory of finite groups. In the 1970s, Illinois was an internationally recognized center for the study of group theory. More recently, research in group theory at Illinois has shifted to geometric group theory.

Research in commutative algebra in Illinois kicked off in the early 1970s when Robert Fossum, Phillip Griffith, and Graham Evans came to the department, joined by Sankar Dutta in the 80s. An accompanying seminar was held regularly for over 30 years, and at least 22 students got their Ph.D. degrees in the subject. Early research activities led to Fossum’s frequently cited text, *The Divisor Class Group of a Krull Domain* (1973) and Evans-Griffith’s solution of the “Syzygy Problem” (published in the *Annals of Math*, 1981). Dutta’s contributions to unresolved questions in J-P Serre’s intersection theory followed. The activities of the research group resulted in the National Science Foundation-funded “Special Year in Commutative Algebra”’ (1983-84) at Illinois, including many distinguished visiting researchers. The department offers a standard one-semester graduate course in commutative algebra, usually followed by an advanced topics course in the subject. These courses also support the program in algebraic geometry, now a very active research area in the department.

**Current faculty in algebra are**

- Chris Dodd (Geometric representation theory)
- Sankar Dutta (Commutative algebra)
- William Haboush (Algebraic groups)
- Sergei Ivanov (Geometric group theory)
- Rinat Kedem (Representation theory, combinatorics, and integrable systems)
- Igor Mineyev (Geometric group theory)

**Faculty in related areas include the following...**