Commutative Algebra: Making an impact worldwide

The University of Nebraska-Lincoln has a campaign called "Put Yourself on the Map," that connects successful alumni, faculty and students. Certainly, the Department of Mathematics is an international leader in commutative algebra that has put itself on the map with far reaching, worldwide connections. Indeed, mathematicians who do research in commutative algebra have made Lincoln, Neb., a "must-visit" place for scholars interested in the field.

Currently, the department has seven full professors working in commutative algebra and related fields such as algebraic geometry, algebraic K-theory, representation theory and homological algebra. The commutative algebra group has also hosted nine postdoctoral faculty, including two who are currently on the UNL faculty (Susan Cooper and Ananth Hariharan, Page 11), and mentored nine "early-career faculty" who have visited UNL for the department's IMMERSE program (Page 7). Currently, there are 14 Ph.D. students doing research in commutative algebra.

UNL's high visibility in commutative algebra can be traced to the arrival of Roger and Sylvia Wiegand in 1972. Their research achievements, their frequent travel to professional meetings and even their outgoing personalities contributed to UNL's growing reputation in the area.

Roger, however, is quick to note that he and Sylvia were attracted to UNL because there were faculty already at UNL doing research, namely Max Larson, Jim Lewis and Tom Shores. While Max, Jim and Tom turned to other interests, Brian Harbourne, Tom Marley and Mark Walker joined the Wiegands and the department's influence in the area continued to grow.

The highly visible research group became one of the best in the world when Luchezar Avramov joined the faculty in 2002 as the Dale Jensen Professor of Mathematics. Not long after "Lucho" joined the department, so did Srikanth Iyengar, who had earned his Ph.D. under Lucho's direction at Purdue.

Thirty-three mathematicians have earned their Ph.D. at UNL in commutative algebra, with 17 earning their degree in the 2000s. Roger Wiegand, Willa Cather Professor of Mathematics, has been the most productive thesis advisor, with 15 students graduated so far, and three more in the works.

One important trait of the faculty in commutative algebra is their commitment to working with one another and with scholars around the world, especially with mathematicians who earned their Ph.D. at UNL (or at Purdue under Lucho's direction before he joined the UNL faculty).

This willingness to "talk mathematics" with anyone interested in commutative algebra and to pursue research projects with many of them has also resulted in several important conferences held in Lincoln. Since 2005, the UNL campus has been host to the annual University of Kansas, University of Missouri and University of Nebraska (KUMUNU) Algebra Day and commutative algebra conferences held in honor of the Wiegands in 2005 and Avramov's 60th birthday in 2008. UNL will play host next spring to KUMUNU 2011, and David Eisenbud will give the 2011 Rowlee Lecture. Slightly farther into the future, Iyengar is on the organizing committee for a special year in commutative algebra in 2012-2013 at the Mathematical Sciences Research Institute.

Collaborations among our commutative algebra faculty and with visitors have resulted in many high-impact projects. Articles that are particularly noteworthy include:

  • L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar and C. Miller, "Homology of perfect complexes", Advances in Mathematics 223 (2010).
  • L. L. Avramov, R.-O. Buchweitz and S. B. Iyengar, "Class and rank of differential modules", Inventiones Mathematicae 169 (2007).
  • C. Bocci and B. Harbourne, "Comparing Powers and Symbolic Powers of Ideals", J. Algebraic Geometry, (2010).
  • C. Huneke, D. Katz and T. Marley, "On the support of local cohomology", Journal of Algebra (2009).
  • F. Moore, G. Piepmeyer, S. Spiroff and M. Walker, "Hochster's theta function and the Hodge-Riemann bilinear relations", Advances in Mathematics 226 (2011).

Avramov, Harbourne, Iyengar, Marley and Walker are UNL faculty. Moore is a UNL Ph.D., and Miller and Piepmeyer were postdoctoral scholars at UNL. Other notable collaborations include Iyengar's five papers with Henning Krause (University of Bielefeld). A productive collaboration among our postdocs and a young researcher in Japan (who will be visiting UNL for two years beginning in January 2011) resulted in a beautiful result, often called the "Postdoc Theorem":

  • L. W. Christensen, G. Piepmeyer, J. Striuli and R. Takahashi, "Finite Gorenstein representation type implies simple singularity", Advances in Mathematics (2008).

The commutative algebra group is involved in major book projects. In 2007 the American Mathematical Society published Twenty-four hours of local cohomology by Iyengar and six co-authors, including Graham Leuschke (Syracuse), a UNL Ph.D., and Claudia Miller (Syracuse), a UNL postdoc. Roger Wiegand and Graham Leuschke have completed about 80 percent of Cohen-Macaulay Representations. Sylvia Wiegand is collaborating with William Heinzer (Purdue) and Christel Rotthaus (Michigan State) on Power Series over Noetherian Rings.

Over the past seven semesters, the commutative algebra faculty have hosted over 200 visitors, with nearly 100 speaking in the group's active seminar or giving a featured talk at a conference. These visitors, together with Ph.D. alumni, postdocs and early-career faculty are represented on the map.