## Research Blurbs for Faculty in Commutative Algebra and Related Areas

Department of Mathematics

University of Nebraska-Lincoln

February 21, 2013

Dale M. Jensen Professor **Lucho Avramov** works on the homological algebra of commutative rings. Recent topics include the structure of ring homomorphisms, finiteness of André-Quillen homology and of Hochschild homology, behavior of infinite resolutions, and the vanishing of Tor and Ext, particularly over complete intersections and Gorenstein rings.

Professor **Brian Harbourne** works in commutative algebra and algebraic geometry. He has studied the geometry of rational surfaces, and his recent focus is on Hilbert functions and resolutions of homogeneous ideals defining fat point subschemes of the projective plane. He is also studying the connections between algebra and geometry that arise in comparing symbolic powers of ideals with their ordinary powers.

Willa Cather Professor **Srikanth Iyengar** is interested in homotopical and homological algebra, mainly of commutative rings, but he also likes to talk to like-minded algebraic topologists and representation theorists. His recent research include square-zero matrices, levels in triangulated categories and its applications to the study of finite free complexes, and the homotopy category of commutative rings. His current research interests are commutative algebra, and the modular representation theory of finite groups.

Professor **Tom Marley** is interested in homological algebra over commutative Noetherian local or graded rings. Specifically, he studies finiteness properties of local cohomology, such as Hartshorne's concept of cofiniteness and the Huneke-Hochster conjecture that the set of associated prime ideals of a local cohomology module is finite. In addition, he works on applications of local cohomology to the theory of Hilbert functions and the depths of Rees algebras.

Edith T. Hitz research assistant professor **Alexandra Seceleanu** is interested in homological problems in local commutative algebra, but also likes to think about graded structures. Her thesis focused on extending lower bounds on ranks of syzygies to the case of certain hypersurface rings in mixed characteristic and separately on studying the Weak Lefschetz property in instances that can be related to the geometry of fat point schemes. She is also interested in computational algebra problems.

Aaron Douglas Professor & Chair **Judy Walker** works in algebraic coding theory. Much of her work uses techniques from number theory, algebraic geometry and graph theory. She has worked with algebraic geometric codes over rings and the relationship between weight measures on these codes and exponential sums. Currently, she studies low density parity check codes, focusing especially on their pseudocodeword structure.

Professor **Mark Walker** works in algebraic K-theory and motivic cohomology. He is an expert in ``semi-topological" K-theory, a blend of the algebraic and topological theories; this approach is shedding new light on some fundamental questions in algebraic K-theory. Recently, much of his work has concerned the algebraic and geometric properties of ``matrix factorizations". It turns out that matrix factorizations reveal a lot of information about the geometry of hypersurface rings, and also about the collection of maximal Cohen-Macaulay modules over them. He is interested in exploiting algebro-geometric and K-theoretic techniques to better understand the connection between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings.

Emeritus professor **Roger Wiegand** works on homology and represenation theory of finitely generated modules over local rings. On the homological side, he is interested in questions concerning vanishing of Tor_{i}^{R}(M,N) for i >> 0 and related questions concerning torsion in the tensor product of M and N. In representation theory, he studies rings with finite representation type and has shown, under very mild hypotheses, that a local ring has finite representation type if and only if its completion does. He also studies non-uniqueness of direct-sum decompositions of finitely generated modules over local rings. (For example, one can have indecomposable finitely generated modules A, B, and C such that A ⊕ B is isomorphic to the direct sum of a million copies of C.) His book ``Cohen-Macaulay Representations" written with Graham Leuschke, was recently published by the American Mathematical Society. This year he has been commutating between Lincoln and MSRI in Berkeley for the Special Year in Commutative Algebra.

Emerita Professor **Sylvia Wiegand** involved in an on-going investigation of the rings between a local ring and its completion. She also works in representation theory and on the partially ordered set of prime ideals in Noetherian rings of low dimension. She has also just completed a book, ``Power series over Noetherian rings" with William Heinzer of Purdue and Christel Rotthaus of Michigan State, a culmination of twenty years of work.
Another area of interest is the classification of indecomposable modules for low-dimensional rings. She has been traveling a fair amount, especially to attend parts of the Special Year in Commutative Algebra in Berkeley.

Assistant Professor **Wenliang Zhang** works in commutative algebra and algebraic geometry.
He received his PhD from the University of Minnesota and was a postdoc at University of Michigan before joining UNL in 2012. His current research interests include: local cohomology, algebraic D-modules, and singularities in characteristic p.