Algebra and Number Theory

The Department of Mathematics at Temple University has a strong tradition of research in algebra and number theory. Under the leadership of Emil Grosswald, a member of our faculty from 1968 to 1980, research in our department developed a particular focus in analytic number theory. Grosswald's memory is honored by our ongoing distinguished lecturer series which carries his name. More recently, research in the Algebra and Number Theory group has diversified and acquired further strengths in several other areas, notably in noncommutative algebra.

Faculty

Boris Datskovsky's research interests lie in algebraic and analytic number theory. He has worked on the distribution of discriminants of abelian and nonabelian extensions of an algebraic number field, the theory of zeta functions associated with prehomogeneous vector spaces, and arithmetic aspects of the theory of modular forms.

Vasily Dolgushev's research interests are in noncommutative geometry, homological algebra, category theory and mathematical physics. Most of the problems he is working on are motivated by questions in mathematical physics. His approach to these problems often involves higher algebraic structures such as homotopy algebras, higher operads, and higher categories. The solutions of these problems are applied to questions in Lie theory, algebraic geometry, and algebraic topology. Vasily's most recent works are on formality theorems and versions of the Deligne conjecture for Hochschild complexes.

Edward Letzter's research interest is in noncommutative rings, their representations (i.e, actions by linear transformations on vector spaces) and internal structure (primarily two-sided ideal theory). His most recent work concerns the structure of skew power series rings. He has also recently studied algorithmic aspects of finite dimensional representations.

Martin Lorenz's research interests encompass several areas of noncommutative algebra. He has worked on topics in ring theory, group theory, Hopf algebras, algebraic K-theory and other fields. For about ten years, Lorenz's research was focused on multiplicative invariant theory. His monograph with that title was published by Springer-Verlag in 2005. Subsequently, Lorenz has worked on applications of Koszul algebras to algebraic combinatoris and, most recently, on algebraic group actions on noncommutative spaces.

Chelsea Walton's research interests are in noncommutative algebraic geometry, noncommutative invariant theory, quantum groups, and all things noncommutative. She is particularly interested in Quantum Symmetries, i.e. Hopf (co)actions on quantum algebras. She also studies subalgebras of invariants (elements that remain fixed) under the Hopf (co)action under consideration, and deformations of related algebraic structures. Moreover, Walton likes to use methods of noncommutative projective algebraic geometry to analyze noncommutative graded algebras, especially those whose origins lie in physics.

Xingting Wang is a postdoc at the Department of Mathematics. He is currently working with Chelsea Walton and his research is concerned with noncommutative algebras, Poisson algebras, noncommutative projective algebraic geometry, cohomology theory, and noncommutative invariant theory.



Research Profile

The Algebra and Number Theory group is active in a variety of research areas including:

  • operads
  • homotopy algebras
  • deformation theory
  • algebraic number theory
  • algorithmic approaches to finite-dimensional representation theory
  • noetherian rings
  • invariant theory (noncommutative and commutative)
  • actions of algebraic transformation groups and Hopf algebras on noncommutative spaces
  • applications of Koszul algebras to algebraic combinatorics
  • ring-theoretic structure and representation theory of quantum groups and related algebras
For a short slide presentation about the research of the Algebra and Number Theory group (from November 2010), click here.

Seminars

Algebra Seminar: Besides serving as a forum for our faculty, students and invited speakers to report on their latest research activities, the weekly departmental Algebra Seminar regularly offers longer and more leisurely paced series of lectures on various algebraic topics. Our graduate students are the primary targeted audience of these mini-courses, but the format has met with great success among our faculty as well. Here is a list of some recent mini-courses:

Number Theory Seminar: The oldest continously running seminar in our department, the Number Theory Seminar was organized for many years by the late Marvin Knopp. The seminar continues to provide a venue for lively interactions between number theorists from the greater Philadelphia area.

Graduate Studies

Several graduate students have completed Ph.D.s in algebra and number theory in recent years. Interested graduate students are encouraged to take advanced topics courses in these and related areas and to attend the above listed weekly seminars. Summer research stipends (from NSA and NSF) are currently available for eligible graduate students.

General information about graduate study in mathematics at Temple, including course descriptions, can be found on the graduate program website. Below is a listing, with brief descriptions, of the central courses specifically in Algebra and Number Theory. These courses provide the basic toolkit for aspiring algebraists or number theorists. More advanced courses are also offered frequently; these cover various topics including deformation theory, computational methods in algebra, algebraic geometry, invariant theory, Lie groups etc.

Central Courses

8011/8012. Abstract Algebra I / II, a two-semester sequence that is offered every year, is the foundational course in abstract algebra; it gives an introduction to the terminology and methods of modern abstract algebra. The course sequence should preferably be taken during the first year of graduate studies, since all other courses on algebraic topics build on it. The main topics covered are: groups, rings, fields, Galois theory, modules, and (multi-)linear algebra.

9012/13. Representation Theory I / II. This is an ideal follow-up course to the 8011/12 sequence. Representations of groups, Lie algebras, and other algebraic structures feature in many areas of mathematics besides algebra, yet the basic methods and results are quite accessible. This two-semester course is offered regularly. The first semester generally focuses on representations of finite groups, while the second semester is mainly devoted to finite-dimensional Lie algebras.

9011. Homological Algebra can also be taken directly after the basic 8011/12 course. The topic is more abstract than representation theory and requires slightly greater mathematical maturity. Homological algebra finds widespread use in pure mathematics, including many areas of analysis. The material covered in this one-semester course includes chain complexes, the rudiments of category theory, derived functors, and spectral sequences.

9014/15. Commutative Algebra and Algebraic Geometry I / II. This is a year-long course on the fundamental concepts of commutative algebra and classical as well as modern algebraic geometry. The 8011/12 course sequence suffices for background, but some knowledge of rudimentary point-set topology will be helpful. Topics for the first semester include: ideals of commutative rings, modules, Noetherian and Artinian rings, Noether normalization, Hilbert's Nullstellensatz, rings of fractions, primary decomposition, discrete valuation rings and the rudiments of dimension theory. Topics for the second semester include: affine and projective varieties, morphisms of algebraic varieties, birational equivalence, and basic intersection theory. In the second semester, students will also learn about schemes, morphisms of schemes, coherent sheaves, and divisors.