# 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:

• Introduction to group representations (5 lectures, Fall 2009)
• Molecular vibration: group representation theory in chemistry and physics (6 lectures, Spring 2010)
• Operads (semester-long mini-course, Fall 2010)
• Representations of $S_n$: the method of Okounkov and Vershik (3 lectures, Spring 2011)
• A taste of algebraic geography (4 lectures, Spring 2012)
• Algebraic operads and homotopy algebras (semester-long mini-course, Fall 2012)
• Rational homotopy theory (semester-long mini-course, Spring 2013)

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.