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
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
diversified and acquired further strengths in several
other areas, notably in noncommutative algebra.
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.
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,
and algebraic topology. Vasily's most recent works are on formality
theorems and versions of
the Deligne conjecture for Hochschild complexes.
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.
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.
'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.
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.
The Algebra and Number Theory group is active in a variety of
research areas including:
For a short slide presentation about the research of the
Algebra and Number Theory group (from November 2010),
- homotopy algebras
- deformation theory
- algebraic number theory
- algorithmic approaches to finite-dimensional representation
- noetherian rings
- invariant theory (noncommutative and commutative)
- actions of algebraic transformation groups and Hopf algebras on noncommutative
- applications of Koszul algebras to algebraic combinatorics
- ring-theoretic structure and representation theory of quantum groups and related algebras
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,
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.
Several graduate students have completed Ph.D.s in algebra and number
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,
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.
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
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
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.