Algebra and Number Theory is a research area at the core of modern mathematics, having many connections to other areas within mathematics as well as applications to physics, computer science, biology, and chemistry. Members of the group work on several topics in modern algebra and algebraic number theory that involve modular forms, Galois theory, representation theory, invariant theory, arithmetic groups, and operads. These topics have compelling connections to hyperbolic geometry, low-dimensional topology, algebraic geometry and number theory. Upon completing their studies, our PhD students find either academic positions or jobs in industry (e.g. Dexai Robotics, LinkedIn, The Vanguard Group). Undergraduate students who interact with members of our group go to graduate programs in Mathematics or Computer Science, or find jobs in industry (e.g. IBM, Lockheed Martin, The Vanguard Group).
Research interests: algebraic and analytic number theory.
Webpage. Research interests: geometric Galois actions, operads and questions related to finite type invariants of knots.
Webpage. Research interests: number theory and machine learning.
Webpage. Research interests: modular forms, Galois representations, elliptic curves.
Webpage. Research interests: noncommutative algebra, representation theory and invariant theory.
Webpage. Research interests: algebraic geometry, low-dimensional topology, hyperbolic geometry, dynamics, and number theory.
Webpage. Advisor: Jaclyn Lang. Research interests: modular forms, Galois representations, elliptic curves.
Temple University Algebra Seminar usually takes place on Mondays 1:30-2:30pm in Wachman 617. Current contacts for the seminar are Vasily Dolgushev, Jaclyn Lang and Martin Lorenz.
Philadelphia Area Number Theory Seminar rotates between Bryn Mawr, Swarthmore, and Temple. If you would like to be added to our mailing list or if you are interested in being a speaker, please contact one of the organizers: Jaclyn Lang, Catherine Hsu, Ian Whitehead, and Djordje Milicevic.
Several graduate students have completed Ph.D.s under the direction of members of our group. Interested graduate students are encouraged to take advanced topics courses in these and related areas and to attend our seminars. General information about graduate study in mathematics at Temple University can be found on the graduate program website.
Selected Courses
This is the foundational year-long course that 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 are: groups, rings, fields, Galois theory, modules, and (multi-)linear algebra.
This is a year-long course on the fundamental concepts of commutative algebra and classical as well as modern algebraic geometry. 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.
This is a course on the principal methods and results of algebraic representation theory. It starts with an introduction to the fundamental notions, tools and general results of representation theory in the setting of associative algebras. This is followed by a thorough coverage of the classical representation theory of finite groups over an algebraically closed field of characteristic zero. If time permits, then the semester concludes with a brief introductory discussion of the representation theory of the general linear group. Other possible additional topics depend on the instructor.
The course is devoted to fundamental notions of homological algebra: chain complexes, abelian categories, derived functors, and spectral sequences. A portion of this course is also devoted to rudiments of category theory. Students will learn how to apply constructions of homological algebra and category theory to questions from abstract algebra, topology and deformation theory.
Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory and algebraic geometry.