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Department of Mathematics

Algebra and Number Theory Seminars

Tiling

Group theory is the formal mathematical study of symmetry. Groups are among the foundational objects composing abstract algebra, yet they also pervade nearly every discipline in pure mathematics as well as many areas of science and engineering. One striking result of group theory shows that there are exactly 17 different types of planar symmetry. This image illustrates one of these types of symmetry in a section of tilework at the Alhambra Palace in Granada, Spain. This particular symmetry is characterized by 3-fold rotational symmetry with no reflections (Photo credit The_Alhambra_and_The_Alcazar).

The Algebra and Number Theory Seminar is structured to include talks on a broad range of mathematical areas that are of interest to algebraists and number theorists, including analytic and algebraic number theory, algebra, combinatorics, algebraic and arithmetic geometry, cryptography, representation theory, and more. Talks are given by a variety of local, national, and international speakers in number theory and related areas.

See below for upcoming seminars or access the seminar archive.

Organizers

Mary Flahive, Clayton Petsche, Thomas Schmidt and Holly Swisher

Timing

We traditionally meet every Tuesday at 11:00 a.m.

Free abelian groups detected by the Weil height

STAG 263

Speaker: Jeff Vaaler

In 1947 Skolem proved that the multiplicative group of an algebraic number field K modulo its torsion subgroup is a free abelian group. We outline a proof that this remains true for infinite algebraic extensions of the rationals provided the infinite extension satisfies the Bogomolov property. In contrast to these results, the multiplicative group of all nonzero algebraic numbers modulo its torsion subgroup is known to be a vector space over the rationals, and therefore it is a divisible abelian group. Read more.

Rankin-Cohen Type Differential Operators on Automorphic Forms

STAG 263

Speaker: Francis Dunn

In the classical setting, the derivative of a holomorphic modular form of integral weight on the complex upper half-plane is not in general a modular form since the derivative fails to satisfy the correct transformation properties. However, R. A. Rankin and H. Cohen were able to construct particular bilinear differential operators sending modular forms to modular forms. These Rankin—Cohen operators have several interesting properties and have been studied by D. Zagier, Y. Choie, T. Ibukiyama, and others.In this talk I will discuss the classical Rankin—Cohen operators, and some of their generalizations to automorphic forms in higher dimension, including ​​constructing Rankin—Cohen​ type differential operators on Hermitian modular forms of signature (n,n). Read more.