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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.

Category O for Lie superalgebras

STAG 261 (Note Room Change!)

Speaker: Daniel K. Nakano

Abstract: In this talk, the speaker will define a general Category $\mathcal{O}$ for any quasi-simple Lie superalgebra. Our construction encompasses (i) the parabolic Category ${\mathcal O}$ for complex semisimple Lie algebras, and (ii) the known constructions of Category ${\mathcal O}$ for specific examples of classical Lie superalgebras. In particular, the speaker will develop a parabolic Category ${\mathcal O}$ for classical Lie superalgebras. Connections between the categorical cohomology and the relative Lie superalgebra cohomology will be firmly established. These results will be used to show that the Category ${\mathcal O}$ is standardly co-stratified. The definition of standardly co-stratified used in this context is generalized from the original definition of Cline, Parshall, and Scott. An explicit description of the categorical cohomology ring will be given, and finite generation results will be presented. Furthermore, it will be shown that the complexity of modules in… Read more.

Heap Birkhoff polytopes

Zoom

Speaker: Emily Gunawan

Abstract: For each orientation c of a type A Dynkin quiver, we define a c-Birkhoff polytope Birk(H) and show that it is integrally equivalent to the order polytope for poset H, the heap of the c-sorting word of the longest permutation. A consequence of this result is that the volume of the c-Birkhoff polytope is the number of the longest chains in the type A c-Cambrian lattice. We will also discuss current work in type B and a generalization of our result to other Birkhoff subpolytopes Birk(H) corresponding to heaps H of other reduced words of an element in the symmetric group. Read more.