
Combinatorics/Algebra Seminar Fall 2016 Archive:

December 5

Jeremiah Goertz: An analogy between real reflection groups and complexhyperbolic reflection groups.
Abstract:
We present an analogy between reflection groups of positivedefinite lattices over Z, and reflection groups of complexhyperbolic lattices over Z[i]. Such Zlattice reflection groups are wellknown to be generated by reflections in simple roots; these roots form familiar CoxeterDynkin diagrams. There are no "simple roots" for complex lattices, but, in the cases discussed here, there are analogous roots which generate their reflection groups and which have appealing Coxeterlike diagrams. We also highlight the role played by "height reduction arguments" in the complexhyperbolic case.

November 28

Mehmet Dagli: The Ricci curvature of circulant graphs.
Abstract:
The (coarse) Ricci curvature for graphs was proposed by Lin, Lu and Yau, based on a metric space concept due to Ollivier. It provides a useful isomorphism invariant. Our goal is to build up a catalog of Ricci curvatures for circulant graphs. We will review the basic definitions, and exhibit several circulant families with constant Ricci curvature.

November 14

Stefanie Wang: Catalan and periCatalan numbers: Counting the effects of nonas
socativity.
Abstract:
Catalan numbers have many interpretations in mathematics. To
name a few, the nth Catalan number counts the different number of ways
to triangulate a convex (n + 1)gon, the number of rooted binary trees with
n leaves, and the number of ways to bracket a nonassociative product of n factors.
A quasigroup has a nonassociative multiplication that is cancelative, so it comes with right and left divisions. We will introduce periCatalan
numbers that count the reduced quasigroup words in a single argument appearing n times.

November 7

Sejeong Bang (Yeungnam University): Geometric distanceregular graphs.
Abstract:
A noncomplete distanceregular graph is called geometric if there exists a set of Delsarte cliques C such that each edge of the graph lies in a unique clique in C. In this talk, we introduce distanceregular graphs and geometric distanceregular graphs. In particular, we give a diameter bound for geometric distanceregular graphs with smallest eigenvalue  m. Moreover, we characterize geometric distanceregular graphs with smallest eigenvalue at least  4.

October 24, 31

Jonas Hartwig: Lie superalgebras and superdifferential operators.
Abstract:
In the first lecture I plan to give an overview of Lie superalgebras, including Kac's classification of simple Lie superalgebras. In the second lecture I'll talk about natural representations by superdifferential operators. If time permits I will state some open problems related to Howe duality and invariant theory.

October 10, 17

Adnan Abdulwahid: Cofree objects in centralizer and center categories.
Abstract:
We study cocompleteness, cowellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, cowellpoweredness and generators in these monoidal categories can be inherited from their base monoidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.

September 26, October 3

Jonathan Smith: Onesided Hopf algebras and quasigroups.
Abstract:
In a Hopf algebra structure, the antipode plays a role analogous to that of the inverse in a group. Taft and his coworkers have studied socalled left or right Hopf algebras, where the inversion property of the antipode only holds on one side. We will review the construction of a left Hopf algebra which is not twosided, and discuss the possibility of an analogous construction for Hopf quasigroups in the sense of Majid.

September 19

Anna Romanowska (Warsaw University of Technology): Dyadic intervals and dyadic triangles.
Abstract:
Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are respectively defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The onedimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean.
In this talk, I will discuss some older and some newer results concerning the algebraic structure of dyadic intervals and triangles. In contrast with real intervals and triangles, there are infinitely many (pairwise nonisomorphic) dyadic intervals and (pairwise nonisomorphic) dyadic triangles. I will present some characterizations of these objects, and a classification of dyadic intervals and dyadic triangles.

September 12

Alex Nowak: Local representation theory of finite groups.
Abstract:
Employing techniques from both the ordinary (representations over the complex numbers) and modular (representations over finite fields of prime characteristic p) programs, local representation theory reveals connections between the representations of a finite group and the normalizers of its psubgroups, referred to as local subgroups. We describe some of the fundamental constructions in this area; namely, blocks and defect groups are introduced from module and character theoretic perspectives. After establishing the necessary language, we formulate conjectures of Alperin and McKay and review progress made towards their proofs.
Archive of earlier seminars
Back to the Mathematics Institute
Back to Main Street

