Combinatorics/Algebra Seminar
Fall 2016 Archive:

December 5
Jeremiah Goertz: An analogy between real reflection groups and complex-hyperbolic reflection groups.

We present an analogy between reflection groups of positive-definite lattices over Z, and reflection groups of complex-hyperbolic lattices over Z[i]. Such Z-lattice reflection groups are well-known to be generated by reflections in simple roots; these roots form familiar Coxeter-Dynkin 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 Coxeter-like diagrams. We also highlight the role played by "height reduction arguments" in the complex-hyperbolic case.

November 28
Mehmet Dagli: The Ricci curvature of circulant graphs.

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 peri-Catalan numbers: Counting the effects of nonas- socativity.

Catalan numbers have many interpretations in mathematics. To name a few, the n-th 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 peri-Catalan numbers that count the reduced quasigroup words in a single argument appearing n times.

November 7
Sejeong Bang (Yeungnam University): Geometric distance-regular graphs.

A non-complete distance-regular 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 distance-regular graphs and geometric distance-regular graphs. In particular, we give a diameter bound for geometric distance-regular graphs with smallest eigenvalue
- m. Moreover, we characterize geometric distance-regular graphs with smallest eigenvalue at least - 4.

October 24, 31
Jonas Hartwig: Lie superalgebras and super-differential operators.

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 super-differential 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.

We study cocompleteness, co-wellpoweredness 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, co-wellpoweredness 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: One-sided Hopf algebras and quasigroups.

In a Hopf algebra structure, the antipode plays a role analogous to that of the inverse in a group. Taft and his co-workers have studied so-called 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 two-sided, 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.

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 one-dimensional 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 non-isomorphic) dyadic intervals and (pairwise non-isomorphic) 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.

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 p-subgroups, 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.

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