Combinatorics/Algebra Seminar:
Fall 2003 Archive

December 8

Ling Long: Elliptic surfaces, A-D-E Dynkin diagrams, and their applications.

Abstract:

An elliptic curve is a smooth genus-one compact curve. Geometrically, it is a one-dimensional torus. Elliptic surfaces are elliptic curves over function fields. Every elliptic surface has a smooth minimal compactification due to Kodaira and Néron by adding special fibers in a continuous way. The special fibers of elliptic surfaces correspond to Dynkin diagrams of A-D-E types. The A-D-E Dynkin diagrams are simple connected graphs. We will discuss basic properties of these diagrams, the associated modular forms, and some applications in graph theory and sphere packing.

December 1

Alex Burstein: Restricted Dumont permutations.

November 24

Thanksgiving Break: no seminar.

November 17

Chad Brewbaker: Enumeration of (0,1)-matrices with unique row and column sum vectors

Abstract:

We examine the set of (0,1)-matrices uniquely determined by their row and column sums. Their cardinality is shown to be the poly-Bernoulli numbers of negative index. The graphs generated by adjacency matrices of this type are then discussed. (Labeled directed graphs with unique in/out degrees, labeled linear orders, and a labeled graph with a looped and totally connected center, and nonadjacent unlooped outer vertices.)

November 10

Ryan Martin: You've got Erdõs-Ko-Radó: Intersecting hypergraphs online

Abstract:

In this talk we discuss a random process in which sets of size r (hyperedges) are chosen from a ground set of size n, uniformly at random. We accept a hyperedge if it has nonempty intersection with each previously-accepted edge. Otherwise, it is rejected. The process concludes when there are no remaining edges to choose.

When the process concludes, we count the number of accepted edges. The famous Erdõs-Ko-Radó theorem gives an upper bound for this quantity. In this talk, we see that the probability this upper bound is achieved can be found asymptotically, and that the threshold is sharp.

November 3

Christian Roettger: Vsevolod Lev's work on restricted set addition in abelian groups.

Abstract: If you consider restricted addition of sets X, Y omitting the sums a + a, a being both in X and Y, you can still prove theorems similar to Cauchy-Davenport. In a way, the theory is just as beautiful as the classical theory of set addition.

October 13 and 27

Richard Ng: Frobenius-Schur indicators for semisimple quasi-Hopf algebras.

Abstract: The Frobenius-Schur (FS) indicator of an irreducible representation V of a finite group G is a scalar c in {1,-1, 0}, which indicates whether V admits a non-degenerate G-invariant symmetric bilinear form, anti-symmetric bilinear form or neither. Moreover, the indicator c is given by the formula c = ¹/_|G| S_{g e G} c(g²) where c is the character of V. It was little known that the set of FS indicators for G is, indeed, an invariant of the monoidal category C[G]-mod. In these talks, I will give a short review of FS indicators for a finite group. Thereafter, I will define the generalized notion of Frobenius-Schur indicators for semisimple quasi-Hopf algebras and show that they are invariants of the monoidal categories of their representations.

October 20

Sergey Kitaev (University of Kentucky): 2-separated paths in the n-cube.

Abstract:

A 2-separated path in the n-dimensional unit cube C_n is a sequence of adjacent vertices of the cube such that the distance between any two non-adjacent vertices is at least two (the distance is the number of coordinates in which the vertices differ). Let L(n) be the maximal length of such a path in C_n. The problem is to find L(n) for each n.

In 1969, Evdokimov found the order of L(n), which is equal to a constant times 2ⁿ. The problem was not solved by any complicated geometric considerations, but rather by considering the following equivalent problem: Find a word W of maximal length in an n-letter alphabet, such that any subword S of W (of length greater than one) has at least two letters that occur an odd number of times in S.

October 6

Cliff Bergman & Jonathan Smith:
Report from the Boulder CO AMS session Algebras, Lattices and Varieties.

September 29

Maria Axenovich & Ryan Martin:
Report from the 37th Midwestern Graph Theory Conference.

September 22

Benard Kivunge: Report on the LOOPS '03 conference.

September 15

Maria Axenovich: On rainbow arithmetic progressions.
(Joint work with Dmitri Fon-Der-Flaass.)

Abstract: Consider natural numbers { 1 ,..., n } colored in three colors. We prove that if each color appears on at least (n + 4)/6 numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. We also discuss a many-color version of this result providing a generalization of Van der Waerden's theorem.

September 8

Anna Romanowska: Central groupoids and de Bruijn graphs.

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