
Combinatorics/Algebra Seminar: Fall 2003 Archive

December 8

Ling Long: Elliptic surfaces, ADE Dynkin diagrams, and their applications.
Abstract:
An elliptic curve is a smooth genusone compact curve.
Geometrically, it is a onedimensional 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 ADE types.
The ADE 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 polyBernoulli 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 ErdõsKoRadó: 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 previouslyaccepted 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 ErdõsKoRadó 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 CauchyDavenport. In a way, the theory is just as beautiful as the classical theory of set addition.

October 13 and 27

Richard Ng: FrobeniusSchur indicators for semisimple quasiHopf algebras.
Abstract:
The FrobeniusSchur (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 nondegenerate Ginvariant symmetric bilinear form, antisymmetric bilinear form or neither. Moreover, the indicator c is given by the formula
c = ^{1}/_{G} S_{g e G} c(g^{2})
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 FrobeniusSchur indicators for semisimple quasiHopf algebras and show that they are invariants of the monoidal categories of their representations.

October 20

Sergey Kitaev (University of Kentucky): 2separated paths in the ncube.
Abstract:
A 2separated path in the ndimensional unit cube C_{n} is a sequence of
adjacent vertices of the cube such that the distance between any two
nonadjacent 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 2^{n}. 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 nletter 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 FonDerFlaass.)
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 threeterm arithmetic progression
whose elements are colored in distinct colors.
We also discuss a manycolor 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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