Speaker:
Louis Billera
Title: Eulerian posets, enumeration and Quasisymmetric functions
Abstract:
We describe some links between enumeration of chains in
Eulerian posets and questions about representations of certain
quasisymmetric functions. The commutative peak algebra Pi of
Stembridge is generated by quasisymmetric functions arising from
enriched P-partitions. The noncommutative algebra A_E consists
of all chain-enumeration functional on Eulerian posets. Both have
Hilbert series given by the Fibonacci numbers. Bergeron, Mykytiuk,
Sottile and van Willigenburg have shown that, with natural coproducts,
Pi and A_E are dual Hopf algebras. We describe some consequences of
their result for the cd-index of Eulerian posets, a simple application
to enumeration of faces in hyperplane arrangements and a related
random walk on "peak sets".
This is joint work with Sam Hsiao and Steph van Willigenburg:
http://www.math.cornell.edu/~billera/papers/peaks.ps
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Speaker:
Igor Pak
Title: Groups, graphs, and random walks
Abstract:
I will present a somewhat biased survey of classical
and recent results on Cayley graphs of finite groups.
We start with the most simple questions related to
properties of 'random' Cayley graphs, and then move
to quite difficult, sometimes unsolved, combinatorial
and probabilistic problems. We then discuss applications
to computational group theory, including generation of
random group elements. The talk assumes no previous
knowledge of the subject.
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Speaker:
Victor Reiner
Title: Coxeter-like complexes for group presentations
Abstract:
(joint work with E. Babson)
We consider a highly symmetric simplicial cell complex associated to
any presentation of a group by a minimal generating set. This
construction generalizes the Coxeter complex associated to a Coxeter
presentation, and is closely related to other "coset complex"
constructions, such as Tits buildings and chessboard
complexes.
We look at the homology representations arising from these complexes,
particularly in the case of the symmetric group generating by a set of
transpositions which need not be Coxeter generators.
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Speaker:
Joel Spencer
Title: Phase Transitions for Random Processes
Abstract: In has been forty years since the discovery by Paul
Erdös and Alfred Rényi that the random graph undergoes a phase
transition (they called it: double jump) near p=1/n, many small
components rapidly melding into the ``giant component." We now view
this as a percolation effect, with classical connections to
Mathematical Physics. In recent years a host of random processes
coming from the fecund intersection of Discrete Math, Computer Science
and Probability have shown strong percolation behavior. One example:
each time unit two random edges are generated on n vertices. The
first is added to an evolving graph (initially empty) if it is
isolated, otherwise the second is added. We feel there will be an
explicit t0 such that near time t0 n/2 a giant
component suddenly emerges but the behavior near criticality remains
mysterious.
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Speaker:
Rekha Thomas
Title: A Disconnected Toric Hilbert Scheme
Abstract:
Recently, Santos has constructed an example of a disconnected toric
Hilbert scheme providing the first example of a disconnected
multi-graded Hilbert scheme in the sense of Haiman and Sturmfels
(2002). The connectedness of the scheme can be phrased as a
combinatorial problem. In this talk I will describe this combinatorial
approach and Santos' construction and then discuss some further
combinatorial questions that arise in this setting.
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Speaker:
Michelle Wachs
Title: Homology of the matching complex and the chessboard complex
Abstract: The matching complex is the simplicial complex of
partial matchings on a finite set and the chessboard complex is the
simplicial complex of rook placements on a chessboard. These
complexes have arisen in various contexts in the literature and have a
number of interesting topological properties. Topological properties
of the matching complex were first examined by Bouc in connection with
Quillen complexes, and topological properties of the chessboard
complex were first examined by Garst in connection with Coxeter
complexes. In this talk, I will survey old and new results on the
homology of these complexes. I will present joint work with John
Shareshian in which a conjecture of Bj\"orner, Lov\'asz, Vre\'cica and
\u Zivaljevi\'c dealing with connectivity of these complexes, is
settled. Torsion in the bottom nonvanishing integral homology is also
determined. This involves the use of symmetric group representation
theory and tableau combinatorics.
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Speaker:
David Wagner
Title: Generalizing Electrical Network Theory from Graphs to Matroids
Abstract:
The theory of linear electrical networks
(going back to Kirchhoff) involves an interesting interplay of complex
function theory and the combinatorics of finite graphs. As
motivation, I will briefly review Kirchhoff's formula for the
effective admittance of a network, and two physically transparent
properties of this response function -- real--part positivity and
Rayleigh monotonicity. This function may be defined for any element e
of any matroid M, although in this generality the physical
interpretation is at best problematic. Nonetheless, we may ask for
conditions under which the "physical" properties above still hold.
The short answer is "sometimes they do, sometimes they don't, and a
characterization seems impossible". The longer answer consists of
several necessary or sufficient conditions involving many familiar
classes of matroids (matching and transversal matroids, balanced
matroids, sixth-root-of-unity matroids, etc.) Most of the talk is
joint work with YoungBin Choe, James Oxley, and Alan Sokal.
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