Speaker:
William Trotter
Title: A Combinatorial Approach to Correlation
Inequalities
Abstract:In joint research with Graham Brightwell, we have
initiated work into finding combinatorial proofs of correlation
inequalities for finite partially ordered sets (posets). Last year,
we gave a new proof of the (strong) XYZ-theorem of L. Shepp and
P.C. Fishburn.
Quite recently, we found a new proof of the first
non-trivial case of the following result of R. Stanley. For a fixed
element x in a poset P on n elements, let hi denote the
number of linear extensions of P in which x appears at height i.
Using the Alexandrov/Fenchel inequalities for mixed volumes, Stanley
proved that the sequence {hi: i between 1 and n} is
log-concave, i.e., hihi+2 is less than or equal
to h2i+1 for all i=1,2,...,n-2. We prove this
result for the special case where there are exactly three non-zero
terms in the sequence. Our proof is purely combinatorial in the sense
that it does not require any geometric machinery, although it does
depend on the Ahlswede/Daykin four functions theorem. Despite its
specialized nature, our proof gives valuable structural information
and holds hope of being extended to the general case.
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Speaker:
Guenter Ziegler
Title: Decompositions of simplicial balls and knots
consisting of few edges
Abstract: Constructibility is a condition on pure simplicial
complexes that is weaker than shellability. We show that
non-constructible triangulations of the d-dimensional sphere exist for
every d greater than or equal to 3. This answers a question of
Danaraj & Klee from 1978. It also strengthens a result of
Lickorish about non-shellable spheres.
Furthermore, we provide a hierarchy of combinatorial decomposition
properties that follow from the existence of a non-trivial knot with
``few edges'' in a 3-sphere or 3-ball, and a similar hierarchy for
3-balls with a knotted spanning arc that consists of ``few edges.''
(Joint work with Masahiro Hachimori.)
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Speaker:
Margaret Bayer
Title: Flag Numbers of Polytopes and Partially Ordered
Sets
Abstract: The f-vector of a convex polytope gives the number of
i-dimensional faces of the polytope for all i. The problem of
characterizing the f-vectors of all polytopes has proved intractable.
More combinatorial information is encoded in the flag vector, a useful
extension of the f-vector. In this talk I will define flag vectors,
the cd-index and Eulerian posets, and will survey results on linear
inequalities.
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Speaker:
Dominic Welsh
Title: Counting Polynomials for Generalised
Arrangements
Abstract: I will present recent work with Geoff Whittle on
counting problems associated with arrangements (interpreted in a very
general sense), hypergraph colourings and lattice point enumeration.
In particular we obtain a natural generalisation of the classical
characteristic polynomial and Tutte polynomial of a combinatorial
geometry and graph
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Speaker:
Joseph Kung
Title: Critical problems for matroids and polymatroids
Abstract: A polymatroid is a matroid in which a single element
do not necessarily have rank zero or one. The model for a polymatroid
is a collection of subspaces, or in the dual space, a subspace
arrangement. Many of the results and methods for matroids do not work
for polymatroids. In this talk, we will discuss the similarities and
differences between the critical problem for matroids and the critical
problem for polymatroids. In particular, we will develop a uniform
geometric approach for constructing blocks and tangential blocks.
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Speaker:
James Oxley
Title: The interplay between graphs and matroids
Abstract: Historically, one of the most profound influences on
the development of matroid theory has been the interaction between
graphs and matroids. Indeed, a powerful statement of the strength of
this link is contained in the following words of Tutte: ``If a
theorem about graphs can be expressed in terms of edges and circuits
alone it probably exemplifies a more general theorem about
matroids.'' This talk will survey some recent extremal results for
graphs and matroids that illustate how the subjects of graph theory
and matroid theory are mutually enriching.
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Speaker:
Catherine Yan
Title: Integral Apollonian Packings
Abstract:Apollonian circle packings arise by repeatedly filling
the interstices between mutually tangent circles with further tangent
circles. It is possible for every circle in such a packing to have
integer radius of curvature. In fact, it is even possible for a
packing to be oriented in the Euclidean plane so that for each circle
Ci in the packing with center (xi,yi)
and radius of curvature ri, all the quantities
ri, ri xi and ri
yi are integers.
In this talk we study these remarkable packings via the Descartes
equation, which relates the radii of curvature of four mutually
tangent circles:
2 ( x2 + y2 + z2 + w2) -
(x + y + z + w)2 = 0
We establish a generalization of the Descartes equations which
characterizes Descartes configurations consisting of four
mutually tangent circles with disjoint interiors. There are two
natural group actions on Descartes configurations: the Moebius
transformations and the isochronous Lorentz group. We analyze the
Apollonian packings by studying these group actions on the set of
ordered Descartes configurations. In particular, we introduce a set of
Apollonian operators and prove that they generate the isochronous
Lorentz group (up to a conjugate).
An integral Apollonian packing is classified by a certain root
quadruple that corresponds to the smallest quadruple of circles in
the packing. We determine the asymptotics of the number of root
quadruples, and discuss some congruence restrictions and conjectures
on the integers represented by an Apollonian packing. Finally, we
touch on higher-dimensional analogs of these packings.
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