Speaker:
Helene Barcelo
Title: Foundations of a Connectivity Theory for Simplicial Complexes
Abstract:
We lay the foundations of a combinatorial homotopy theory, called
A-theory, for simplicial complexes, which reflects their connectivity
properties. This theory differs from the classical homotopy theory
for simplicial complexes. For example A-theory is not invariant under
diverse triangulations. On the other hand, in many instances it
behaves like the classical one: we derive a Seifert-Van Kampen type
theorem, and a long exact sequence of relative A-groups. A related
theory for graphs is also constructed. This theory provides a general
framework encompassing homotopy methods used to prove connectivity
results about order complexes, graphs and matroids.
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Speaker:
Mihai Ciucu
Title: Rotational invariance of quadromer correlations on
the hexagonal lattice
Abstract:
In 1963 Fisher and Stephenson conjectured that the monomer-monomer
correlation on the square lattice is rotationally invariant. In this
paper we prove a closely related statement on the hexagonal
lattice. Namely, we consider correlations of two quadromers
(four-vertex subgraphs consisting of a monomer and its three
neighbors) and show that they are rotationally invariant.
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Speaker:
Robin Forman
Title: Finite-Type Invariants for Graphs, and Graph
Reconstruction Problems
Abstract:
We present a mathematical soap opera, in which the speaker, beginning
with an open problem in knot theory, is inexorably (and unknowingly)
led to one of the most notorious open problems in graph theory. The
point of view suggested by the theory of knot invariants leads to some
new insights in the theory of graph reconstructions.
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Speaker:
Jesus de Loera
Title: Optimization problems in the space of triangulations of a
convex polytope
Abstract:Suppose you are given a finite set of points S in R^d
and weights or costs for each of the possible d-simplices determined
by (d+1) of these points. Motivated by problems in optimization and
computer graphics it is natural to consider the problem of finding a
triangulation of S which minimizes/maximizes the total weight of the
simplices involved. An example of such optimization problems is that
of finding minimal triangulations of cubes. In this talk we discuss
the structure of optimal triangulations and present several resultss
related to their computation. Joint work with Alexander Below and
Juergen Richter-Gebert (ETH-Zurich)
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Speaker:
Richard Stanley
Title: Polynomials with real zeros
Abstract: The theory of polynomials whose zeros are all real
goes back to Descartes and Newton. Further progress was made by a host
of other researchers and is closely related to the theory of total
positivity. More recently this subject has found surprising
connections with the representation theory of the symmetric group.
There are many interesting cases of combinatorially defined
polynomials whose zeros are all real. For instance, the Eulerian
polynomial An(x) has only real zeros. There are
also probabilistic aspects of polynomials with real zeros, e.g., what
is the probability that a polynomial of degree n (chosen from
some distribution) has all real zeros? We will give a general survey
of the theory of polynomials with real zeros.
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Speaker:
Neil White
Title: Coxeter Matroids
Abstract:
Coxeter matroids are a generalization of ordinary matroids, which are based
on a finite Coxeter group. Ordinary matroids are the case in which the Coxeter
group is the symmetric group. We present Coxeter matroids from three equivalent
points of view: a geometric one in terms of polytopes, an algebraic on, and a
combinatorial one. We also present representations of Coxeter matroids in
vector spaces and Tits buildings.
This survey talk will not require a background in either matroid theory or
Coxeter groups.
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Speaker:
Doron Zeilberger
Title: Enumeration and Symbol-Crunching
Abstract: Instead of trying to SOLVE a given Enumeration
problem, it is more profitable (and, for me, more fun), to try and
PROGRAM the computer, to solve the problem for us. Often, for almost
the same effort, we can program a whole infinite class of related, but
increasingly complex, problems.
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