3,409 research outputs found
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings
of G. We give an affirmative answer to the conjecture of Read and
Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic
polynomial form a log-concave sequence. We define a sequence of numerical
invariants of projective hypersurfaces analogous to the Milnor number of local
analytic hypersurfaces. Then we give a characterization of correspondences
between projective spaces up to a positive integer multiple which includes the
conjecture on the chromatic polynomial as a special case. As a byproduct of our
approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor
number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So
The Patterson-Sullivan embedding and minimal volume entropy for outer space
Motivated by Bonahon's result for hyperbolic surfaces, we construct an
analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann
outer space into the space of projectivized geodesic currents on a
free group. We prove that this map is a topological embedding. We also prove
that for every the minimum of the volume entropy of the universal
covers of finite connected volume-one metric graphs with fundamental group of
rank and without degree-one vertices is equal to and that
this minimum is realized by trivalent graphs with all edges of equal lengths,
and only by such graphs.Comment: An updated versio
Lifts of convex sets and cone factorizations
In this paper we address the basic geometric question of when a given convex
set is the image under a linear map of an affine slice of a given closed convex
cone. Such a representation or 'lift' of the convex set is especially useful if
the cone admits an efficient algorithm for linear optimization over its affine
slices. We show that the existence of a lift of a convex set to a cone is
equivalent to the existence of a factorization of an operator associated to the
set and its polar via elements in the cone and its dual. This generalizes a
theorem of Yannakakis that established a connection between polyhedral lifts of
a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts
of convex sets can also be characterized similarly. When the cones live in a
family, our results lead to the definition of the rank of a convex set with
respect to this family. We present results about this rank in the context of
cones of positive semidefinite matrices. Our methods provide new tools for
understanding cone lifts of convex sets.Comment: 20 pages, 2 figure
A tropical proof of the Brill-Noether Theorem
We produce Brill-Noether general graphs in every genus, confirming a
conjecture of Baker and giving a new proof of the Brill-Noether Theorem, due to
Griffiths and Harris, over any algebraically closed field.Comment: 17 pages, 5 figures; v3: added a new Section 3, detailing how the
classical Brill-Noether theorem for algebraic curves follows from Theorem
1.1. Update references, minor expository improvement
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