12,006 research outputs found
Uniformizing surfaces via discrete harmonic maps
We show that for any closed surface of genus greater than one and for any
finite weighted graph filling the surface, there exists a hyperbolic metric
which realizes the least Dirichlet energy harmonic embedding of the graph among
a fixed homotopy class and all hyperbolic metrics on the surface. We give
explicit examples of such hyperbolic surfaces through a new interpretation of
the Nielsen realization problem for the mapping class groups.Comment: 31 pages, 5 figure
On the complexity of optimal homotopies
In this article, we provide new structural results and algorithms for the
Homotopy Height problem. In broad terms, this problem quantifies how much a
curve on a surface needs to be stretched to sweep continuously between two
positions. More precisely, given two homotopic curves and
on a combinatorial (say, triangulated) surface, we investigate the problem of
computing a homotopy between and where the length of the
longest intermediate curve is minimized. Such optimal homotopies are relevant
for a wide range of purposes, from very theoretical questions in quantitative
homotopy theory to more practical applications such as similarity measures on
meshes and graph searching problems.
We prove that Homotopy Height is in the complexity class NP, and the
corresponding exponential algorithm is the best one known for this problem.
This result builds on a structural theorem on monotonicity of optimal
homotopies, which is proved in a companion paper. Then we show that this
problem encompasses the Homotopic Fr\'echet distance problem which we therefore
also establish to be in NP, answering a question which has previously been
considered in several different settings. We also provide an O(log
n)-approximation algorithm for Homotopy Height on surfaces by adapting an
earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the
planar setting
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Splitting homomorphisms and the Geometrization Conjecture
This paper gives an algebraic conjecture which is shown to be equivalent to
Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It
generalizes the Stallings-Jaco theorem which established a similar result for
the Poincare Conjecture. The paper also gives two other algebraic conjectures;
one is equivalent to the finite fundamental group case of the Geometrization
Conjecture, and the other is equivalent to the union of the Geometrization
Conjecture and Thurston's Virtual Bundle Conjecture.Comment: 11 pages, Some typos are correcte
Yang-Mills theory over surfaces and the Atiyah-Segal theorem
In this paper we explain how Morse theory for the Yang-Mills functional can
be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem.
Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma)
of a compact Lie group to the complex K-theory of the classifying
space . For infinite discrete groups, it is necessary to take into
account deformations of representations, and with this in mind we replace the
representation ring by Carlsson's deformation --theory spectrum \K
(\Gamma) (the homotopy-theoretical analogue of ). Our main theorem
provides an isomorphism in homotopy \K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)
for all compact, aspherical surfaces and all . Combining this
result with work of Tyler Lawson, we obtain homotopy theoretical information
about the stable moduli space of flat unitary connections over surfaces.Comment: 43 pages. Changes in v4: improved results in Section 7, simplified
arguments in the Appendix, various minor revision
A local to global argument on low dimensional manifolds
For an oriented manifold whose dimension is less than , we use the
contractibility of certain complexes associated to its submanifolds to cut
into simpler pieces in order to do local to global arguments. In particular, in
these dimensions, we give a different proof of a deep theorem of Thurston in
foliation theory which says that the natural map between classifying spaces
induces a
homology isomorphism where denotes the group of
homeomorphisms of made discrete. Our proof shows that in low dimensions,
Thurston's theorem can be proved without using foliation theory. Finally, we
show that this technique gives a new perspective on the homotopy type of
homeomorphism groups in low dimensions. In particular, we give a different
proof of Hacher's theorem that the homeomorphism groups of Haken -manifolds
with boundary are homotopically discrete without using his disjunction
techniques.Comment: Thoroughly revised. To appear in Transactions of the AM
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