639 research outputs found
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
Cuts and flows of cell complexes
We study the vector spaces and integer lattices of cuts and flows associated
with an arbitrary finite CW complex, and their relationships to group
invariants including the critical group of a complex. Our results extend to
higher dimension the theory of cuts and flows in graphs, most notably the work
of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut
and flow spaces, interpret their coefficients topologically, and give
sufficient conditions for them to be integral bases of the cut and flow
lattices. Second, we determine the precise relationships between the
discriminant groups of the cut and flow lattices and the higher critical and
cocritical groups with error terms corresponding to torsion (co)homology. As an
application, we generalize a result of Kotani and Sunada to give bounds for the
complexity, girth, and connectivity of a complex in terms of Hermite's
constant.Comment: 30 pages. Final version, to appear in Journal of Algebraic
Combinatoric
K-area, Hofer metric and geometry of conjugacy classes in Lie groups
Given a closed symplectic manifold we introduce a certain
quantity associated to a tuple of conjugacy classes in the universal cover of
the group by means of the Hofer metric on
. We use pseudo-holomorphic curves involved in the
definition of the multiplicative structure on the Floer cohomology of a
symplectic manifold to estimate this quantity in terms of actions
of some periodic orbits of related Hamiltonian flows. As a corollary we get a
new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of
products of unitary matrices. As another corollary we get a new proof of the
geodesic property (with respect to the Hofer metric) of Hamiltonian flows
generated by certain autonomous Hamiltonians. Our main technical tool is K-area
defined for Hamiltonian fibrations over a surface with boundary in the spirit
of L.Polterovich's work on Hamiltonian fibrations over .Comment: Corrected final version, accepted for publication in Inventiones
Mathematica
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