110 research outputs found
A variational approach to the study of the existence of invariant Lagrangian graphs
This paper surveys some results by the author and collaborators on the
existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The
presentation is based on an invited talk by the author at XIX Congresso Unione
Matematica Italiana (Bologna, 12-17 Sept. 2011).Comment: 28 page
Axioms for infinite matroids
We give axiomatic foundations for non-finitary infinite matroids with
duality, in terms of independent sets, bases, circuits, closure and rank. This
completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig
Expanders and right-angled Artin groups
The purpose of this article is to give a characterization of families of
expander graphs via right-angled Artin groups. We prove that a sequence of
simplicial graphs forms a family of expander
graphs if and only if a certain natural mini-max invariant arising from the cup
product in the cohomology rings of the groups
agrees with the Cheeger constant of the
sequence of graphs, thus allowing us to characterize expander graphs via
cohomology. This result is proved in the more general framework of \emph{vector
space expanders}, a novel structure consisting of sequences of vector spaces
equipped with vector-space-valued bilinear pairings which satisfy a certain
mini-max condition. These objects can be considered to be analogues of expander
graphs in the realm of linear algebra, with a dictionary being given by the cup
product in cohomology, and in this context represent a different approach to
expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.Comment: 21 pages. Accepted version. To appear in J. Topol. Ana
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
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