88 research outputs found
Geodetic topological cycles in locally finite graphs
We prove that the topological cycle space C(G) of a locally finite graph G is
generated by its geodetic topological circles. We further show that, although
the finite cycles of G generate C(G), its finite geodetic cycles need not
generate C(G).Comment: 1
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
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Measured Group Theory
The workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as -invariants, the rank gradient, cost, torsion growth, entropy-type invariants and invariants coming from random walks and percolation theory. The participants came from a wide range of mathematics: asymptotic group theory, geometric group theory, ergodic theory, -theory, graph convergence, representation theory, probability theory, descriptive set theory and algebraic topology
Quantization, Classical and Quantum Field Theory and Theta - Functions
In the abelian case (the subject of several beautiful books) fixing some
combinatorial structure (so called theta structure of level k) one obtains a
special basis in the space of sections of canonical polarization powers over
the jacobians. These sections can be presented as holomorphic functions on the
"abelian Schottky space". This fact provides various applications of these
concrete analytic formulas to the integrable systems, classical mechanics and
PDE's. Our practical goal is to do the same in the non abelian case that is to
give an answer to the Beauville's question. In future we hope to extend this
digest to a mathematical mohograph with title "VBAC".Comment: To Igor Rostislavovich Shafarevich on his 80th birthday (will be
published by CRS, Canada
On the homology of locally finite graphs
We show that the topological cycle space of a locally finite graph is a
canonical quotient of the first singular homology group of its Freudenthal
compactification, and we characterize the graphs for which the two coincide. We
construct a new singular-type homology for non-compact spaces with ends, which
in dimension~1 captures precisely the topological cycle space of graphs but
works in any dimension.Comment: 30 pages. This is an extended version of the paper "The homology of a
locally finite graph with ends" (to appear in Combinatorica) by the same
authors. It differs from that paper only in that it offers proofs for Lemmas
3, 4 and 10, as well as a new footnote in Section
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