415 research outputs found
The local Gromov-Witten theory of CP^1 and integrable hierarchies
In this paper we begin the study of the relationship between the local
Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line
and the theory of integrable hierarchies. We first of all construct explicitly,
in a large number of cases, the Hamiltonian dispersionless hierarchies that
govern the full descendent genus zero theory. Our main tool is the application
of Dubrovin's formalism, based on associativity equations, to the known results
on the genus zero theory from local mirror symmetry and localization. The
hierarchies we find are apparently new, with the exception of the resolved
conifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For this
example the relevant dispersionless system turns out to be related to the
long-wave limit of the Ablowitz-Ladik lattice. This identification provides us
with a complete procedure to reconstruct the dispersive hierarchy which should
conjecturally be related to the higher genus theory of the resolved conifold.
We give a complete proof of this conjecture for genus g<=1; our methods are
based on establishing, analogously to the case of KdV, a "quasi-triviality"
property for the Ablowitz-Ladik hierarchy at the leading order of the
dispersive expansion. We furthermore provide compelling evidence in favour of
the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing
it successfully in the primary sector for g=2.Comment: 30 pages; v2: an issue involving constant maps contributions is
pointed out in Sec. 3.3-3.4 and is now taken into account in the proofs of
Thm 1.3-1.4, whose statements are unchanged. Several typos, formulae,
notational inconsistencies have been fixed. v3: typos fixed, minor textual
changes, version to appear on Comm. Math. Phy
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Discrete Differential Geometry
This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry
The Local Gromov-Witten Theory of and Integrable Hierarchies
In this paper we begin the study of the relationship between the local Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full-descendent genus zero theory. Our main tool is the application of Dubrovin's formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold in the equivariantly Calabi-Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz-Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g≤ 1; our methods are based on establishing, analogously to the case of KdV, a "quasi-triviality” property for the Ablowitz-Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing it successfully in the primary sector for g=
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
An almost-linear time algorithm for uniform random spanning tree generation
We give an -time algorithm for generating a uniformly
random spanning tree in an undirected, weighted graph with max-to-min weight
ratio . We also give an -time algorithm for
generating a random spanning tree with total variation distance from
the true uniform distribution. Our second algorithm's runtime does not depend
on the edge weights. Our -time algorithm is the first
almost-linear time algorithm for the problem --- even on unweighted graphs ---
and is the first subquadratic time algorithm for sparse weighted graphs.
Our algorithms improve on the random walk-based approach given in
Kelner-M\k{a}dry and M\k{a}dry-Straszak-Tarnawski. We introduce a new way of
using Laplacian solvers to shortcut a random walk. In order to fully exploit
this shortcutting technique, we prove a number of new facts about electrical
flows in graphs. These facts seek to better understand sets of vertices that
are well-separated in the effective resistance metric in connection with Schur
complements, concentration phenomena for electrical flows after conditioning on
partial samples of a random spanning tree, and more
Triangulations
The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods
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