23,360 research outputs found
On rational homology disk smoothings of valency 4 surface singularities
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has
a complete list of resolution graphs of weighted homogeneous complex surface
singularities admitting a rational homology disk ("QHD") smoothing, i.e., one
with Milnor number 0. They fall into several classes, the most interesting of
which are the three classes whose resolution dual graph has central vertex with
valency 4. We give a uniform "quotient construction" of the QHD smoothings for
these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise
description of the Milnor fibre and its non-abelian fundamental group. This had
already been done for two of these classes in a previous paper; what is new
here is the construction of the third class, which is far more difficult. In
addition, we explain the existence of two different QHD smoothings for the
first class.
We also prove a general formula for the dimension of a QHD smoothing
component for a rational surface singularity. A corollary is that for the
valency 4 cases, such a component has dimension 1 and is smooth. Another
corollary is that "most" H-shaped resolution graphs cannot be the graph of a
singularity with a QHD smoothing. This result, plus recent work of
Bhupal-Stipsicz, is evidence for a general
Conjecture: The only complex surface singularities with a QHD smoothing are
the (known) weighted homogeneous examples.Comment: 28 pages: title changed, typos fixed, references and small
clarifications adde
Belyi-extending maps and the Galois action on dessins d'enfants
We study the absolute Galois group by looking for invariants and orbits of
its faithful action on Grothendieck's dessins d'enfants. We define a class of
functions called Belyi-extending maps, which we use to construct new Galois
invariants of dessins from previously known invariants. Belyi-extending maps
are the source of the ``new-type'' relations on the injection of the absolute
Galois group into the Grothendieck-Teichmuller group. We make explicit how to
get from a general Belyi-extending map to formula for its associated invariant
which can be implemented in a computer algebra package. We give an example of a
new invariant differing on two dessins which have the same values for the other
readily computable invariants.Comment: 13 pages, 7 figures; submitted for publication; revisions are that
the paper now deals only with Galois invariants of dessins, and that material
is slightly expande
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
Genus Ranges of 4-Regular Rigid Vertex Graphs
We introduce a notion of genus range as a set of values of genera over all
surfaces into which a graph is embedded cellularly, and we study the genus
ranges of a special family of four-regular graphs with rigid vertices that has
been used in modeling homologous DNA recombination. We show that the genus
ranges are sets of consecutive integers. For any positive integer , there
are graphs with vertices that have genus range for all
, and there are graphs with vertices with genus range
for all or . Further, we show that
for every there is such that is a genus range for graphs with
and vertices for all . It is also shown that for every ,
there is a graph with vertices with genus range , but there
is no such a graph with vertices
Ramanujan graphs in cryptography
In this paper we study the security of a proposal for Post-Quantum
Cryptography from both a number theoretic and cryptographic perspective.
Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the
hardness of finding paths in Ramanujan graphs. One is based on
Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on
Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks
the hash function based on LPS graphs. On the Supersingular Isogeny Graphs
proposal, recent work has continued to build cryptographic applications on the
hardness of finding isogenies between supersingular elliptic curves. A 2011
paper by De Feo-Jao-Pl\^{u}t proposed a cryptographic system based on
Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In
this paper we show that the security of the SIDH proposal relies on the
hardness of the SIG path-finding problem introduced in [CGL06]. In addition,
similarities between the number theoretic ingredients in the LPS and Pizer
constructions suggest that the hardness of the path-finding problem in the two
graphs may be linked. By viewing both graphs from a number theoretic
perspective, we identify the similarities and differences between the Pizer and
LPS graphs.Comment: 33 page
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