47,587 research outputs found
Irregular holonomic kernels and Laplace transform
Given a (not necessarily regular) holonomic D-module defined on the product
of two complex manifolds, we prove that the associated correspondence commutes
(in some sense) with the De Rham functor. We apply this result to the study of
the classical Laplace transform. The main tools used here are the theory of
ind-sheaves and its enhanced version.Comment: 62 pages. 2nd version typoes correcte
Riemann-Hilbert correspondence for holonomic D-modules
The classical Riemann-Hilbert correspondence establishes an equivalence
between the triangulated category of regular holonomic D-modules and that of
constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence
for holonomic D-modules which are not necessarily regular. The construction of
our target category is based on the theory of ind-sheaves by Kashiwara-Schapira
and influenced by Tamarkin's work. Among the main ingredients of our proof is
the description of the structure of flat meromorphic connections due to
Mochizuki and Kedlaya.Comment: 114pages; v.2 minor changes, 114 p
Regular and irregular holonomic D-modules
This is a survey paper based on a series of lectures given at the IHES in
February/March 2015. In a first part, we recall the main results on the
tempered holomorphic solutions of D-modules in the language of indsheaves and,
as an application, the Riemann-Hilbert correspondence for regular holonomic
modules. In a second part, we present the enhanced version of the first part,
treating along the same lines the irregular holonomic case.Comment: 114 page
A microlocal approach to the enhanced Fourier-Sato transform in dimension one
Let be a holonomic algebraic -module on the affine
line. Its exponential factors are Puiseux germs describing the growth of
holomorphic solutions to at irregular points. The stationary
phase formula states that the exponential factors of the Fourier transform of
are obtained by Legendre transform from the exponential factors
of . We give a microlocal proof of this fact, by translating it in
terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.Comment: 56 page
Detecting palindromes, patterns, and borders in regular languages
Given a language L and a nondeterministic finite automaton M, we consider
whether we can determine efficiently (in the size of M) if M accepts at least
one word in L, or infinitely many words. Given that M accepts at least one word
in L, we consider how long a shortest word can be. The languages L that we
examine include the palindromes, the non-palindromes, the k-powers, the
non-k-powers, the powers, the non-powers (also called primitive words), the
words matching a general pattern, the bordered words, and the unbordered words.Comment: Full version of a paper submitted to LATA 2008. This is a new version
with John Loftus added as a co-author and containing new results on
unbordered word
Finite type invariants and fatgraphs
We define an invariant of pairs M,G, where M is a 3-manifold
obtained by surgery on some framed link in the cylinder , S is a
connected surface with at least one boundary component, and G is a fatgraph
spine of S. In effect, is the composition with the maps of
Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin
computed relative to choices determined by the fatgraph G; this provides a
basic connection between 2d geometry and 3d quantum topology. For each fixed G,
this invariant is shown to be universal for homology cylinders, i.e.,
establishes an isomorphism from an appropriate vector space
of homology cylinders to a certain algebra of Jacobi diagrams. Via
composition for any pair of fatgraph spines
G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the
combinatorial model for the fundamental path groupoid of Teichmuller space, as
a group of automorphisms of this algebra. The space comes equipped
with a geometrically natural product induced by stacking cylinders on top of
one another and furthermore supports related operations which arise by gluing a
homology handlebody to one end of a cylinder or to another homology handlebody.
We compute how interacts with all three operations explicitly in
terms of natural products on Jacobi diagrams and certain diagrammatic
constants. Our main result gives an explicit extension of the LMO invariant of
3-manifolds to the Ptolemy groupoid in terms of these operations, and this
groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the
Morita-Penner cocycle representing the first Johnson homomorphism using a
variant/generalization of .Comment: 39 page
- …