57 research outputs found
The vanishing cohomology of non-isolated hypersurface singularities
We employ the perverse vanishing cycles to show that each reduced cohomology
group of the Milnor fiber, except the top two, can be computed from the
restriction of the vanishing cycle complex to only singular strata with a
certain lower bound in dimension. Guided by geometric results, we alternately
use the nearby and vanishing cycle functors to derive information about the
Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead
to the description of the reduced cohomology groups, except the top two, in
terms of the vanishing cohomology of the nearby section. We use it to compute
explicitly the lowest (possibly nontrivial) vanishing cohomology group of the
Milnor fiber.Comment: comments are very welcome! v2: mild changes and clarifications of the
text; v3: more clarifications to the text are added, an obvious typo in the
statement of Theorem 5.5(a) is corrected, the computations in Example 6.4 are
revised; v4: the exposition in Section 5 is simplified and Example 6.5 is
adde
Describing general cosmological singularities in Iwasawa variables
Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description
of the asymptotic behavior of a generic solution of Einstein equations near a
spacelike singularity could be drastically simplified by considering that the
time derivatives of the metric asymptotically dominate (except at a sequence of
instants, in the `chaotic case') over the spatial derivatives. We present a
precise formulation of the BKL conjecture (in the chaotic case) that consists
of basically three elements: (i) we parametrize the spatial metric by
means of \it{Iwasawa variables} ); (ii) we define, at
each spatial point, a (chaotic) \it{asymptotic evolution system} made of
ordinary differential equations for the Iwasawa variables; and (iii) we
characterize the exact Einstein solutions whose asymptotic
behavior is described by a solution of the
previous evolution system by means of a `\it{generalized Fuchsian system}' for
the differenced variables , , and by requiring that and tend to zero on the singularity. We also show that, in spite of the
apparently chaotic infinite succession of `Kasner epochs' near the singularity,
there exists a well-defined \it{asymptotic geometrical structure} on the
singularity : it is described by a \it{partially framed flag}. Our treatment
encompasses Einstein-matter systems (comprising scalar and p-forms), and also
shows how the use of Iwasawa variables can simplify the usual (`asymptotically
velocity term dominated') description of non-chaotic systems.Comment: 50 pages, 4 figure
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics
Higher Segal spaces I
This is the first paper in a series on new higher categorical structures
called higher Segal spaces. For every d > 0, we introduce the notion of a
d-Segal space which is a simplicial space satisfying locality conditions
related to triangulations of cyclic polytopes of dimension d. In the case d=1,
we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal
spaces. The starting point of the theory is the observation that Hall algebras,
as previously studied, are only the shadow of a much richer structure governed
by a system of higher coherences captured in the datum of a 2-Segal space. This
2-Segal space is given by Waldhausen's S-construction, a simplicial space
familiar in algebraic K-theory. Other examples of 2-Segal spaces arise
naturally in classical topics such as Hecke algebras, cyclic bar constructions,
configuration spaces of flags, solutions of the pentagon equation, and mapping
class groups.Comment: 221 page
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Non-abelian symmetries in tensor networks: a quantum symmetry space approach
A general framework for non-abelian symmetries is presented for
matrix-product and tensor-network states in the presence of orthonormal local
as well as effective basis sets. The two crucial ingredients, the
Clebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckart
theorem for operators, are accounted for in a natural, well-organized, and
computationally straightforward way. The unifying tensor-representation for
quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with
standard renormalization group algorithms such as the numerical renormalization
group (NRG), the density matrix renormalization group (DMRG), or also more
general tensor networks such as the multi-scale entanglement renormalization
ansatz (MERA). In this paper, the focus is on the application of the
non-abelian framework within the NRG. A detailed analysis is given for a fully
screened spin-3/2 three-channel Anderson impurity model in the presence of
conservation of total spin, particle-hole symmetry, and SU(3) channel symmetry.
The same system is analyzed using several alternative symmetry scenarios. This
includes the more traditional symmetry setting SU(2)^4, the larger symmetry
SU(2)*U(1)*SU(3), and their much larger enveloping symplectic symmetry
SU(2)*Sp(6). These are compared in detail, including their respective dramatic
gain in numerical efficiency. In the appendix, finally, an extensive
introduction to non-abelian symmetries is given for practical applications,
together with simple self-contained numerical procedures to obtain
Clebsch-Gordan coefficients and irreducible operators sets. The resulting
QSpace tensors can deal with any set of abelian symmetries together with
arbitrary non-abelian symmetries with compact, i.e. finite-dimensional,
semi-simple Lie algebras.Comment: 25 pages (main text) + 32 pages (appendix - more detailed pedagogical
introduction to the non-abelian framework); accepted by Annals of Physics and
available online since Jul 2012, with final publication in the Dec 2012 issu
Algorithms for Mappings and Symmetries of Differential Equations
Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points
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