851 research outputs found
Asymptotics of the colored Jones function of a knot
To a knot in 3-space, one can associate a sequence of Laurent polynomials,
whose th term is the th colored Jones polynomial. The paper is concerned
with the asymptotic behavior of the value of the th colored Jones polynomial
at e^{\a/n}, when \a is a fixed complex number and tends to infinity.
We analyze this asymptotic behavior to all orders in when \a is a
sufficiently small complex number. In addition, we give upper bounds for the
coefficients and degree of the th colored Jones polynomial, with
applications to upper bounds in the Generalized Volume Conjecture. Work of
Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically
optimal. Moreover, we give results for the Generalized Volume Conjecture when
\a is near . Our proofs use crucially the cyclotomic expansion of
the colored Jones function, due to Habiro.Comment: 31 pages, 13 figure
Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method
This dissertation investigates the geometric combinatorics of convex
polytopes and connections to the behavior of the simplex method for linear
programming. We focus our attention on transportation polytopes, which are sets
of all tables of non-negative real numbers satisfying certain summation
conditions. Transportation problems are, in many ways, the simplest kind of
linear programs and thus have a rich combinatorial structure. First, we give
new results on the diameters of certain classes of transportation polytopes and
their relation to the Hirsch Conjecture, which asserts that the diameter of
every -dimensional convex polytope with facets is bounded above by
. In particular, we prove a new quadratic upper bound on the diameter of
-way axial transportation polytopes defined by -marginals. We also show
that the Hirsch Conjecture holds for classical transportation
polytopes, but that there are infinitely-many Hirsch-sharp classical
transportation polytopes. Second, we present new results on subpolytopes of
transportation polytopes. We investigate, for example, a non-regular
triangulation of a subpolytope of the fourth Birkhoff polytope . This
implies the existence of non-regular triangulations of all Birkhoff polytopes
for . We also study certain classes of network flow polytopes
and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California,
Davis. 183 pages, 49 figure
Monte Carlo simulations of 4d simplicial quantum gravity
Dynamical triangulations of four-dimensional Euclidean quantum gravity give
rise to an interesting, numerically accessible model of quantum gravity. We
give a simple introduction to the model and discuss two particularly important
issues. One is that contrary to recent claims there is strong analytical and
numerical evidence for the existence of an exponential bound that makes the
partition function well-defined. The other is that there may be an ambiguity in
the choice of the measure of the discrete model which could even lead to the
existence of different universality classes.Comment: 16 pages, LaTeX, epsf, 4 uuencoded figures; contribution to the JMP
special issue on "Quantum Geometry and Diffeomorphism-Invariant Quantum Field
Theory
Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications
Lattice polytopes arise naturally in many different branches of pure and applied mathematics such as number theory, commutative algebra, combinatorics, toric geometry, optimization, and mirror symmetry. The miniworkshop on “Lattice polytopes: methods, advances, applications” focused on two current hot topics: the classification of lattice polytopes with few lattice points and unimodality questions for Ehrhart polynomials. The workshop consisted of morning talks on recent breakthroughs and new methods, and afternoon discussion groups where participants from a variety of different backgrounds explored further applications, identified open questions and future research directions, discussed specific examples and conjectures, and collaboratively tackled open research problems
Polyhedra in loop quantum gravity
Interwiners are the building blocks of spin-network states. The space of
intertwiners is the quantization of a classical symplectic manifold introduced
by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to
interpret generic configurations in this space as bounded convex polyhedra in
Euclidean space: a polyhedron is uniquely described by the areas and normals to
its faces. We provide a reconstruction of the geometry of the polyhedron: we
give formulas for the edge lengths, the volume and the adjacency of its faces.
At the quantum level, this correspondence allows us to identify an intertwiner
with the state of a quantum polyhedron, thus generalizing the notion of quantum
tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent
intertwiners result to be peaked on the classical geometry of polyhedra. We
discuss the relevance of this result for loop quantum gravity. In particular,
coherent spin-network states with nodes of arbitrary valence represent a
collection of semiclassical polyhedra. Furthermore, we introduce an operator
that measures the volume of a quantum polyhedron and examine its relation with
the standard volume operator of loop quantum gravity. We also comment on the
semiclassical limit of spinfoams with non-simplicial graphs.Comment: 32 pages, many figures. v2 minor correction
Classification of empty 4-simplices and other lattice polytopes
RESUMEN: Un d-politopo es la envolvente convexa de un conjunto finito de puntos en R^d. En particular, si un d-politopo está generado por exactamente d + 1 puntos se dice que es un símplice o un d-símplice. Además, si tomamos los puntos con coordenadas enteras, se dice que el politopo es reticular.
A lo largo de esta tesis doctoral se estudian los politopos reticulares y, más concretamente, se estudian dos tipos de estos que son los politopos reticulares vacíos (cuyos únicos puntos reticulares son los vértices) y los politopos reticulares huecos, politopos reticulares que no poseen puntos reticulares en su interior relativo, es decir, todos sus puntos reticulares se encuentran en la frontera. Los politopos huecos, también vacíos, aparecen como el ejemplo más sencillo de politopos reticulares al no tener puntos enteros en el interior de su envolvente convexa.
El principal resultado de la tesis doctoral es la clasificación de símplices vacíos en dimensión 4. Mientras los casos en dimensión 1 y 2 son triviales y el caso de dimensión 3 estaba concluido desde 1964 con el trabajo de White [Whi64], con este trabajo se completa esta clasificación en dimensión 4. Artículos como el de Mori, Morrison y Morrison [MMM88] en 1988 consiguen describir algunas familias de 4-símplices vacíos de volumen primo en términos de quíntuplas. Otros trabajos como el de Haase y Ziegler [HZ00] en el 2000, obtienen resultados parciales de esta clasificación. En particular, en ese trabajo se conjeturó una lista completa de 4-símplices vacíos con anchura mayor que dos, la cual se prueba completa en esta tesis.
Empleando técnicas de geometría convexa, geometría de números y resultados previos sobre la relación entre la anchura de un politopo y su volumen, somos capaces de establecer unas cotas superiores para los 4-símplices vacíos que deseamos clasificar. Con estas cotas para el volumen de los símplices y una gran cantidad de computación de estos politopos reticulares en dimensión 4 somos capaces de completar la clasificación, explicando el método general utilizado para describir las familias de símplices vacíos que aparecen en la clasificación.ABSTRACT: A d-polytope is the convex hull of a finite set of points in R^d. In particular, if a d-polytope is generated by exactly d + 1 points, it is said to be a simplex or a d-simplex. In addition, if we take the points with integer coordinates, the polytope is a lattice polytope.
Throughout this thesis, lattice polytopes are studied and, more specifically, two types of these, which are empty lattice polytopes (whose only integer points are its vertices) and hollow polytopes, lattice polytopes that do not have integer points in their interior, that is, all their integer points are in their facets. Hollow polytopes, also empty, appear as the simplest example of lattice polytopes because they have no integer points inside their convex hull.
The main result of the thesis is the classification of empty simplices in dimension 4. While cases in dimension 1 and 2 are trivial and the case of dimension 3 has been completed since 1964 with the work of White [Whi64], this work completes this classification in dimension 4. Papers such as Mori, Morrison and Morrison [MMM88] in 1988 manage to describe some families of empty 4-simplices of prime volume in terms of quintuples. Other works, such as Haase and Ziegler [HZ00] in 2000, obtain partial results tor this classification. In particular, this work conjecture a complete list of empty 4-simplices of width greater than two, which is verified in this thesis.
With convex geometry tools, geometry of numbers and previous results that rely on the relationship between the width of a polytope and its volume, we are able to to set upper bounds for the volume of hollow 4-simpolices, that we want to classify. With these upper bounds for the volume of the simplices and a lot of computation of these lattice polytopes in dimension 4 we are able to complete the classification, explaining the general method used to describe the families of empty simplices that appear in the classification.This thesis has been developed under the following scholarships and project grants:
MTM2014-54207-P, MTM2017-83750-P and BES-2015-073128 of the Spanish Ministry
of Economy and Competitiveness
Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants
ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation
Resolving numerically Vlasov-Poisson equations for initially cold systems can
be reduced to following the evolution of a three-dimensional sheet evolving in
six-dimensional phase-space. We describe a public parallel numerical algorithm
consisting in representing the phase-space sheet with a conforming,
self-adaptive simplicial tessellation of which the vertices follow the
Lagrangian equations of motion. The algorithm is implemented both in six- and
four-dimensional phase-space. Refinement of the tessellation mesh is performed
using the bisection method and a local representation of the phase-space sheet
at second order relying on additional tracers created when needed at runtime.
In order to preserve in the best way the Hamiltonian nature of the system,
refinement is anisotropic and constrained by measurements of local Poincar\'e
invariants. Resolution of Poisson equation is performed using the fast Fourier
method on a regular rectangular grid, similarly to particle in cells codes. To
compute the density projected onto this grid, the intersection of the
tessellation and the grid is calculated using the method of Franklin and
Kankanhalli (1993) generalised to linear order. As preliminary tests of the
code, we study in four dimensional phase-space the evolution of an initially
small patch in a chaotic potential and the cosmological collapse of a
fluctuation composed of two sinusoidal waves. We also perform a "warm" dark
matter simulation in six-dimensional phase-space that we use to check the
parallel scaling of the code.Comment: Code and illustration movies available at:
http://www.vlasix.org/index.php?n=Main.ColDICE - Article submitted to Journal
of Computational Physic
Lectures on 0/1-polytopes
These lectures on the combinatorics and geometry of 0/1-polytopes are meant
as an \emph{introduction} and \emph{invitation}. Rather than heading for an
extensive survey on 0/1-polytopes I present some interesting aspects of these
objects; all of them are related to some quite recent work and progress.
0/1-polytopes have a very simple definition and explicit descriptions; we can
enumerate and analyze small examples explicitly in the computer (e.g. using
{\tt polymake}). However, any intuition that is derived from the analysis of
examples in ``low dimensions'' will miss the true complexity of 0/1-polytopes.
Thus, in the following we will study several aspects of the complexity of
higher-dimensional 0/1-polytopes: the doubly-exponential number of
combinatorial types, the number of facets which can be huge, and the
coefficients of defining inequalities which sometimes turn out to be extremely
large. Some of the effects and results will be backed by proofs in the course
of these lectures; we will also be able to verify some of them on explicit
examples, which are accessible as a {\tt polymake} database.Comment: 45 pages, many figures; to appear in "Polytopes - Combinatorics and
Computation" (G. Kalai and G.M. Ziegler, eds.), DMV Seminars Series,
Birkh"auser Base
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