5,498 research outputs found
On restricting the ambiguity in morphic images of words
For alphabets Delta_1, Delta_2, a morphism g : Delta_1* to Delta_2* is ambiguous with respect to a word u in Delta_1* if there exists a second morphism h : Delta_1* to Delta_2* such that g(u) = h(u) and g not= h. Otherwise g is unambiguous. Hence unambiguous morphisms are those whose structure is fully preserved in their morphic images.
A concept so far considered in the free monoid, the first part of this thesis considers natural extensions of ambiguity of morphisms to free groups. It is shown that, while the most straightforward generalization of ambiguity to a free monoid results in a trivial situation, that all morphisms are (always) ambiguous, there exist meaningful extensions of (un)ambiguity which are non-trivial - most notably the concepts of (un)ambiguity up to inner automorphism and up to automorphism.
A characterization is given of words in a free group for which there exists an injective morphism which is unambiguous up to inner automorphism in terms of fixed points of morphisms, replicating an existing result for words in the free monoid. A conjecture is presented, which if correct, is sufficient to show an equivalent characterization for unambiguity up to automorphism. A rather counterintuitive statement is also established, that for some words, the only unambiguous (up to automorphism) morphisms are non-injective (or even periodic).
The second part of the thesis addresses words for which all non-periodic morphisms are unambiguous. In the free monoid, these take the form of periodicity forcing words. It is shown using morphisms that there exist ratio-primitive periodicity forcing words over arbitrary alphabets, and furthermore that it is possible to establish large and varied classes in this way. It is observed that the set of periodicity forcing words is spanned by chains of words, where each word is a morphic image of its predecessor. It is shown that the chains terminate in exactly one direction, meaning not all periodicity forcing words may be reached as the (non-trivial) morphic image of another. Such words are called prime periodicity forcing words, and some alternative methods for finding them are given.
The free-group equivalent to periodicity forcing words - a special class of C-test words - is also considered, as well as the ambiguity of terminal-preserving morphisms with respect to words containing terminal symbols, or constants. Moreover, some applications to pattern languages and group pattern languages are discussed
A Diagrammatic Temperley-Lieb Categorification
The monoidal category of Soergel bimodules categorifies the Hecke algebra of
a finite Weyl group. In the case of the symmetric group, morphisms in this
category can be drawn as graphs in the plane. We define a quotient category,
also given in terms of planar graphs, which categorifies the Temperley-Lieb
algebra. Certain ideals appearing in this quotient are related both to the
1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We
demonstrate how further subquotients of this category will categorify the cell
modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio
Twisty itsy bitsy topological field theory
We extend the topological field theory (``itsy bitsy topological field
theory"') of our previous work from mod-2 to twisted coefficients. This
topological field theory is derived from sutured Floer homology but described
purely in terms of surfaces with signed points on their boundary (occupied
surfaces) and curves on those surfaces respecting signs (sutures). It has
information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'')
aspects. In the process we extend some results of sutured Floer homology,
consider associated ribbon graph structures, and construct explicit admissible
Heegaard decompositions.Comment: 52 pages, 26 figure
A note on toric Deligne-Mumford stacks
We give a new description of the data needed to specify a morphism from a
scheme to a toric Deligne-Mumford stack. The description is given in terms of a
collection of line bundles and sections which satisfy certain conditions. As
applications, we characterize any toric Deligne-Mumford stack as a product of
roots of line bundles over the rigidified stack, describe the torus action,
describe morphisms between toric Deligne-Mumford stacks with complete coarse
moduli spaces in terms of homogeneous polynomials, and compare two different
definitions of toric stacks.Comment: Version2: typos corrected; we add sections 6 and 7 where we relate
the present work with the paper of Iwanari: arXiv:math/0610548 and that of
Fantechi-Mann-Nironi: arXiv:0708.1254. Accepted for publication in the Tohoku
Math.
Configuration of points and strings
Let be a smooth projective variety of dimension . It is shown
that a finite configuration of points on subject to certain geometric
conditions possesses rich inner structure. On the mathematical level this inner
structure is a variation of Hodge-like structure. As a consequence one can
attach to such point configurations: (i) Lie algebras and their representations
(ii) Fano toric variety whose hyperplane sections are Calabi-Yau varieties.
These features lead to a picture which is very suggestive of quantum gravity
according to string theory.Comment: 32 pages, 1 figure this version will appear in Journal of Geometry
and Physic
Extended quantum field theory, index theory and the parity anomaly
We use techniques from functorial quantum field theory to provide a geometric
description of the parity anomaly in fermionic systems coupled to background
gauge and gravitational fields on odd-dimensional spacetimes. We give an
explicit construction of a geometric cobordism bicategory which incorporates
general background fields in a stack, and together with the theory of symmetric
monoidal bicategories we use it to provide the concrete forms of invertible
extended quantum field theories which capture anomalies in both the path
integral and Hamiltonian frameworks. Specialising this situation by using the
extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners
due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity
anomaly. We compute explicitly the 2-cocycle of the projective representation
of the gauge symmetry on the quantum state space, which is defined in a
parity-symmetric way by suitably augmenting the standard chiral fermionic Fock
spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that
naturally appear in the index theorem. We describe the significance of our
constructions for the bulk-boundary correspondence in a large class of
time-reversal invariant gauge-gravity symmetry-protected topological phases of
quantum matter with gapless charged boundary fermions, including the standard
topological insulator in 3+1 dimensions.Comment: 63 pages, 3 figures; v2: clarifying comments and references added;
Final version to be published in Communications in Mathematical Physic
- …