60 research outputs found
The equations of Rees algebras of ideals of almost-linear type
In this dissertation, we tackle the problem of describing the equations of
the Rees algebra of I for I =(J,y), with J being of linear type. Throughout,
such ideals are referred to as ideals of almost-linear type.
In Theorem A, we give a full description of the equations of Rees algebras of
ideals of the form I = (J,y), with J satisfying an homological vanishing
condition. Theorem A permits us to recover and extend well-known results about
families of ideals of almost-linear type due to W.V. Vasconcelos, S. Huckaba,
N.V. Trung, W. Heinzer and M.-K. Kim, among others.
In Theorem B, we prove that the injectivity of a single component of the
canonical morphism from the symmetric algebra of I to the Rees algebra of I,
propagates downwards, provided I is of almost-linear type. In particular, this
result gives a partial answer to a question posed by A.B. Tchernev.
Packs of examples are introduced in each section, illustrating the scope and
applications of each of the results presented. The author also gives a
collection of computations and examples which motivate ongoing and future
research.Comment: Research report based on PhD Thesis, Universitat Polit\`ecnica de
Catalunya, October 2011. Advisor: Francesc Planas-Vilanov
The general caloron correspondence
We outline in detail the general caloron correspondence for the group of
automorphisms of an arbitrary principal -bundle over a manifold ,
including the case of the gauge group of . These results are used to define
characteristic classes of gauge group bundles. Explicit but complicated
differential form representatives are computed in terms of a connection and
Higgs field.Comment: 25 pages. New section added containing example
Symmetric Instantons and Skyrme Fields
By explicit construction of the ADHM data, we prove the existence of a charge
seven instanton with icosahedral symmetry. By computing the holonomy of this
instanton we obtain a Skyrme field which approximates the minimal energy charge
seven Skyrmion. We also present a one parameter family of tetrahedrally
symmetric instantons whose holonomy gives a family of Skyrme fields which
models a Skyrmion scattering process, where seven well-separated Skyrmions
collide to form the icosahedrally symmetric Skyrmion.Comment: 22 pages plus 1 figure in GIF forma
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
is equal to the bottleneck distance . This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page
A "Periodic Table" for Supersymmetric M-Theory Compactifications
We develop a systematic method for classifying supersymmetric orbifold
compactifications of M-theory. By restricting our attention to abelian
orbifolds with low order, in the special cases where elements do not include
coordinate shifts, we construct a "periodic table" of such compactifications,
organized according to the orbifolding group (order up to 12) and dimension (up
to 7). An intriguing connection between supersymmetric orbifolds and
G2-structures is explored.Comment: 34 pages, late
Obstruction theory on 8-manifolds
This note gives a uniform, self-contained, and fairly direct approach to a
variety of obstruction-theoretic problems on 8-manifolds. We give necessary and
sufficient cohomological critera for the existence of almost complex and almost
quaternionic structures on the tangent bundle and for the reduction of the
structure group to U(3) by the homomorphism U(3) --> O(8) given by the Lie
algebra representation of PU(3).Comment: 19 page
I-Brane Inflow and Anomalous Couplings on D-Branes
We show that the anomalous couplings of -brane gauge and gravitational
fields to Ramond-Ramond tensor potentials can be deduced by a simple anomaly
inflow argument applied to intersecting -branes and use this to determine
the eight-form gravitational coupling.Comment: 8 pages, harvmac, no figure
- …