8,371 research outputs found
Polyfolds: A First and Second Look
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding
commonalities in the analytic framework for a variety of geometric elliptic
PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to
systematically address the common difficulties of compactification and
transversality with a new notion of smoothness on Banach spaces, new local
models for differential geometry, and a nonlinear Fredholm theory in the new
context. We shine meta-mathematical light on the bigger picture and core ideas
of this theory. In addition, we compiled and condensed the core definitions and
theorems of polyfold theory into a streamlined exposition, and outline their
application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version,
to appear in the EMS Surv. Math. Sc
A type theory for synthetic -categories
We propose foundations for a synthetic theory of -categories
within homotopy type theory. We axiomatize a directed interval type, then
define higher simplices from it and use them to probe the internal categorical
structures of arbitrary types. We define Segal types, in which binary
composites exist uniquely up to homotopy; this automatically ensures
composition is coherently associative and unital at all dimensions. We define
Rezk types, in which the categorical isomorphisms are additionally equivalent
to the type-theoretic identities - a "local univalence" condition. And we
define covariant fibrations, which are type families varying functorially over
a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a
directed form of the usual elimination rule for identity types. We conclude by
studying homotopically correct adjunctions between Segal types, and showing
that for a functor between Rezk types to have an adjoint is a mere proposition.
To make the bookkeeping in such proofs manageable, we use a three-layered
type theory with shapes, whose contexts are extended by polytopes within
directed cubes, which can be abstracted over using "extension types" that
generalize the path-types of cubical type theory. In an appendix, we describe
the motivating semantics in the Reedy model structure on bisimplicial sets, in
which our Segal and Rezk types correspond to Segal spaces and complete Segal
spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the
Mathematics Research Community on Homotopy Type Theory; v3 incorporates many
expository improvements suggested by the referee; v4 is the final journal
version to appear in Higher Structures with a more precise syntax for our
type theory with shape
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
Cohomology at infinity and the well-rounded retract for general Linear Groups
Let be a reductive algebraic group defined over \Q, and let
be an arithmetic subgroup of \bold G(\Q). Let be the symmetric
space for , and assume is contractible. Then the cohomology
(mod torsion) of the space is the same as the cohomology of
. In turn, will have the same cohomology as , if
is a ``spine'' in . This means that (if it exists) is a deformation
retract of by a -equivariant deformation retraction, that
is compact, and that equals the virtual cohomological
dimension (vcd) of . Then can be given the structure of a cell
complex on which acts cellularly, and the cohomology of can
be found combinatorially
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