1,720,964 research outputs found
Global Fukaya category I
Get PDFLet denote the Frechet Lie group of Hamiltonian
symplectomorphisms of a monotone symplectic manifold . Let be the -nerve of the Fukaya category , and let denote the component of the ``space of -categories'' .
Using Floer-Fukaya theory for a monotone we construct a natural
up to homotopy classifying map \begin{equation*}
BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{equation*} This
verifies one sense of a conjecture of Teleman on existence of action of on the Fukaya category of . This construction is very
closely related to the theory of the Seidel homomorphism and the quantum
characteristic classes of the author, and this map is intended to be the
deepest expression of their underlying geometric theory. In part II the above
map is shown to be nontrivial by an explicit calculation. In particular, we
arrive at a new non-trivial ``quantum'' invariant of any smooth manifold, which
motives the statement of a kind of ``quantum'' Novikov conjecture.Comment: To appear in IMRN, 61 page
Elliptic curves in lcs manifolds and metric invariants
Get PDFWe study invariants defined by count of charged, elliptic -holomorphic
curves in locally conformally symplectic manifolds. We use this to define
-valued deformation invariants of certain complete Riemann-Finlser
manifolds and their isometries and this is used to find some new phenomena in
Riemann-Finlser geometry. In contact geometry this Gromov-Witten theory is used
to study fixed Reeb strings of strict contactomorphisms. Along the way, we
state an analogue of the Weinstein conjecture in lcs geometry, directly
extending the Weinstein conjecture, and discuss various partial verifications.
A counterexample for a stronger, also natural form of this conjecture is given.Comment: 34 pages. Improved the results. arXiv admin note: text overlap with
arXiv:2102.0582
Hamiltonian elements in algebraic K-theory
Get PDFRecall that topological complex -theory associates to an isomorphism class
of a complex vector bundle over a space an element of the complex
-theory group of . Or from algebraic -theory perspective, one assigns
a homotopy class , where is the ring of
compact operators on the Hilbert space. We show that there is an analogous
story for algebraic -theory of a general commutative ring , replacing
complex vector bundles by certain Hamiltonian fiber bundles. The construction
actually first assigns elements in a certain categorified algebraic -theory,
analogous to To\"en's secondary -theory of . And there is a natural map
from this categorified algebraic -theory to the classical variant.Comment: 11 pages. Comments welcom
A conformal symplectic Weinstein conjecture
Get PDFWe introduce a direct generalization of the Weinstein conjecture to closed,
Lichnerowicz exact, locally conformally symplectic manifolds, (for short \lcs
manifolds). This conjectures existence of certain 2-curves in the manifold,
which we call Reeb 2-curves. The conjecture readily holds for all closed exact
lcs surfaces. In higher dimensions, we give partial verifications of this
conjecture, based on certain extended (
valued) Gromov-Witten, elliptic curve counts in \lcs manifolds. As a basic
application we get some novel results in classical Reeb dynamics. The most
basic such result gives sufficient conditions for a strict contactomorphism to
fix the image of some closed Reeb orbit on a closed contact manifold. Along the
way we give a Gromov-Witten theoretic construction of the classical dynamical
Fuller index (for Reeb vector field), which among other things explains its
rationality.Comment: This is mostly superseded by arXiv:2309.0984
A remark on deformation of Gromov non-squeezing
Get PDFWe prove that in dimension 4 the Gromov non-squeezing phenomenon is
persistent with respect to symplectic perturbations of the symplectic
form on the range. This motivates an intriguing question on further deforming
non-squeezing to general nearby forms. Our methods consist of a certain trap
idea for holomorphic curves, analogous to traps in dynamical systems, and
Hofer-Wysocki-Zehnder polyfold regularization in Gromov-Witten theory,
especially as recently worked out in this present context by the team of
Franziska Beckschulte, Ipsita Datta, Irene Seifert, Anna-Maria Vocke, and
Katrin Wehrheim.Comment: 3 pages. Minor clarifications, comments welcom
Incompleteness theorems via Turing category
Get PDFWe give a reframing of Godel\u27s first and second incompleteness theorems that applies even to some undefinable theories of arithmetic. The usual Hilbert-Bernays provability conditions and the diagonal lemma are replaced by a more direct diagonalization argument, from first principles, based in category theory and in a sense analogous to Cantor\u27s original argument. To this end, we categorify the theory Gödel encodings, which might be of independent interest. In our setup, the Gödel sentence is computable explicitly by construction even for theories (likely extending to ). In an appendix, we study the relationship of our reframed second incompleteness theorem with arguments of Penrose.This also corrects and supersedes 2208.0475
Untwisted Gromov-Witten invariants of Riemann-Finsler manifolds
Get PDFWe define a -valued deformation invariant of certain complete Riemann-Finsler manifolds, in particular of complete Riemannian manifolds with non positive sectional curvature. It is proved that every rational number is the value of this invariant for some compact Riemannian manifold. We use this to find the first and mostly sharp generalizations, to non-compact products and fibrations, of Preissman\u27s theorem on non-existence of negative sectional curvature metrics on compact products. For example, admits a metric of negative sectional curvature, where is a non-compact possibly infinite type surface, if and only if has genus zero. We also give novel estimates on counts of closed geodesics with restrictions on multiplicity. Along the way, we also prove that sky catastrophes of smooth dynamical systems are not geodesible by a certain class of forward complete Riemann-Finsler metrics, in particular by complete Riemannian metrics with non-positive sectional curvature. This partially answers a question of Fuller and gives important examples for our theory here.18 pages. New results, improved the presentation. Title adjuste
Non-computability of human intelligence
Get PDFWe revisit the question (most famously) initiated by Turing: can human intelligence be completely modeled by a Turing machine?
We show that the answer is \emph{no}, assuming a certain weak soundness hypothesis. More specifically we show that at least some meaningful thought processes of the brain cannot be Turing computable. In particular some physical processes are not Turing computable, which is not entirely expected. There are some similarities of our argument with the well known Lucas-Penrose argument, but we work purely on the level of Turing machines, and do not use G\"odel's incompleteness theorem or any direct analogue. Instead we construct directly and use a weak analogue of a G\"odel statement for a certain system which involves our human, this allows us to side-step some (possible) meta-logical issues with their argument
Locally conformally symplectic deformation of Gromov non-squeezing
Get PDFWe prove one deformation theoretic extension of the Gromov non-squeezing
phenomenon to structures, or locally conformally symplectic structures,
which suitably generalize both symplectic and contact structures. We also
conjecture an analogue in geometry of contact non-squeezing of
Eliashberg-Polterovich and discuss other related questions.Comment: Version to appear in Archiv der Mathematik. 10 pages. arXiv admin
note: text overlap with arXiv:2102.0582
Turing analogues of G\"odel statements and computability of intelligence
Get PDFWe show that there is a mathematical obstruction to complete Turing computability of intelligence. This obstruction can be circumvented only if human reasoning is fundamentally unsound. The most compelling original argument for existence of such an obstruction was proposed by Penrose, however G\"odel, Turing and Lucas have also proposed such arguments. We first partially reformulate the argument of Penrose. In this formulation we argue that his argument works up to possibility of construction of a certain G\"odel statement. We then completely re-frame the argument in the language of Turing machines, and by partially defining our subject just enough, we show that a certain analogue of a G\"odel statement, or a G\"odel string as we call it in the language of Turing machines, can be readily constructed directly, without appeal to the G\"odel incompleteness theorem, and thus removing the final objection
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