arXiv subjects

Yasha Savelyev

Publications and source records attributed to Yasha Savelyev.

At least 19 recordsLinked to original sources

arXiv2026

Geodesic string counting invariants and arithmetic of multiplicities

We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let $g$ be a generic Finsler metric on $T ^{2}$, sufficiently $C ^{\infty }$-close to a flat metric, and fix a prime $p$ and a nontrivial free homotopy class $\beta $. If there is a class $\beta$ $g$-geodesic string with multiplicity divisible by $p$, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack $[LX/S^1]$.

math.DG↗

arXiv2025

On moduli spaces of uniformly negatively curved metrics

Let $\Neg(X)$ denote the space of complete Riemannian metrics with uniformly negative curvature on a surface $X$, equipped with the intrinsic uniform $C^2$ topology. Let $\mathcal M(X)=\Neg(X)/\Diff(X)$ be the corresponding moduli space, with the quotient topology. We construct elementary locally constant functionals on $\mathcal M(X)$, with values in finite symmetric products of $\{0,1\}$, based on geodesic string counts. As an upshot we show that $\mathcal M(\mathbb{R}\times S^1)$ is disconnected. This is perhaps surprising: the two metrics we separate are joined by an explicit path of metrics with constant curvature $-1$. The point is that this path is only continuous in the weak Whitney topology. More generally, if $X$ is a finite-type surface of hyperbolic type with $n$ punctures, then the pure moduli space has at least $2^n$ connected components, while $\mathcal M(X)$ has at least $n+1$ connected components.

math.DG↗

arXiv2025

Global Fukaya category II: applications

To paraphrase, part I constructs a bundle of $A _{\infty}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal application is differential geometric, and the other is about algebraic $K$-theory of the integers and the rationals. We find new curvature constraint phenomena for smooth and singular $\mathcal{G}$-connections on principal $\mathcal{G}$-bundles over $S ^{4}$, where $\mathcal{G}$ is $\operatorname {PU} (2)$ or $\operatorname {Ham} (S ^{2} )$. Even for the classical group $\operatorname {PU} (2)$ these phenomena are inaccessible to known techniques like the Yang-Mills theory. The above mentioned computation is the geometric component used to show that the categorified algebraic $K$-theory of the integers and the rationals, defined in ~\cite{cite_SavelyevAlgKtheory} following To\"en, admits a $\mathbb{Z} $ injection in degree $4$. This gives a path from Floer theory to number theory.

math.SG↗

arXiv2025

Strict contactomorphisms are scarce

The notion of non-projectible contact forms on a given compact manifold $M$ is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms, both in the case with a fixed contact structure and in the case without it. In this paper, we prove that for any non-projectible contact form $\lambda$ the set, denoted by $\text{\rm Cont}^{\text{\rm st}}(M,\lambda)$, consisting of strict contactomorphisms of $\lambda$ is a a countable disjoint union of real lines $\mathbb R$, one for each connected component.

math.SG↗

arXiv2025

Quantum Maslov classes

We give a construction of ``quantum Maslov characteristic classes'', generalizing to higher dimensional cycles the Hu-Lalonde-Seidel morphism. We also state a conjecture extending this to an $A _{\infty}$ functor from the exact path category of the space of monotone Lagrangian branes to the Fukaya category. Quantum Maslov classes are used here for the study of Hofer geometry of Lagrangian equators in $S ^{2}$, giving a rigidity phenomenon for the Hofer metric 2-systole, which stands in contrast to the flexibility phenomenon of the closely related Hofer metric girth studied by Rauch ~\cite{cite_Itamar}, in the same context of Lagrangian equators of $S ^{2}$. More applications appear in ~\cite{cite_SavelyevGlobalFukayacategoryII}.

math.SG↗

arXiv2024

Incompleteness theorems via Turing category

We give a reframing of Godel's 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's original argument. To this end, we categorify the theory G\"odel encodings, which might be of independent interest. In our setup, the G\"odel sentence is computable explicitly by construction even for $\Sigma ^{0} _{2}$ theories (likely extending to $\Sigma ^{0} _{n}$). In an appendix, we study the relationship of our reframed second incompleteness theorem with arguments of Penrose.

math.LO↗

arXiv2024

Hamiltonian elements in algebraic K-theory

A Hamiltonian bundle $M \hookrightarrow P \to X$ (with monotone compact fibers) induces via Floer theory a type of ``bundle of $A _{\infty}$ categories'' over $X$, with fiber given by the Fukaya category of $M$. Morita theory of $A _{\infty} $ categories, the above picture for $X=S ^{m}$, and geometric representation theory yield the following: if $G$ is a compact Lie group and $R$ is a commutative ring then there is a natural group homomorphism $\pi _{m} (BG) \to K ^{Cat}_{m}(R) $, where $K ^{Cat} _{m} (R)$ are a type of categorified algebraic $K$-theory groups of $R$, analogous to To\"en's secondary $K$-theory. We also construct underlying maps of this type to classical algebraic $K$-theory of $R$. This framework gives a geometry-powered proof that $K ^{Cat} _{2} (\mathbb{Z} )$ is infinitely generated (with the details to appear in a future work). This is in contrast to Quillen's finite generation result for standard algebraic $K$-theory of $\mathbb{Z} $. Taking the Langlands dual of $G$, we explore a conjectural relationship between the images of the corresponding homomorphisms above.

math.AT↗

arXiv2023

Elliptic curves in lcs manifolds and metric invariants

We study invariants defined by count of charged, elliptic $J$-holomorphic curves in locally conformally symplectic manifolds. We use this to define $\mathbb{Q} $-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.

math.SG↗

arXiv2023

Untwisted Gromov-Witten invariants of Riemann-Finsler manifolds

We define a $\mathbb{Q}$-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's theorem on non-existence of negative sectional curvature metrics on compact products. For example, $\Sigma \times T ^{n}$ admits a metric of negative sectional curvature, where $\Sigma$ is a non-compact possibly infinite type surface, if and only if $\Sigma$ 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.

math.DG↗

arXiv2023

A remark on deformation of Gromov non-squeezing

Let $R,r$ be as in the classical Gromov non-squeezing theorem, and let $\epsilon = (\pi R ^{2} - \pi r ^{2})/ \pi r ^{2} $. We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range $C ^{0}$ (w.r.t. the standard metric) $\epsilon $-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when $R < \sqrt 2 r$. Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.

math.SG↗

arXiv2022

Locally conformally symplectic deformation of Gromov non-squeezing

We prove one deformation theoretic extension of the Gromov non-squeezing phenomenon to $lcs$ structures, or locally conformally symplectic structures, which suitably generalize both symplectic and contact structures. We also conjecture an analogue in $lcs$ geometry of contact non-squeezing of Eliashberg-Polterovich and discuss other related questions.

math.SG↗

arXiv2022

Incompleteness for stably computable formal systems

We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer solutions, and in particular such sets are generally not computably enumerable. And so this gives the first extension of the second incompleteness theorem to non classically computable formal systems. Let's motivate this with a somewhat physical application. Let $\mathcal{H} $ be the suitable infinite time limit (stabilization in the sense of the paper) of the mathematical output of humanity, specializing to first order sentences in the language of arithmetic (for simplicity), and understood as a formal system. Suppose that all the relevant physical processes in the formation of $\mathcal{H} $ are Turing computable. Then as defined $\mathcal{H} $ may \emph{not} be computably enumerable, but it is stably computably enumerable. Thus, the classical G\"odel disjunction applied to $\mathcal{H} $ is meaningless, but applying our incompleteness theorems to $\mathcal{H} $ we then get a sharper version of G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H} $ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in particular is not sound) or $\mathcal{H} $ cannot prove a certain true statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $ is 2-consistent).

math.LO↗

arXiv2021

Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups

We give the construction of the universal, natural up to homotopy Chern-Weil differential graded algebra homomorphism: $$cw: \mathcal{I} (G) \to \Omega ^{\bullet } (BG, \mathbb{R})$$ for infinite dimensional Milnor regular Lie groups $G$, where $\Omega ^{\bullet}(BG, \mathbb{R})$ is a certain de Rham algebra of $BG$ (Milnor $BG$ up to a natural weak homotopy equivalence) and where $\mathcal{I} (G)$ is the algebra of continuous, $Ad _{G}$ invariant, symmetric multilinear functionals on the Lie algebra. In particular, this applies to the group of compactly generated Hamiltonian symplectomorphisms, using which we verify a conjecture of Reznikov. For the construction of $cw$ we introduce a basic geometric-categorical notion of a smooth simplicial set. Loosely, this is to Chen spaces as simplicial sets are to spaces. We then give a new construction of the classifying space of $G$ as a smooth Kan complex, with the geometric realization weakly equivalent to the Milnor $BG$.

math.AT↗

arXiv2021

Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds

For each contact diffeomorphism $\phi: (Q,\xi) \to (Q,\xi)$ of $(Q,\xi)$, we equip its mapping torus $M_\phi$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{\rm lcs}$ mapping torus} of contact diffeomorphism $\phi$. In the present paper, we consider the product $Q \times S^1= M_{id}$ (corresponding to $\phi = id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $$ \overline{\partial}^\pi w = 0, \quad w^*\lambda\circ j = f^*d\theta $$ for the map $u = (w,f): \dot \Sigma \to Q \times S^1$ for the $\lambda$-compatible almost complex structure $J$ and a punctured Riemann surface $(\dot \Sigma, j)$. In particular, $w$ is a \emph{contact instanton} in the sense of [OW2, OW3]. We develop a scheme of treating the non-vanishing charge by introducing the notion of \emph{charge class} in $H^1(\dot \Sigma,\mathbb Z)$ and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of compactification of the moduli space on the $\mathfrak{lcs}$-fication of $(Q,\lambda)$ (more generally on arbitrary locally conformal symplectic manifolds).

math.SG↗

arXiv2021

Gromov-Witten theory of a locally conformally symplectic manifold

We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holomorphic sky catastrophes, an analogue for pseudo-holomorphic curves of sky catastrophes in dynamical systems originally discovered by Fuller. We are able to rule these out in some situations, particularly for certain $\lcs$ 4-folds, and as one application we show that in dimension 4 the classical Gromov non-squeezing theorem has certain $C ^{0} $ rigidity or persistence with respect to $\lcs$ deformations, this is one version of $\lcs$ non-squeezing a first result of its kind. In a different direction we study Gromov-Witten theory of the $\lcsm$ $C \times S ^{1} $ induced by a contact manifold $(C, λ)$, and show that the Gromov-Witten invariant (as defined here) counting certain elliptic curves in $C \times S ^{1} $ is identified with the classical Fuller index of the Reeb vector field $R ^λ $. This has some non-classical applications, and based on the story we develop, we give a kind of `holomorphic Seifert/Weinstein conjecture' which is a direct extension for some types of $\lcsm$'s of the classical Seifert/Weinstein conjecture. This is proved for $\lcs$ structures $C ^{\infty} $ nearby to the Hopf $\lcs$ structure on $S ^{2k+1} \times S ^{1} $.

math.SG↗

arXiv2021

A conformal symplectic Weinstein conjecture

We 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 ($\mathbb{Q} ^{} \sqcup \{\pm \infty\}$ 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.

math.SG↗

arXiv2020

Non-computability of human intelligence

We 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 Gödel's incompleteness theorem or any direct analogue. Instead we construct directly and use a weak analogue of a Gödel statement for a certain system which involves our human, this allows us to side-step some (possible) meta-logical issues with their argument.

cs.AI↗

arXiv2020

Incompleteness for stably consistent formal systems

We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably $1,2$-consistent formal systems. Our argument in particular re-proves the original incompleteness theorems from first principles, using Turing machine language to (computably) construct our "G\"odel sentence" directly, in particular we do not use the diagonal lemma, nor any meta-logic, with the proof naturally formalizable in set theory. In practice such a stably consistent formal system could be meant to represent the mathematical output of humanity evolving in time, so that the above gives a formalization of a famous disjunction of G\"odel, obstructing computability of intelligence.

cs.LO↗