arXiv subjects

Yasha Savelyev

Publications and source records attributed to Yasha Savelyev.

36 records · Page 2Linked to original sources

arXiv2022

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

For each contact diffeomorphism $ϕ: (Q,ξ) \to (Q,ξ)$ of $(Q,ξ)$, we equip its mapping torus $M_ϕ$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{\rm lcs}$ mapping torus} of contact diffeomorphism $ϕ$. In the present paper, we consider the product $Q \times S^1= M_{id}$ (corresponding to $ϕ= id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $$ \overline{\partial}^πw = 0, \quad w^*λ\circ j = f^*dθ$$ for the map $u = (w,f): \dot Σ\to Q \times S^1$ for the $λ$-compatible almost complex structure $J$ and a punctured Riemann surface $(\dot Σ, j)$. 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 $H^1(\dot Σ,\mathbb Z)$ 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 $(Q,λ)$ (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↗

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↗

arXiv2019

Mean curvature versus diameter and energy quantization

We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed curves, of all genus, in a compact locally conformally symplectic manifold.

math.DG↗

arXiv2019

Extended Fuller index, sky catastrophes and the Seifert conjecture

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields $X$ on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at $X$ has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on $S ^{2k+1} $. A sky catastrophe, is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller's foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

math.DS↗

arXiv2019

K-theoretic invariants of Hamiltonian fibrations

We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.

math.SG↗

arXiv2016

Floer theory and topology of $Diff (S^2)$

We say that a fixed point of a diffeomorphism is non-degenerate if 1 is not an eigenvalue of the linearization at the fixed point. We use pseudo-holomorphic curves techniques to prove the following: the inclusion map $$i: \text{Diff} ^{1} (S ^{2} ) \to \text{Diff} (S^2)$$ vanishes on all homotopy groups, where $\text{Diff} ^{1} (S^{2} ) \subset \text {Diff} (S^{2} )$ denotes the space of orientation preserving diffeomorphisms of $S ^{2} $ with a prescribed non-degenerate fixed point. This complements the classical results of Smale and Eels and Earl.

math.SG↗

arXiv2016

On the Hofer geometry injectivity radius conjecture

We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev \cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of $Ham (S^2)$ and $Ham(Σ, ω)$, for $Σ$ a closed positive genus surface. In particular we show that any loop in $Ham (S^2)$, respectively $Ham(Σ, ω)$ with $L ^{+}$ Hofer length less than $area(S ^{2} )/2$, respectively any $L ^{+} $ length is contractible through ($L ^{+} $) Hofer shorter loops, in the $C ^{\infty} $ topology. We also prove some stronger variants of this statement on the loop space level. One dynamical type corollary is that there are no smooth, positive Morse index (Ustilovsky) geodesics, in $Ham (S^2)$, respectively in $Ham(Σ, ω)$ with $L ^{+} $ Hofer length less than $area (S ^{2} )/2$, respectively any length. The above condition on the geodesics can be expanded as an explicit and elementary dynamical condition on the associated Hamiltonian flow. We also give some speculations on connections of this later result with curvature properties of the Hamiltonian diffeomorphism group of surfaces.

math.SG↗

arXiv2015

Yang Mills theory and jumping curves

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over $ S ^{2} $ to show that any non-trivial, smooth Hermitian vector bundle $E $ over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense, (when $c_1 (E)$ is zero). We also use this to give a new proof a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse-Bott chain complex for the Yang-Mills functional over $S ^{2} $.

math.DG↗

arXiv2014

Floer-Fukaya theory and topological elliptic objects

Inspired by Segal-Stolz-Teichner project for geometric construction of elliptic (tmf) cohomology, and ideas of Floer theory and of Hopkins-Lurie on extended TFT's, we geometrically construct some $Ring$-valued representable cofunctors on the homotopy category of topological spaces. Using a classical computation in Gromov-Witten theory due to Seidel we show that for one version of these cofunctors $π_{2}$ of the representing space is non trivial, provided a certain categorical extension of Kontsevich conjecture holds for the symplectic manifold $ \mathbb{CP} ^{n}$, for some some $n \geq 1$. This gives further evidence for existence of generalized cohomology theories built from field theories living on a topological space.

math.AT↗

arXiv2014

Morse theory for the Hofer length functional

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, ω) $. This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.

math.SG↗

arXiv2013

Proof of the index conjecture in Hofer geometry

Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.

math.SG↗

arXiv2012

Gromov K-area and jumping curves in CP^n

We give here some extensions of Gromov's and Polterovich's theorems on $\karea$ of $ \mathbb{CP} ^{n}$, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov-Witten theory, and some connections with Bott periodicity, and loop groups. The argument is closely connected with study of jumping curves in $ \mathbb{CP} ^{n}$, and as an upshot we prove a new symplectic geometric theorem on these jumping curves.

math.SG↗

arXiv2012

Bott periodicity and stable quantum classes

We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on $BU$. This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimensional, Hamiltonian rigidity aspects of Hofer geometry of $ \mathbb{CP} ^{n}$, one of which we discuss here.

math.SG↗

arXiv2012

On configuration spaces of stable maps

We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X=BU,$ the rational homology of the spherical mapping space injects into the rational homology of the space of stable curves. We also give here a definition of what we call $q$-complete symplectic manifolds, which roughly speaking means Gromov-Witten theory captures all information about homology of the space of smooth stable maps.

math.SG↗

arXiv2011

Spectral geometry of the group of Hamiltonian symplectomorphisms

We introduce here a natural functional associated to any $b \in QH_* (M, ω)$: \emph{spectral length functional}, on the space of "generalized paths" in $ \text {Ham}(M, ω)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its domain of definition, and moreover the nature of extremals of this functional suggests that it may be variationally complete, in the sense that any suitably generic element of $ \widetilde{\text {Ham}}(M, ω)$ is connected to $id$ by a generalized path minimizing spectral length. Rather strong evidence is given for this when $M=S ^{2}$, where we show that all the Lalonde-McDuff Hamiltonian symplectomorphisms are joined to id by such a path. We also prove that the associated norm on $ {\text {Ham}}(M, ω)$ is non-degenerate and bounded from below by the the spectral norm. If the spectral length functional is variationally complete the associated norm reduces to the spectral norm.

math.DG↗

arXiv2010

Virtual Morse theory on $ΩHam(M,ω)$

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on $\ls$. As an application we prove a theorem which can be interpreted as stating that this functional behaves "virtually" as a perfect Morse-Bott functional with a flow. This can be applied to study topology and Hofer geometry of $ \text {Ham}(M, ω)$. We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz.

math.SG↗

arXiv2008

Quantum characteristic classes and the Hofer metric

Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $L \text {Ham}(M, ω)$ with values in $QH_* (M)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{*} (L\text {Ham}(M, ω), \mathbb{Q})$, with its Pontryagin product to $QH_{2n+*} (M)$ with its quantum product. As an application we prove an extension of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action, to higher dimensional geometry of the loop space $L \text {Ham}(M, ω)$.

math.SG↗