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Dan Burghelea

Publications and source records attributed to Dan Burghelea.

At least 37 records · Page 2Linked to original sources
arXiv2012

Graph Representations and Topology of Real and Angle Valued Maps

In this paper we review the definition of the invariants "bar codes" and "Jordan cells" of real and angle valued tame maps as proposed in Burghelea and Dey and Carlsson et al and prove the homotopy invariance of the sum # B^c_r +#B^o_{r-1}$ and of the Jordan cells. Here B^c_r resp. B^o_r denote the sets of closed resp. open bar codes in dimension r and # denotes cardinality. In addition we provide calculation of some familiar topological invariants in terms of bar codes and Jordan cells. The presentation provides a different perspective on Morse-Novikov theory based on critical values, bar codes and Jordan cells rather than on critical points instantons and closed trajectories of a gradient of a real or angle valued map.

math.AT↗
arXiv2012

Topology Of Real And Angle Valued Maps And Graph Representations (A Brief Survey)

(lecture delivered at the Congress of the Romanian mathematicians, Brasov, June 2011) Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to Morse-Novicov theory for real or angle valued Morse maps. The invariants are "barcodes" and "Jordan cells". From them one can derive all familiar topological invariants which can be derived via Morse-Novikov theory, like the Betti numbers and in the case of angle valued maps also the Novikov Betti numbers and the monodromy. Stability results for (some) bar codes and the homotopy invariance of the Jordan cells are the key results, and two new polynomials for any nonnegative integer (up to the dimension of the source) associated to a continuous nonzero complex valued map provide potentially interesting refinements of the Betti numbers and of the Novikov Betti numbers. In our theory the bar codes which are intervals with ends critical values/angles, the Jordan cells and the " canonical long exact sequence" of a tame map are the analogues of instantons between rest points, closed trajectories and of the Morse-Smale complex of the gradient of a Morse function in the Morse-Novikov theory.

math.AT↗
arXiv2011

Defining and Computing Topological Persistence for 1-cocycles

The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various directions. However, no attempt has been made so far to extend the concept of topological persistence to a generalization of `maps' such as cocycles which are discrete analogs of closed differential forms, a well known concept in differential geometry. We define a notion of topological persistence for 1-cocycles in this paper and show how to compute its relevant numbers. It turns out that, instead of the standard persistence, one of its variants which we call level persistence can be leveraged for this purpose. It is worth mentioning that 1-cocyles appear in practice such as in data ranking or in discrete vector fields.

math.AT↗
arXiv2011

On the Space of Trajectories of a Generic Vector Field

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex.

math.DS↗
arXiv2010

Dynamics, Spectral Geometry and Topology

The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting in the case of a generic vector field which has some additional properties satisfied by a still very large class of vector fields.

math.DS↗
arXiv2009

Cyclic theory for commutative differential graded algebras and s-cohomology

In this paper one considers three homotopy functors on the category of manifolds, $hH^\ast, cH^\ast, sH^\ast,$ and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, $HH^\ast, CH^\ast, SH^\ast.$ If $P$ is a smooth 1-connected manifold and the algebra is the de-Rham algebra of $P$ the two pairs of functors agree but in general do not. The functors $ HH^\ast $ and $CH^\ast$ can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The third $SH^\ast ,$ although inspired from negative cyclic homology, can not be identified with any sort of cyclic homology of any algebra. The functor $sH^\ast$ might play some role in topology. Important tools in the construction of the functors $HH^\ast, CH^\ast $and $SH^\ast ,$ in addition to the linear algebra suggested by cyclic theory, are Sullivan minimal model theorem and the "free loop" construction described in this paper.

math.AT↗
arXiv2009

Euler structures, the variety of representations and the Milnor-Turaev torsion

In this paper we extend and Poincare dualize the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler-Poincare characteristic, to arbitrary manifolds. We use the Poincare dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray-Singer torsion by providing a slightly modified object which is a topological invariant. We show that when the co-Euler structure is integral then the modified Ray-Singer torsion when regarded as a function on the variety of generically acyclic complex representations of the fundamental group of the manifold is the absolute value of a rational function which we call in this paper the Milnor-Turaev torsion.

math.DG↗
arXiv2008

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space.

math.FA↗
arXiv2007

Complex valued Ray-Singer torsion

In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations.

math.DG↗
arXiv2006

Complex valued Ray--Singer torsion II

In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsion with the complex valued analytic torsion, for odd dimensional manifolds. This is done along the lines of Burghelea, Friedlander and Kappeler's proof of the Cheeger-Müller theorem.

math.DG↗
arXiv2006

Torsion, as a function on the space of representations

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical torsion. They are associated essentially to a closed smooth manifold equipped with a (co)Euler structure and a Riemannian metric in the first case, a smooth triangulation in the second case, and a smooth flow of type described in section 2 in the third case. In this paper we define these functions, describe some of their properties and calculate them in some case. We conjecture that they are essentially equal and have analytic continuation to rational functions on the variety of representations. We discuss the case of one dimensional representations and other relevant situations when the conjecture is true. As particular cases of our torsions, we recognize familiar rational functions in topology such as the Lefschetz zeta function of a diffeomorphism, the dynamical zeta function of closed trajectories, and the Alexander polynomial of a knot. A numerical invariant derived from Ray-Singer torsion and associated to two homotopic acyclic representations is discussed in the last section.

math.DG↗
arXiv2005

Dynamics, Laplace transform and Spectral geometry

We consider a vector field $X$ on a closed manifold which admits a Lyapunov one form. We assume $X$ has Morse type zeros, satisfies the Morse--Smale transversality condition and has non-degenerate closed trajectories only. For a closed one form $η$, considered as flat connection on the trivial line bundle, the differential of the Morse complex formally associated to $X$ and $η$ is given by infinite series. We introduce the exponential growth condition and show that it guarantees that these series converge absolutely for a non-trivial set of $η$. Moreover the exponential growth condition guarantees that we have an integration homomorphism from the deRham complex to the Morse complex. We show that the integration induces an isomorphism in cohomology for generic $η$. Moreover, we define a complex valued Ray--Singer kind of torsion of the integration homomorphism, and compute it in terms of zeta functions of closed trajectories of $X$. Finally, we show that the set of vector fields satisfying the exponential growth condition is $C^0$--dense.

math.DG↗
arXiv2005

The topology of the monodromy map of the second order ODE

We consider the following question: given $A \in SL(2,R)$, which potentials $q$ for the second order Sturm-Liouville problem have $A$ as its Floquet multiplier? More precisely, define the monodromy map $μ$ taking a potential $q \in L^2([0,2π])$ to $μ(q) = \tildeΦ(2π)$, the lift to the universal cover $G = \widetilde{SL(2,R)}$ of $SL(2,R)$ of the fundamental matrix map $Φ: [0,2π] \to SL(2,R)$, \[ Φ(0) = I, \quad Φ'(t) = \begin{pmatrix} 0 & 1 q(t) & 0 \end{pmatrix} Φ(t). \] Let $H$ be the real infinite dimensional separable Hilbert space: we present an explicit diffeomorphism $Ψ: G_0 \times H \to H^0([0,2π])$ such that the composition $μ\circ Ψ$ is the projection on the first coordinate. The key ingredient is the correspondence between potentials $q$ and the image in the plane of the first row of $Φ$, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let $C_1 \subset L^2([0,2π])$ be the set of potentials $q$ for which the equation $-u'' + qu = 0$ admits a nonzero periodic solution: $C_1$ is diffeomorphic to the disjoint union of a hyperplane and cartesian products of the usual cone in $R^3$ with $H$.

math.CA↗
arXiv2005

Laplace transform, dynamics and spectral geometry

We consider vector fields $X$ on a closed manifold $M$ with rest points of Morse type. For such vector fields we define the property of exponential growth. A cohomology class $ξ\in H^1(M;\mathbb R)$ which is Lyapunov for $X$ defines counting functions for isolated instantons and closed trajectories. If $X$ has exponential growth property we show, under a mild hypothesis generically satisfied, how these counting functions can be recovered from the spectral geometry associated to $(M,g,ω)$ where $g$ is a Riemannian metric and $ω$ is a closed one form representing $ξ$. This is done with the help of Dirichlet series and their Laplace transform.

math.DG↗
arXiv2001

Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators

We consider the nonlinear Sturm-Liouville differential operator $F(u) = -u'' + f(u)$ for $u \in H^2_D([0, π])$, a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity $f: \RR \to \RR$ we show that there is a diffeomorphism in the domain of $F$ converting the critical set $C$ of $F$ into a union of isolated parallel hyperplanes. For the proof, we show that the homotopy groups of connected components of $C$ are trivial and prove results which permit to replace homotopy equivalences of systems of infinite dimensional Hilbert manifolds by diffeomorphisms.

math.FA↗
arXiv2001

Non-contractible periodic trajectories of symplectic vector fields, Floer cohomology and symplectic torsion

For a closed symplectic manifold $(M,ω)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $γ$ we define a Floer complex based on 1-periodic trajectories of $Z$ in the homotopy class $γ$. We suppose that the closed 1-form $i_{Z_t}ω$ represents a cohomology class $β(Z):=β$, independent of $t$. We show how to associate to $(M,ω,γ,β)$ and to two pairs $(Z_i,J_i)$, $i=1,2$ with $β(Z_i)=β$ an invariant, the relative symplectic torsion, which is an element in the Whitehead group $Wh(Λ_0)$, of a Novikov ring $Λ_0$ associated with $(M,ω,Z,γ)$. If the cohomology of the Floer complex vanishes or if $γ$ is trivial we derive an invariant, the symplectic torsion for any pair $(Z,J)$. We prove, that when $β(γ)\neq 0$, or when $γ$ is non-trivial and $β$ is 'small', the cohomology of the Floer complex is trivial, but the symplectic torsion can be non-trivial. Using the first fact we conclude results about non-contractible 1-periodic trajectories of 1-periodic symplectic vector fields. In this version we will only prove the statements for closed weakly monotone manifolds, but note that they remain true as formulated for arbitrary closed symplectic manifolds.

math.SG↗
arXiv2001

A short course on Witten Helffer-Sjöstrand theory

Witten-Helffer-Sjöstrand theory is an addition to Morse theory and Hodge-de Rham theory for Riemannian manifolds and considerably improves on them by injecting some spectral theory of elliptic operators. It can serve as a general tool to prove results about comparison of numerical invariants associated to compact manifolds analytically, i.e. by using a Riemannian metric, or combinatorially, i.e. by using a triangulation. It can be also refined to provide an alternative presentation of Novikov Morse theory and improve on it in many respects. In particular it can be used in symplectic topology and in dynamics. This material represents my Notes for a three lectures course given at the Goettingen summer school on groups and geometry, June 2000.

math.DG↗