Linear Stability in an Ideal Incompressible Fluid

 We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 RID="*" ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. RID="**" ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084. Acknowledgements. The authors thank Susan Friedlander for useful discussions. Communicated by P. Constantin     

Semi-Focusing Billiards: Hyperbolicity

In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of ℝ³. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of ℝ³ with divided phase space. The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.     

A geometric criterion for positive topological entropy

We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincaré-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point is subexponential.The first author was partially supported by a Sloan Foundation Fellowship and an N.S.F. grant. The second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship. Both authors would like to thank MSRI for their support during the period that much of this paper was written.     

Regularized Laplacian determinants of self-similar fractals

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.     

Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad).The author is supported in part by NSF grant DMS-9104519     

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835     

Energy Decay for the Damped Wave Equation Under a Pressure Condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This result holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.     

Constructing Locally Connected Non-Computable Julia Sets

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.     

A criterion for a topological group to admit a continuous embedding in a locally compact group

It is proved that a topological group admits a continuous embedding in a locally compact group if and only if the group in question admits a separating family of unitary representations defining a symmetric Hopf-von Neumann algebra whose intrinsic group coincides with the set of nonzero characters of the predual algebra. To the blessed memory of Leonid Romanovich Volevich The author is indebted to Victor Kac for a pdf-file of the paper [13]. The research was partially supported by the Russian Foundation for Basic Research under grant no. 08-01-00034.     

Computer calculation of Witten's 3-manifold invariant

Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, and a Presidential Young Investigators award. The second author is supported by NSF grant DMS-8902153     

Rotation Sets of Billiards with One Obstacle

We investigate the rotation sets of billiards on the m-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.The first author was partially supported by NSF grant DMS 0456748.The second author was partially supported by NSF grant DMS 0456526.The third author was partially supported by NSF grant DMS 0457168.     

Spectral Order Automorphisms of the Spaces of Hilbert Space Effects and Observables

We describe the general form of bijective maps on the space of all Hilbert space effects or all observables on a finite dimensional space which preserve the spectral order in both directions. The first author was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T043080, T046203, and the second author was partially supported by a grant from the Ministry of Science of Slovenia. Both authors were supported by a joint Hungarian-Slovene grant, Reg. No. SLO-5/05.     

Geometrical derivation of collective operators and paths in TDHF

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.     

The Asymptotic Behavior of the Takhtajan-Zograf Metric

We obtain the asymptotic behavior of the Takhtajan-Zograf metric on the Teichmüller space of punctured Riemann surfaces. The first author is partially supported by JSPS Grant-in-Aid for Exploratory Research 2005-2007. The second author is partially supported by the research grant R-146-000-106-112 from the National University of Singapore and the Ministry of Education.     

On Computational Complexity of Siegel Julia Sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. The first and third authors are partially supported by NSERC Discovery grants. The second author is partially supported by NSERC Postgraduate Scholarship     

Transversal Dirac families in Riemannian foliations

We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation.Work supported in part by a grant from the National Science Foundation     

Carleman Estimates and Absence of Embedded Eigenvalues

Let L = ?Δ? W be a Schrödinger operator with a potential $W\in L^{\frac{n+1}{2}}(\mathbb{R}^n)Let L = −Δ− W be a Schr?dinger operator with a potential , . We prove that there is no positive eigenvalue. The main tool is an Carleman type estimate, which implies that eigenfunctions to positive eigenvalues must be compactly supported. The Carleman estimate builds on delicate dispersive estimates established in [7]. We also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials.The first author was partially supported by DFG grant KO1307/1 and also by MSRI for Fall 2005The second author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005     

An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics

We report on an “anti-Gleason” phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories—symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians. M. Entov partially supported by E. and J. Bishop Research Fund and by the Israel Science Foundation grant # 881/06. L. Polterovich partially supported by the Israel Science Foundation grant # 11/03.     

Quantized conductance at the Majorana phase transition in a disordered superconducting wire

Superconducting wires without time-reversal and spin-rotation symmetries can be driven into a topological phase that supports Majorana bound states. Direct detection of these zero-energy states is complicated by the proliferation of low-lying excitations in a disordered multimode wire. We show that the phase transition itself is signaled by a quantized thermal conductance and electrical shot noise power, irrespective of the degree of disorder. In a ring geometry, the phase transition is signaled by a period doubling of the magnetoconductance oscillations. These signatures directly follow from the identification of the sign of the determinant of the reflection matrix as a topological quantum number.     

On the location of poles of Ruelle's zeta function

We discuss examples of one-dimensional lattice spin systems of classical statistical mechanics whose generalized zeta function has all its poles and zeros on the real axis. The close relation between certain hyperbolic dynamical systems and these spin systems lets one expect that this is also true for some of the dynamical systems. In fact, we have found several one-dimensional expansive systems, among them the Gauss map whose zeta functions have their zeros, respectively their poles, on the real axis. Such a behaviour is closely related to the spectral properties of the sytems transfer operator which in the cases considered is a positive nuclear operator in a Banach space of holomorphic functions. We formulate a general conjecture concerning the spectrum of this class of operators.