2,092 research outputs found
Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems
No full textMorse Index Theorems for elliptic boundary value problems in multi-dimensions are proved under various boundary conditions. The theorems work for star-shaped domains and are based on a new idea of measuring the "oscillation" of the trace of the set of solutions on a shrinking boundary. The oscillation is measured by formulating a Maslov index in an appropriate Sobolev space of functions on this boundary. A fundamental difference between the cases of Dirichlet and Neumann boundary conditions is exposed through a monotonicity that holds only in the former case
Maslov S^{1} Bundles and Maslov Data
No full textIn this paper we consider the Maslov S^{1}bundles of a symplectic manifold
(M,\omega), which refer to both the determinant bundle (denoted by \Gamma_{J})
of the unitary frame bundle and the bundle
\Gamma_{J}^{2}=\Gamma_{J}\big/\{\pm1\}. The sympletic action of a compact Lie
group G on M can be lifted to group actions on the principal S^{1} bundles
\Gamma_{J} and \Gamma_{J}^{2}. In this work we study the interplay between the
geometry of the Maslov S^{1} bundles and the dynamics of the group action on M.
We show that when M is a homogeneous G-space and the first real Chern class
c_{\Gamma} is nonvanishing, \Gamma_{J} and \Gamma_{J}^{2} are also homogeneous
G-spaces. We also show that when [\omega]=r\cdot c_{\Gamma} for some real
number r, then the G action is Hamiltonian and the Hamiltonians assume
particular forms. In the end, we study a function called the \beta-Maslov data
of a symplectic S^{1} action with respect to a connection 1-form \beta on
\Gamma_{J}^{2}, which serves as the nonintegrable version of the notion of
Maslov indices when \Gamma_{J}^{2} is not a trivial bundle
Maslov index in semi-Riemannian submersions
No full textWe study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.M.I.U.R.[PRIN07]M.I.U.R.Regional Junta AndaluciaRegional Junta Andalucia[P06-FQM-01951]Fundacion Seneca[04540/GERM/06]Fundacion SenecaSpanish MEC[MTM2009-10418]Spanish MECCapes, Brazil[BEX 1509/08-0]Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES
Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index
No full textWe study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the mo- menta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0, +∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymp- totic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector
Economic law of increase of Kolmogorov complexity. Transition from financial crisis 2008 to the zero-order phase transition (social explosion)
No full textIn Maslov (2003), a two level model of the occurrence of financial pyramid (bubbles) has been considered. We also considered the mathematical analogy of this model to Bose condensation. In the present paper, we explain why Ponzi schemes and bubbles result in a crisis in real economics. In Maslov (2005), the law of increase of entropy in financial systems, and consequently increase of Kolmogorov complexity, is formulated. If this law is broken, the financial system makes a phase transition to a different state. In Maslov (2005) the author considered a two level model of the zeroth-order phase transition which was interpreted in Maslov (2006) as an analog of social catastrophe. In the present paper we also examine this model.
Maslov indices of resonant tori
No full textIt is well known that the resonant tori of an integrable, classical Hamiltonian system break up under a small perturbation into new tori which wind around the remaining nonresonant tori. It turns out that the Maslov indices of one of these satellite tori depend on the Maslov indices of its unperturbed, resonant parent, and on the integers which enter into the resonance condition. They do not, however, depend on the functional form of the Hamiltonian or the perturbation. Our results are valid for any number of degrees of freedom
New way to compute Maslov indices
No full textA new formula is presented for computing Maslov indices in integrable and near-integrable Hamiltonian systems. For several kinds of applications the new formula is particularly easy to use. It does not rely on counting caustics or other kinds of discontinuities. Its theoretical justification calls on wave-packet concepts and the topological properties of the group of symplectic matrices. Techniques are also presented for manipulating the Maslov index in analytical expressions
Maslov index and Spectral Flow
No full textIn [1], Arnol'd discussed the Maslov index as an intersection number of Lagrangian loop with the Maslov cycle, whereas in [17], Robbin and Salamon gave a de nition of the Maslov index in term of the signature of crossing form. Furthermore, the Maslov index is characterized by axioms. The index is a homotopy invariant with xed endpoints, and is additive for the concatenation of Lagrangian paths. Application wise, in [8], Floer studied the case where the Hessian (a second order di erential operator) of the symplectic action functional A is taken along a gradient ow line x(t) i.e. a solution satisfying the gradient ow equation x_ = rA (x) connecting two critical endpoints x = limt! 1 x(t): Both the gradient rA and Hessian A = r2A are taken with respect to suitable metric on the underlying manifold. In particular, Floer de ned a relative index at x and showed that the spectral ow of the Hessian of A is equal to the relative Morse index between x which is also equal to the dimension of the space of trajectories of gradient ow between x : In Morse theory (as nite dimensional case of Floer theory), such A is bounded below and the Hessian H has only nitely many negative eigenvalues, when treated as matrix. Then the spectral ow of A(t) is the number of negative eigenvalues (counted with multiplicity) of A at x+ minus the number of negative eigenvalues of A at x?? which is equal to the Fredholm index of linearization operator DA: Moreover, it happens to be the case that the unstable manifold Wu(x??) intersects the stable manifold Ws(x+) transversally if and only if DA is surjective. Then, the moduli space M= Wu(x??) \Ws(x+) is a nite dimensional manifold of dimension equal to the relative Morse index. In this thesis, we will mainly focus on both the notion of Maslov index and spectral ow and their coincidence. Roughly speaking, Maslov index can be seen as the number of times the Lagrangian paths crosses the Maslov cycle and the spectral ow can be seen as the number of eigenvalues of A(t) crossing zero from negative to positive from t = ??1 to t = 1: The organization of this thesis will be distributed as follows: In Chapter 1 we brie y review some backgrounds in Symplectic Geometry, that include the notion of symplectic vector space, symplectic manifold and Darboux Theorem. To prepare for later chapter, we study the relation between Sp(2n) and U(n) and their actions on the Lagrangian Grassmannian (n): In Chapter 2 we study the notion and computational tool for the Maslov index in the sense of crossing operator(adopting the method introduced in [17]). Other related indices such as H ormander index and the well-known Conley-Zehnder index will also be introduced. In Chapter 3, we study the notion of Spectral ow of operator satisfying certain conditions and its coincidence with Maslov index and Fredholm index, together with several examples in which this notion applies to will be given
Generalized local systems and the Maslov class of exact Lagrangians
No full textThis PhD-thesis in symplectic topology consists of an introduction and two research papers. The goal of the thesis is to investigate the Maslov class for exact Lagrangians in Weinstein domains. In the introduction we give some history, introduce the research area and explain everything that is needed to understand the main results of the papers. Afterwards we give a short summary of the papers and their main results. Finally we state some open questions that the author was not able to answer during his time as a student. In Paper I we reprove with new methods the well known fact that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. We also further extend that result to closed exact Lagrangians in slightly more general Weinstein domains built by attaching sub-critical handles in a certain way to cotangent bundles. The proof uses Floer theory with coefficients in path local systems which is a theory developed in the paper that builds on previous works by Abouzaid, Barraud and Cornea. In paper II we show that the main result of Paper I does not hold if we allow the handles that build the Weinstein domains to be critical. For instance we show that to any Weinstein domain one can attach a one-handle and a critical handle and then find a closed exact Lagrangian with non vanishing Maslov class. In the proof we construct the Lagrangians with explicit formulas and explicitly compute their Maslov class
Generalized local systems and the Maslov class of exact Lagrangians
No full textThis PhD-thesis in symplectic topology consists of an introduction and two research papers. The goal of the thesis is to investigate the Maslov class for exact Lagrangians in Weinstein domains. In the introduction we give some history, introduce the research area and explain everything that is needed to understand the main results of the papers. Afterwards we give a short summary of the papers and their main results. Finally we state some open questions that the author was not able to answer during his time as a student. In Paper I we reprove with new methods the well known fact that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. We also further extend that result to closed exact Lagrangians in slightly more general Weinstein domains built by attaching sub-critical handles in a certain way to cotangent bundles. The proof uses Floer theory with coefficients in path local systems which is a theory developed in the paper that builds on previous works by Abouzaid, Barraud and Cornea. In paper II we show that the main result of Paper I does not hold if we allow the handles that build the Weinstein domains to be critical. For instance we show that to any Weinstein domain one can attach a one-handle and a critical handle and then find a closed exact Lagrangian with non vanishing Maslov class. In the proof we construct the Lagrangians with explicit formulas and explicitly compute their Maslov class
- …
