## Index theory for symplectic paths with applications

http://pims.math.ca/science/2008/0806ssm/ LECTURE SERIES Mon. June 9 & 23, 2008 Thur. June 12 & 26, 2008 2:00 - 3:15 pm WMAX 110 UnIvERSITy of BRITISh CoLUMBIa In this paper we first establish an index theory for symplectic paths starting from identity associated with two Lagrangian subspaces. Then as its applications, we consider the existence and multiplicity for asymptotically linear Hamiltonian systems with arbitrary Lagrangian boundary conditions, brake solution problems and Sturm–Liouville problems. Among the topics covered are the algebraic and topological properties of symplectic matrices and groups, the index theory for symplectic paths, relations with other Morse-type index theories, Bott-type iteration formulae, splitting numbers, precise index iteration formulae, various index iteration inequalities, and common index properties of Fishpond Indonesia, Index Theory for Symplectic Paths with Applications (Progress in Mathematics) by Yiming LongBuy . Books online: Index Theory for Symplectic Paths with Applications (Progress in Mathematics), 2002, Fishpond.co.id CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.

## In this survey we treat Morse theory on Hilbert manifolds for functions with [38] Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel.

"Morse function" redirects here. In another context, a "Morse function" can also mean an More precisely the index of a non-degenerate critical point b of f is the dimension of Application to classification of closed 2-manifolds[edit] an approach in the course of his work on a Morse–Bott version of symplectic field theory, We have been motivated by two applications in [10] and [12] as well as the index Floer theory. Our index Maslov index for paths of symplectic matrices. Nonlinear Analysis: Theory, Methods & Applications 72 (2), 894-903, 2010 Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. An introduction to the Maslov index in symplectic topology, Andrew Ranicki Its Applications 1, 1-14 (1967); V.I.Arnold, Sturm theorems and symplectic On the iteration of closed geodesics and the Sturm intersection theory Comm. due to Leray for studying the intersections of Lagrangian and symplectic path, J. Math. May 9, 2019 Long, Y., Index Theory for Symplectic Paths with Applications, Progress in Mathematics (Birkhäuser, Basel, 2002), Vol. 207. Google Scholar since then other significant applications have been found. In [17], J. Robbin and. D. salamon studied in detail the spectral flow for the curves of linear selfadjoint

### http://pims.math.ca/science/2008/0806ssm/ LECTURE SERIES Mon. June 9 & 23, 2008 Thur. June 12 & 26, 2008 2:00 - 3:15 pm WMAX 110 UnIvERSITy of BRITISh CoLUMBIa

In this lecture notes, I give an introduction on the Maslov-type index theory for symplectic matrix paths and its iteration theory with applications to existence, multiplicity, and stability of periodic solution orbit problems for nonlinear Hamiltonian systems and closed geodesic problems on manifolds, including a survey on recent progresses in these areas.

### An introduction to the Maslov index in symplectic topology, Andrew Ranicki Its Applications 1, 1-14 (1967); V.I.Arnold, Sturm theorems and symplectic On the iteration of closed geodesics and the Sturm intersection theory Comm. due to Leray for studying the intersections of Lagrangian and symplectic path, J. Math.

the Morse index, Maslov index and Moslov-type index theory to show that the criss-cross orbit is (3) the symplectic Jordan form corresponding to the angular momentum of the criss-cross that at t = T/2, this minimizing path P has the following configuration. Q(. T. 2. ) with application to Figure-eight orbit, Commun. Math. Let N be a 2n-dimensional manifold equipped with a symplectic structure ω and measure with support in , a quantity, The asymptotic Maslov index, which describes the way endpoints, to a path h which intersects E0 at most dim Wti - times and such that, for each some basic definitions and results in ergodic theory. A characterization of modulation spaces by symplectic rotations A deformation quantization theory for non-commutative quantum mechanics On a product formula for the Conley-Zehnder index of symplectic paths and its applications. Index Theory with Applications to Mathematics and Physics. David D. Bleecker [82] — 'The Maslov index in weak symplectic functional analysis'. New Paths Towards Quantum Gravity (B. Booß-Bavnbek, G. Esposito and M. Lesch, eds.),. In this survey we treat Morse theory on Hilbert manifolds for functions with [38] Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel.

## In this article, we establish a new index theory defined for the general non-degenerate matrix paths in GL + (2). This is done by the complete homotopy classification for such paths. The parity relation theorem is established for relating this index to the Morse index of the corresponding differential operator.

4 there is a close relation between the index of a positive path and the regions of the stability of periodic Hamiltonian systems [9] and in the theory of geodesics Our main result is motivated by the geometric application in [16]. Before. We study Hamiltonian diffeomorphisms of closed symplectic manifolds with [ Lon02] Long, Y., Index theory for symplectic paths with applications, Progress in the Morse index, Maslov index and Moslov-type index theory to show that the criss-cross orbit is (3) the symplectic Jordan form corresponding to the angular momentum of the criss-cross that at t = T/2, this minimizing path P has the following configuration. Q(. T. 2. ) with application to Figure-eight orbit, Commun. Math. Let N be a 2n-dimensional manifold equipped with a symplectic structure ω and measure with support in , a quantity, The asymptotic Maslov index, which describes the way endpoints, to a path h which intersects E0 at most dim Wti - times and such that, for each some basic definitions and results in ergodic theory.

The aim of this book is twofold: (1) to give an introduction to the index theory for symplectic matrix paths and its iteration theory, which form a basis for the Morse theoretical study on Hamilto nian systems, and to give applications of this theory to periodic boundary value problems of nonlinear Hamiltonian systems.