5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.

/pdf/symplectic-fixed-points-and-holomorphic-spheres-1j818uo4lx.pdf

Symplectic fixed points and holomorphic spheres

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1

Introduction to Functional Analysis. By Angus E. Taylor. Pp. xvi. 423. $12.50. 1958. (John Wiley and Sons)

I The homotopy index theory- 11 Local semiflows- 12 The no blow-up condition Convergence of semiflows- 13 Isolated invariant sets and isolating blocks- 14 Admissibility- 15 Existence of isolating blocks- 16 Homotopies and inclusion induced maps- 17 Index and quasi-index pairs- 18 Some special maps used in the construction of the Morse index- 19 The Categorial Morse index- 110 The homotopy index and its basic properties- 111 Linear semiflows Irreducibility- 112 Continuation of the homotopy index- II Applications to partial differential equations- 21 Sectorial operators generated by partial differential operators- 22 Center manifolds and their approximation- 23 The index product formula- 24 A one-dimensional example- 25 Asymptotically linear systems- 26 Estimates at zero and nontrivial solution of elliptic equations- 27 Positive heteroclinic orbits of second-order parabolic equations- 28 A homotopy index continuation method and periodic solutions of second-order gradient systems- III Selected topics- 31 Repeller-attractor pairs and Morse decompositions- 32 Block pairs and index triples- 33 A Morse equation- 34 The homotopy index and Morse theory on Hilbert manifolds- 35 Continuation of the categorial Morse index along paths- Bibliographical notes and comments

The Homotopy Index and Partial Differential Equations

ABsRAc'r. In this paper we consider semiflows whose solution operator is eventually a conditional a-contraction. Such semiflows include solutions of retarded and neutral functional differential equations, of parabolic and certain other classes of partial differential equations. We prove existence of (nonsmooth) isolating blocks and index pairs for such semiflows, via the construction of special Lyapunov functionals. We show that index pairs enjoy all the properties needed to define the notion of a homotopy index, thus generalizing earlier results of Conley [2]. Finally, using a result of Maii [9], we prove that, under additional smoothness assumptions on the semiflow, the homotopy index is essentially a finite-dimensional concept. This gives a formal justification of the applicability of Waiewski's Principle to infinite-dimensional problems. Several examples illustrate the theory.

/pdf/on-the-homotopy-index-for-infinite-dimensional-semiflows-2er4vu8ukq.pdf

Krzysztof P. Rybakowski

Papers

The Homotopy Index and Partial Differential Equations

On the homotopy index for infinite-dimensional semiflows

The Effect of Domain Squeezing upon the Dynamics of Reaction-Diffusion Equations

Nonconvergent Bounded Trajectories in Semilinear Heat Equations

Ważewski's Principle for retarded functional differential equations