Prof. Alexander F. Ritter
Associate Professor, University of Oxford



I would be very grateful for your feedback in the form of typos, suggestions or mistakes you found, because these notes will be published as a book.

My contact details: find the contacts here

Combined PDF of all 24 Lectures and Homeworks: all Lectures and homeworks in one package

Combined PDF of all 24 Lectures (no homeworks)

Combined PDF of all Homeworks

Lecture 1: (PDF)(PS) Morse functions, topology of sublevel sets, Morse homology, number of critical points of a generic function, geometry vs functional analysis, Poincare duality via Morse homology

Lecture 2: (PDF)(PS) connections, parallel transport, Levi-Civita, geodesics, exp map, diffeomorphisms, regular maps

Lecture 3: (PDF)(PS) Sard's thm, transversality, intersection numbers, stability, embedding thm, tubular nbhd thm

Lecture 4: (PDF)(PS) Parametric transversality, perturbing a map to make it transverse to a submfd, zeros of sections, Morse functions, properties of Morse functions.

Lecture 5: (PDF)(PS) Banach spaces, C^{infinity} is not Banach, Banach manifolds, Inverse/implicit fn thm, why Fredholm maps

Lecture 6: (PDF)(PS) Fredholm operators, Sard-Smale theorem, zeros of Fredholm sections

Lecture 7: (PDF)(PS) Flowlines, properties of negative gradient flows, energy estimates, cobordisms, h-cobordism thm

Lecture 8: (PDF)(PS) handle attachments, topology of sublevel sets, stable and unstable mfds,

Lecture 9: (PDF)(PS) Morse-Smale metrics, Morse homology = cellular homology for self-indexing Morse functions,

Lecture 10: (PDF)(PS) Moduli spaces, overview: Floer homology, symplectic homology, Lagrangian Floer homology, Fukaya category and mirror symmetry, Heegaard-Floer homology, Instanton homology, Seiberg-Witten homology

Lecture 11: (PDF)(PS) Sobolev spaces, Sobolev embedding thm, Rellich compactness thm, Sobolev on manifolds

Lecture 12: (PDF)(PS) Banach space setup for the transversality thm: checking the spaces are Banach mfds

Lecture 13: (PDF)(PS) showing the section is C^k, proof of the transversality thm

Lecture 14: (PDF)(PS) reducing to the local setup d/ds + A_s, proof it is Fredholm

Lecture 15: (PDF)(PS) formal adjoints, index calculation for d/ds + A_s

Lecture 16: (PDF)(PS) spectral flow for d/ds + A_s with A_s converging to hyperbolic matrices, homotopy invariance of the index

Lecture 17: (PDF)(PS) compactness of the moduli spaces up to breaking

Lecture 18: (PDF)(PS) gluing theorem

Lecture 19: (PDF)(PS) definition of Morse homology, invariance theorem

Lecture 20: (PDF)(PS) invariance of Morse homology under changing Morse function and Riemannian metric

Lecture 21: (PDF)(PS) Poincare' duality, Kunneth Theorem, Morse inequalities, Intersection/cup product via Morse-graph flows

Lecture 22 and 23: (PDF)(PS) (Non-examinable) Spectral sequences, Leray-Serre spectral sequence in Morse homology

Lecture 24: (PDF)(PS) (Non-examinable) Morse-Bott homology, spectral sequence for Morse-Bott theory

(Lecture 24 was the last lecture)

Course Description: description of the Part III course

Syllabus: Initial Syllabus of the course I handed out

Exam Guide: What the key topics are, what is examinable, and what isn't

Prof. Alexander F. Ritter. Contact me.
Associate Professor in Geometry, Mathematical Institute, Oxford.
The Roger Penrose Fellow and Tutor, Wadham College, Oxford.