Course Information

Lectures: MWF at 11am in Olson Hall 117. Attendance not mandatory.

Textbook: Morse Theory by J.W. Milnor.

Office Hours: The office hours of Professor Casals (casals--at--ucdavis.edu) are Fridays 3:30-4:30pm at MSB 3214. Always feel free to ask after class.

Important Dates: First Day (Apr 3) and Final Exam (June 7).

Grade: The grade will be determined by Problem Sets and a Final Project, each part with a 50% weight.

Problem Sets: Weekly assignments due on Fridays at 9pm, submitted through Gradescope.

Problem Sets

1. Problem Set 1: Due Friday Apr 14, available Mar 31.
   
   
2. Problem Set 2: Due Friday May 12, available May 1.
   
   

Math Diaries

1. Monday Apr 3: Introduction to MAT-240B.
   
   Principles in Morse Theory and sample applications. A degenerate example.
   Definition of a non-degenerate critical point and its index. Morse functions.
   Examples of Morse functions on the 2-sphere and the 2-torus.
   Intuition on how to construct Morse functions.
   
   Textbook Reading (Apr 3): Chapter 1.1.
   
   
   
2. Wednesday Apr 5: Local models.
   
   The Morse Lemma: normal form at non-degenerate critical points. Proof.
   Proof that sublevel sets are diffeomorphic if no critical value is crossed.
   Examples of how sublevel sets change if critical values are crossed.
   
   Textbook Reading (Apr 5): Chapters 1.2 and 1.3.
   
   
   
3. Friday Apr 7: Handle attachments and applications.
   
   Precise change of sublevel sets in crossing critical values: handle attachments.
   Corollaries on algebraic topology and smooth functions. Reeb's Theorem.
   Milnor's construction of exotic spheres and Morse function on them.
   Morse functions on adjoint orbits via Killing forms and regular elements.
   
   Textbook Reading (Apr 7): Chapters 1.3 and 1.4.
   
   
   
4. Monday Apr 10: Existence and manipulation of Morse functions.
   
   Statement that distance to a generic point is a Morse function. Examples of degeneracies via focal points.
   Focal points for spheres and ellipses. Proof of existence of Morse functions via distance functions.
   Review of Sard's Theorem on the zero measure of critical values. Manipulation of Morse functions. Heegaard Splittings in 3-dimensions and Kirby diagrams in 4-dimensions.
   Textbook Reading (Apr 10): Chapters 4 and 5 of "4-Manifolds and Kirby Calculus".
   
   
   
5. Wednesday Apr 12: Manipulating Morse functions (Part II).
   
   Existence of self-indexing Morse functions. Canceling Lemma for canceling pairs.
   Examples of canceling pairs in dimensions 1,2 and 3. Data needed for handle attachments.
   Gluing map and trivialization of normal bundles. Classification of normal framings.
   
   Textbook Reading (Apr 12): Chapters 4 and 5 of "4-Manifolds and Kirby Calculus".
   
   
   
6. Friday Apr 14: Handlebody Decompositions.
   
   Exotic gluings. Determination of framing data in terms of homotopy groups.
   On the homotopy type of the general linear group in known ranks. Examples in dimensions 1,2,3 and 4.
   Description of 2-disk bundles over the 2-sphere in terms of handlebody diagrams. Existence of Heegaard splittings.
   
   Textbook Reading (Apr 14): Chapters 4 and 5 of "4-Manifolds and Kirby Calculus".
   
   
   
7. Monday Apr 17: First steps in infinite-dimensional Morse theory.
   
   The path space associated to a smooth manifold. Tangent vectors of the path space as vector fields.
   Energy functional on path space. Example of path space for Euclidean plane and energy of plane curves.
   Definition of a first variation. The first variation of the energy functional.
   
   Textbook Reading (Apr 17): Part III - Chapter 11.
   
   
   
8. Wednesday Apr 19: Connections and geodesics.
   
   The need for connections: deriving vector fields. Physical motivation: accelaration.
   Integrated definition of connections (parallel transport) and infinitesimal definition.
   Coordinate description in terms of Christoffel symbols. Existence of connections.
   Examples of connections in the real plane. Definition of geodesics.
   
   Textbook Reading (Apr 19): Part II - Chapters 8 and 10.
   
   
   
9. Friday Apr 21: Riemannian Metrics and Levi-Civita connections.
   
   Explicit geodesic equations for the 2-sphere, real plane and hyperbolic plane.
   Choice of Christoffel symbols for connections in these cases and geodesic curves.
   Definition of a connection compatible with a Riemannian metric. Properties.
   
   Textbook Reading (Apr 21): Part II - Chapter 8.
   
   
10. Monday Apr 24: First variation of the energy functional.
    
    The energy functional from the Morse Theory viewpoint. Use of Riemannian metric and Levi-Civita connection.
    Path space fibration and the homotopy type of the loop space. Cohomology of loop space (Leray's spectral sequence).
    First variation of the energy functional. Critical points are geodesic curves. Example of loop of geodesics for 2-sphere.
    Issues with the second variation: degeneracy and index. Need for curvature for second variation.
    
    Textbook Reading (Apr 24): Part III - Chapters 12 and 13.
    
    
    
11. Wednesday Apr 26: Curvature tensor.
    
    Second variation of energy involve a curvature term. Jacobi fields.
    Examples of Jacobi fields as variations of a geodesic through geodesics.
    Definition of curvature tensor. Proof that it is a linear 3-tensor. Examples.
    Symmetric and skew-symmetric properties of the curvature tensor and inner products.
    Sectional curvature, Ricci curvature and scalar curvature. Examples.
    
    Textbook Reading (Apr 26): Part II - Chapter 9.
    
    
    
12. Friday Apr 28: Second variation and Riemannian curvature.
    
    Formula for the second variation of the energy functional. Use of curvature tensor and Levi-Civita connection.
    Expanded terms for second variation in terms of Jacobi equation and broken contributions. Examples.
    Kernel of the Hessian: Jacobi fields. Jacobi fields in spheres and negative spaces for second variation.
    
    Textbook Reading (Apr 28): Part III - Chapter 13.
    
    
    
13. Monday May 1: Introduction to index of geodesics.
    
    Review of second variation of energy funtional and Jacobi fields. Intuition for index.
    Examples of Jacobi fields and generic non-existence of Jacobi fields vanishing at endpoints.
    Definition of conjugate points and their multiplicity. Examples in spheres.
    Non-existence of conjugate points in hyperbolic plane.
    
    Textbook Reading (May 1): Part III - Chapter 15.
    
    
    
14. Wednesday May 3: Proof of the Index Theorem.
    
    Statement of the index theorem. Subdivision of a geodesic into minimal-length intervals.
    Decomposition of tangent space at a geodesic into broken Jacobi fields and its complement.
    Hessian of energy functional is positive definition on the complement of broken Jacobi fields.
    Properties of the index function in terms of a parameter sweeping the geodesic.
    
    Textbook Reading (May 3): Part III - Chapter 15.
    
    
    

Further Resources

Books: The book "Riemannian Geometry" by M.P. Do Carmo contains significant parts of the material the we will use and cover, including curvature and Jacobi fields (in Chapters 4 and 5), variations on the energy functional and the index theorem (in Chapters 9 and 11). Nevertheless, it does not include the proof of Bott Periodicity, which we will cover. For Morse theory, in addition to J.W. Milnor's textbook, "An Introduction to Morse Theory" by Y. Matsumoto is a good resource. The first half of "Morse Theory and Floer Homology " by M. Audin and M. Damian is also a solid reference for finite-dimensional Morse theory. See also " An Invitation to Morse Theory " by L. Nicolaescu. Further material on Morse homology can be found in " Lectures on Morse Homology " by A. Banyaga and D. Hurtubise and "Morse Homology" by M. Schwarz. Finally, for applications of Morse theory for smooth 4-manifolds see "4-Manifolds and Kirby Calculus" by A.I. Stipsicz and R.E. Gompf and "4-Manifolds" by S. Akbulut.

Online resources: The lectures on "Morse Theory" by D. Gay are insightful. There are many available introductory talks online, such as "this talk" or "this short introduction".