Morse Theory Online: Intro Resources for Learning

[Alamino]

Does anyone know of resources about Morse Theory online? I'm looking for an introductory text.

 

[mathwonk]

not exactly itro but you might look at this:

http://math.stanford.edu/~ralph/papers.htmli recommend buying milnors book, it is available used for 25-30 dollars. there is no better source. also there is a more elementary one by matsumoto, that looks nice.

 

[mathwonk]

or try

differential topology: first steps, andrew wallace. $10. very elementary and cheap. but nowhere near as ambitious or dep as milnor. but it is an intro to the basic idea of using morse theory to understand geometry.

basically the idea is to look at a single smooth function defined on a surface and ask how many critical points it has, and what the second order term of the taylor series is at each point.

picture a doughnut standing on end and the function is the height function, there are 4 critical points, one max one min, and two sadle points. if you think about it thaqt tells you it is a doughnut!

more preciwely not that as you move a horizontal plane (constant height) up along the surface, the geometry (i.e. the topology) stays the same until the plane crosses a critical level.

morse theiry give you a forlmula for how it chnages in terms of the index of the second order term of the taylor series, thought of as a quadratic form. (i.e. in terms of the signs of the eigenvalues).

the whole thing is to look at the critical points and the flow lines of the gradient. segal et al and witten maybe, make this abstarct by constructing a category with critical points as objects and flow lines as morphisms. there is an abstract simplicial compklex associated to any category, and i supopose it gives abck the manifold up to homotopy. that is an old construction i first ehard from bott in 1971 and he said it waS OLD THEN, ALTHoUGH SEGALS paper on it ahd just come out.

Product Details:
ISBN: 0486453170
Format: Paperback, 144pp
Pub. Date: November 2006
Publisher: Dover Publications

 

[mathwonk]

hre is the bibliography from a course at ucsb:

Useful collateral reading:

1. J. Milnor, Morse Theory, Princeton Univ. Press, Princeton NJ, 1963.

2. R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), 331-358.

3. S. Lang, Differential and Riemannian Manifolds, Springer, New York, 1995.

4. S. Smale, Morse theory and a nonlinear generalization of the Dirichlet problem, Annals of Math, 80 (1964), 382-396.

5. R. Bott and L. Tu, Differential forms in algebraic topology, Springer, New York, 1982.

Items 1, 3 and 5 will be available from the campus book store and will also be put on reserve at the campus library. Items 2 and 4 can be accessed via Math Sci Net: http://www.ams.org/mathscinet/

 

[mathwonk]

bott published some famous lectures, lectures on morse theory old and new, in the bulletin of the AMS in 1982.

unfortunately i have noit been able to find them online, but maybe from a university JSTOR might ahve them. but it seems the BAMS is only online back to 1992 on the AMS site.

milnors book is really fantastic though.

 

[Alamino]

Thanks a lot for the references. I'll look for the books in the university's library (although a first search I made was not very promising). Indeed, maybe you would help me with the question which lead me to look for Morse Theory. I'm trying to identify if a critical point of a functional F[f(x),g(y)] is a maximum or a minimum. I know that if it was a simple function, I should check if the Hessian matrix is positive-definite, but as it is a functional, I guess I should look at the second order functional derivatives
$\frac{\delta^2 F[f(x),g(y)]}{\delta f(z) \delta g(t)},$
as well as the second derivatives w.r.t. f and g only. But it is not sufficient to say that they should be positive, right? What would be the functional analogous of the positive-definiteness of the Hessian in this case?

 

[jmarstonm]

I think some of my Grandfather's writings are still on the shelf

When I am in Princeton next, I could look in the living room...

Actually, that may not be for a while. I have recently been researching my Grandfather's life and works (Marston Morse) and am excited to see that the work he did is still discussed. His work was his life, and I certainly do not understand it (although I saw a recording of one of his lectures) but would like to develop at least a basic understanding of it's pertinence in modern mathematical studies. I look forward to following this thread, thank you all for the added inspiration. You are all remarkable individuals.

Jonathan Marston Morse

 

[mathwonk]

wow! thanks for this wonderful post reminding us how much the internet has brought us together.

i used to think morse theory was related to lefschetz theory of pencils in algebraic geometry as well. do you know if professor morse ever spoke with solomon lefschetz? were they in princeton together?

 

[mathwonk]

in fact i recall now that morse theory is usually used now to prove lefschetz's results. lefschetz was also famous for stating correct results without sufficient argument.

it seems lefschetz was about 8 years older tham morse and preceded him at the institute by about 11 years. morse's theory has many more applications than just to proving lefschetzs results but perhaps we owe to morse the clarification and substantiation of lefschetz's theory of pencils on an algebraic variety.

this is one of the topics is milnors book on morse theory and its applications. milnor also gives bott's aplication of morse theory to the proof of the famous periodicity theorem.

i also enjoyed seeing the film of morse discussing his theory, i believe by simply showing us a rising tide of water, slowly engulfing some islands, and watching how the shoreline changed in shape as the water rose. i was amazed how simple someone could make what i had considered a difficult theory before that.

in my own experience, professor boris moishezon used to show us how to use morse theory to study the topology of surfaces, first fiberinbg them as pencils of curves over the complex "line", then studying the (lefschetz) monodromy of their homology loops as one wandered around on the complex line by different paths.

this leads to braid theory as the monodromy groups are very complicated.

other major developments in ym,olifetime were infinite dimensional mmorse theory, and stratified morse theory, the latter dealing with topology of singular spaces, initiated partly by clinton mccrory now at uga, (previously brown), and developed extensively by goresky and macpherson.

 

[mathwonk]

i am a complete novice here, but it seems on first perusal that the ricci flow technique, as it deals with the evolution and topological changes of a riemannian manifold in time, is a modern analog and outgrowth of morse theory.

this of course is the method of hamilton recently credited with the proof of the poincare conjecture, to be discussed in madrid next week at the ICM meeting, and almost surely to result in a fields medal for perelman.

 

[mathwonk]

indeed milnor makes it clear in the introduction to his book that morse theory inspired the solution [by smale] of the poincare conjecture in dimensions > 4.

 

[jmarstonm]

Solomon Lefschetz

wow! thanks for this wonderful post reminding us how much the internet has brought us together.

i used to think morse theory was related to lefschetz theory of pencils in algebraic geometry as well. do you know if professor morse ever spoke with solomon lefschetz? were they in princeton together?

Both my grandfather and Lefschetz were in Princeton at the same time, but their association was likely limited. According to my father and grandmother (Marston Morse's wife), there was tension between professors at Princeton and those involved with the Institute. Although the two institutions are in the same town, Institute and University professors did not necessarily work together or interact unless their own relationships or projects brought them together. Particularly during the early years of the Institute, University faculty may have been "jealous," and I do not know if that is too strong a word or not strong enough, of Institute professors because Institute faculty made more money and did not have to teach while doing research and writing.

On the other hand, one of my aunts clearly remembers Lefschetz as one of a few grownups in Princeton who was kind to her as a child. I understand that she may have this recollection from a time that Lefschetz came to dinner at the Morse household, indicating that my grandfather and Lefschetz did interact, if on a limited basis. My grandmother was (and still is, I believe!) well known in Princeton for bringing the many academics who lived there together with their families to the Morse home on Battle Rd for social events.

JMM