14,258 reputation
22754
bio website nd.edu/~lnicolae
location University of Notre Dame
age 49
visits member for 2 years, 8 months
seen 37 mins ago
I do mostly geometry and topology, with an analytic bias. For the past few years Morse theory has popped up in my research, but not in a conventional way.

1h
answered Using Discrete Morse Theory to represent hom classes
Sep
13
comment Morse theory Vs degree theory
Let us continue this discussion in chat.
Sep
12
comment Morse theory Vs degree theory
To see that there are only two critical points you don't need Morse theory. It is a very, very simple exercise involving Lagrange multipliers. In any case a point $p$ is critical point for the altitude if and only if the tangent plane at that point horizontal. There are only two such points.
Sep
12
comment Morse theory Vs degree theory
There is no third critical point on $S^2$.
Sep
12
comment Operator theory of initial-value ODE problems
For ordinary differential equations you do not need to go to infinite dimensional vector spaces. It suffices to work with usual matrices. For partial differential evolution equations there you can look at Hille-Yosida theorem.
Sep
12
comment Morse theory Vs degree theory
Let $S^2$ be the unit sphere in $\mathbb{R}^3$, $x^2+y^2+z^2=1$. The function $f: S^2\to\mathbb{R}$ that associates to a point $p\in S^2$ its altitude $z=z(p)$ has exactly two critical points, a maximum (the North Pole $(0,0,1)$ and a minimum (the South Pole $(0,0,-1)$. This can be seen by observing that these are the only two points on the sphere where the tangent plane is horizontal, i.e., parallel with the $xy$-plane.
Sep
12
comment Morse theory Vs degree theory
@ Jean I think that in your example you need to replace $S^2$ with a surface $\Sigma$ of positive genus so that $H_1(\Sigma)\neq 0$. On $S^2$ there exist Morse functions with exactly two critical points, a minimum and a maximum and no saddle points. (Take for example the height function.)
Sep
12
comment Morse theory Vs degree theory
That's the point. Degree theory is ineffective in this example, while Morse theory has plenty to say. The lower bounds on the number of solutions of $\nabla g=0$ are obtained using homology theory.
Sep
12
answered Morse theory Vs degree theory
Sep
12
revised What is the Implicit Function Theorem good for?
edited body
Sep
11
comment Morse theory Vs degree theory
Chang refers to variational equations. For a long time the variational methods were the only ones used to find solutions to (variational) equations. Judging by the number of publications, degree theory does not have as many applications as the variational techniques.
Sep
11
revised Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?
added 3 characters in body
Sep
9
answered Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?
Sep
9
comment $L^p$ norm means
Section 1.8 from the book Special functions by Andrews, Askey and Roy might help. It describes Dirichlet's methods mentioned by WillJagy and gives several useful consequences.
Sep
4
comment Moments of the trace of orthogonal matrices
For $n=3$ you could try using Weyl's integral formula which reduces to the computation of an integral over the maximal torus of $O(3)$. This is an $S^1$ and there is a greater hope you can say something concrete.
Aug
18
revised Homotopy equivalent Morse functions
deleted 3 characters in body
Aug
12
comment Geometric Proof of the Hirzebruch Signature Theorem (i.e. using Connections)
The heat kernel proof of the Atiay-Singer index theorem yields as a special case Hirzebruch's signature theorem. The proof is purely analytical and uses only the Chern-Weil definition of characteristic classes. You can read the details in Roe's book Elliptic operators, topology and asymptotic methods.
Aug
12
answered Is there an effective way to calculate K-theory using Morse functions?
Aug
12
asked Almost sure transversality of smooth random maps
Aug
8
comment Distribution of dropped objects
Perhaps a damping rate ought to be assumed, e.g., the sphere loses a fixed fraction of its energy anytime it touches the ground. Also, in realistic models, there is a difference between a hard sphere and a point particle. The sphere could be spinning, and this affects the bouncing direction. (Think of a professional ping-pong game.)