bio  website  nd.edu/~lnicolae 

location  University of Notre Dame  
age  49  
visits  member for  2 years, 8 months 
seen  37 mins ago  
stats  profile views  5,497 
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 
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Morse theory Vs degree theory
Let us continue this discussion in chat. 
Sep 12 
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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 
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Morse theory Vs degree theory
There is no third critical point on $S^2$. 
Sep 12 
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Operator theory of initialvalue 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 HilleYosida theorem. 
Sep 12 
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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 
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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 
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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 
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What is the Implicit Function Theorem good for?
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Sep 11 
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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?
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Sep 9 
answered  Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians? 
Sep 9 
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$L^p$ 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 
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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
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Aug 12 
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Geometric Proof of the Hirzebruch Signature Theorem (i.e. using Connections)
The heat kernel proof of the AtiaySinger index theorem yields as a special case Hirzebruch's signature theorem. The proof is purely analytical and uses only the ChernWeil 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 Ktheory using Morse functions? 
Aug 12 
asked  Almost sure transversality of smooth random maps 
Aug 8 
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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 pingpong game.) 