14,078 reputation
22754
bio website nd.edu/~lnicolae
location University of Notre Dame
age 49
visits member for 2 years, 8 months
seen 2 hours 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.

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.)
Aug
5
comment Smooth normal forms of vector fields (the path method)
Have look at Brjuno's paper, Normal form of differential equations with a small parameter. (Russian) Mat. Zametki 16 (1974), 407–414 and also his famous work, Analytic form of differential equations. I, II. (Russian) Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239.
Jul
14
awarded  Good Question
Jul
9
awarded  Popular Question
Jul
9
comment Combinatorial Morse functions and random permutations
A most satifying answer. Beautiful.
Jul
9
awarded  Nice Answer
Jul
8
comment Bases for spaces of smooth functions
As basis of $C^\infty(M)$ you can choose the collection of eigenfunctions of the Laplacian of a Riemann metric on $M$.
Jul
2
awarded  Curious
Jun
5
comment How to prove this Weitzenbock formula?
Have a look at section 1.4.2 in www3.nd.edu/~lnicolae/swnotes.pdf
Jun
2
comment $C^{2}$ regularity of a curve of solutions to a family of elliptic equations
You can use the elliptic a priori estimates which imply that for any $k\geq 1$ there exists $C_k>0$ such that for any smooth $u$ that vanishes along the boundary we have $\Vert u\Vert_{H^{k+2}}\leq C_k\Vert \Delta u\Vert_{H^k}$. The map $f_s\to u_s$ is linear and bounded $H^k\to H^{k+2}$. In particular this implies that the map $s\mapsto u_s$ is smooth as the composition of two smooth maps.
May
30
revised The Gauss-Bonnet theorem for Sheaves
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May
30
answered The Gauss-Bonnet theorem for Sheaves
May
28
revised Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?
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May
22
awarded  Nice Answer
May
22
revised How to understand Chern-Simons action
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May
16
comment Everywhere differentiable function that is nowhere monotonic
@ GH from MO: Thanks. Since the derivative of a function satisfies the Darboux property, the question is equivalent to asking whether there exists a differentiable function with a dense set of critical points.