15,993 reputation
23563
bio website nd.edu/~lnicolae
location University of Notre Dame
age 50
visits member for 3 years, 3 months
seen 6 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.

10h
comment Projective family of probability spaces
@Exterior I do not understand your comment. The arrows do matter. Think of the ring of $2$-adics $\mathbb{Z}_{(2)}$ vs. the ring of $3$-adics $\mathbb{Z}_{(3)}$. They're defined by identical diagrams, but the meaning of arrows is different. The result is nonisomorphic local rings.
10h
comment Projective family of probability spaces
How about the category of probability spaces $(\Omega, \mathscr{O}, P)$ where morphisms are measurable maps $(\Omega_0,\mathscr{O}_0,P_0)\to (\Omega_1,\mathscr{O}_1,P_1)$ such that $f_*P_0=P_1$.
13h
comment Integration currents VS Poincaré Dual
@user69938 I think you need to define things precisely. I think that you use a different notion of Poincare duality. I believe you will find the answer in section 7.3 of the notes www3.nd.edu/~lnicolae/Lectures.pdf
13h
comment A Poincare-Type Inequality and its generalization
@Hacino Missed that.
13h
comment A Poincare-Type Inequality and its generalization
The answer is no in both cases. (The first case is a special case of the 2nd question.) In the first case take $f(\theta)=\cos(2\theta)$ and $\phi(\theta)=1$.
14h
comment Generalize Gauss-Bonnet Formula to non-simple closed curves
What do you mean my non-simple closed curve? Do you mean an immersed closed curve?
14h
revised Integration currents VS Poincaré Dual
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14h
revised Integration currents VS Poincaré Dual
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14h
comment Integration currents VS Poincaré Dual
The last equality that you wrote is the definition of the current $T_S$. Also do not confuse $H^{2r}(M)$ with $\Omega^{2r}(M)$. They are different spaces.
15h
answered Integration currents VS Poincaré Dual
16h
revised Integration currents VS Poincaré Dual
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2d
comment Average probability that a random cosine polynomial with bernoulli coefficients is small
Have you tried using the CLT. The variance of $P_n(t)$ is $s(t)^2=\sum_{k=0}^n \cos^2kt$. Then, for fixed $t$, $\frac{1}{s(t)}P_n(t)$ converges weakly to $N(0,1)$
Mar
29
awarded  dg.differential-geometry
Mar
27
revised Why not develop a Hamiltonian-based Morse theory?
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Mar
27
revised Why not develop a Hamiltonian-based Morse theory?
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Mar
27
revised Why not develop a Hamiltonian-based Morse theory?
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Mar
27
answered Why not develop a Hamiltonian-based Morse theory?
Mar
12
revised Reasons for the Arnold conjecture
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Mar
12
comment Estimates of eigenvalues of elliptic operators on compact manifolds
The passage from scalar to vector bundle Laplacians is not difficult. The pseudodifferentia case has a few peculiarities. Hormander's results are the sharpest. They are discussed in Theorems 2.1, 2.3 of paper arxiv.org/pdf/1406.0934v1.pdf (They are proved in the appendix of the paper.) The references in the paper are also useful.
Mar
11
comment More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya
Another useful reference is Mitrinovic's monograph Analytic inequalities. books.google.com/…