Skills (4)
Education
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Sep 1990–
Jun 1994Michigan State University
Mathematics · PhDUSA · East Lansing
Other
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LanguagesRomanian. English, French, Russian
Questions and Answers (5) View all
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Answer added in Geometry and Topology2 Does anyone knows if a latex like code can be used here, in the publications?The owners of this site need to enable MathJacks. Usually there are various command lines, but they vary from site to site. I learned how to d... [more]
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Answer added in Geometry and Topology7 What's the best book for studying characteristic classes from an geometric point of view?@Raul: Please let me know any suggestion or corrections you deem necessary. I plan to teach this topic again in the Fall semester.
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Answer added in Geometry and Topology7 What's the best book for studying characteristic classes from an geometric point of view?You can try my book, Lectures on the geometry of manifolds, http://nd.edu/~lnicolae/Lectures.pdf In chapter 8 i go through Chern-Weil theory in gr... [more]
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Answer added in Geometry and Topology15 Can you explain a simple method for calculating the singular homology group of a topological space ?Hatcher's book is always a good place to start.. I have used it in my basic topology course. At the link below you will find solutions to some of... [more]
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Answer added in Geometry and Topology3 How would a "Fractional Index Theorem" work? By which I mean a version of the index theorem with fractional values.There may already exist such a theorem, due to Atiyah. It has to do with covering spaces. Check Atiyah's paper, "Elliptic operators, discrete group... [more]
Publications (61) View all
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Article: Complexity of random smooth functions on compact manifolds.II
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ABSTRACT: To each even, nonnegative Schwartz function $w(t)$ on the real axis we associate a gaussian random function on a compact Riemann manifold $(M,g)$ of dimension $m$. We investigate the expected distribution of critical values of such a random function and the behavior of this distribution as we rescale $w(t)$ to $w(\varepsilon t)$ and then let $\varepsilon\to 0$. We prove a central limit theorem describing what happens to the expected distribution of critical values when the dimension of the manifold is very large. Finally, we explain how to use the $\varepsilon\to 0$ behavior of the random function to recover the Riemannian geometry of $(M,g)$.09/2012; -
Article: Combinatorial Morse flows are hard to find
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ABSTRACT: We investigate the probability of detecting combinatorial Morse flows on a simplicial complex via a random search. We prove that it is really small, in a quantifiable way.02/2012; -
Article: Complexity of random smooth functions on compact manifolds. I
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ABSTRACT: We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimension of the manifold is very large.01/2012; -
Article: Pixelations of planar semialgebraic sets and shape recognition
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ABSTRACT: We describe an algorithm that associates to each positive real number $r$ and each finite collection $C_r$ of planar pixels of size $r$ a planar piecewise linear set $S_r$ with the following additional property: if $C_r$ is the collection of pixels of size $r$ that touch a given compact semialgebraic set $S$, then the normal cycle of $S_r$ converges to the normal cycle of $S$ in the sense of currents. In particular, in the limit we can recover the homotopy type of $S$ and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory.09/2011; -
Dataset: FLAT CURRENTS AND THEIR SLICES
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ABSTRACT: I hope this description of flat chains and their slices is less intimidating than Federer's [3], though I follow his very closely.
