15,528 reputation
23262
bio website nd.edu/~lnicolae
location University of Notre Dame
age 49
visits member for 2 years, 11 months
seen 14 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.

2d
comment Proof of Kolomogrov-Sinai Theorem
en.wikipedia.org/wiki/Measure-preserving_dynamical_system
2d
comment Reason for the Choice of Line Parameters in the Radon Transform
@Manfred Weis I get it. First,you need to be careful with the concept of distribution/generalized function: on a manifold they are different concepts. That aside, check chapter I of Helgason's book The Radon transform. He computes the Radon transform of the delta-function in section 5 and obtains an invariant description: it is the invariant measure supported along the curve defined by the lines passing through the origin. This description is clearly coordinate independent.
2d
comment Reason for the Choice of Line Parameters in the Radon Transform
@ManfredWeis I do not understand what you mean by the Radon transform of a point. The Radon transform, as I know it, maps (generalized) functions to (generalized) functions.
Dec
17
comment How do these two extensions of Sobolev spaces relate to each other?
The only periodic $L^2$-function on $\mathbb{R}$ is the trivial function. However, for $\beta\geq 0$, any function in $H^\beta_{per}$ is the restriction of a function in $H^\beta(\mathbb{R})$.
Dec
17
comment Cap product à la Poincaré
The cap product $\cap: H^k\times H_m\to H_{m-k}$is defined for any (reasonably) behaved space. Only its intersection theoretic interpretation requires the space be oriented manifold.
Dec
16
comment Schoenberg correspondence on $L^p$
Bochner's theorem on infinite dimensional spaces (such as Banach spaces, or more generally, locally convex spaces) can be tricky. Also, in numerical analysis, there is a condition of conditional positivity that looks a bit different than what you wrote; see e.g. the book Scattered Data Approximation, by H. Wendland.
Dec
15
comment Reason for the Choice of Line Parameters in the Radon Transform
Since the group of isometries of the plane acts transitively on the space of affine lines, up to a multiplicative constant, there is only one measure on the space of affine lines that is invariant with respect to this action. This is the measure used in the Radon formula, and the precise form of the parametrization of the space of affine lines is irrelevant.
Dec
15
revised Measuring the Randomness and Statistics of Convex Polygons
added 19 characters in body
Dec
15
answered Measuring the Randomness and Statistics of Convex Polygons
Dec
9
answered Uhlenbeck's theorem novelty
Dec
9
comment Using Stokes' theorem to define “area” enclosed by a curve
@BrunoLeFloch Sometimes, the projection of the curve onto a $2$-plane can look like a figure $8$. In this case, integrating $xdy-ydx$ over the figure $8$ we can get $0$.
Dec
8
comment Extension of solutions of PDEs with constant coefficients
If $f$ admits an extension to an entire function, then $f$ must be real analytic. Requiring $\mathcal{L}$ to be elliptic will guarantee real analyticity. However you need to phrase your question more carefully. Should we expect all functions in $\ker \mathcal{L}$ to extend or only some of them?
Dec
6
comment Different definitions of spin structures
The Donaldson-Kronheimer definition gathers in it all the facts about a spin structure that are needed in gauge theory. They all follow from the representation theory of Spin(4). In particular, they are needed to define the Dirac operator.
Dec
2
comment Parallel forms and cohomology of symmetric spaces
See Section 7.4.2 of these lectures www3.nd.edu/~lnicolae/Lectures.pdf
Dec
1
comment Variance of the roots of a complex polynomial
Another not so helpful comment:$\sum_k |\alpha_k|^2$ is a semialgebraic function in the variables $a_j,\bar{a}_k$.
Dec
1
comment Variance of the roots of a complex polynomial
It suffices to solve the problem for $\sum_k|\alpha_k|^2$, not that it is any easier.
Dec
1
comment Perron-Frobenius theory for reducible matrices
math.miami.edu/~armstrong/685fa12/…
Nov
30
reviewed Leave Open Riemann Mapping Theorem in Higher Dimensions for Continuous funcions
Nov
30
reviewed Leave Open Generalization of Jordan Curve Theorem
Nov
29
awarded  Notable Question