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Liviu I. Nicolaescu

Publications and source records attributed to Liviu I. Nicolaescu.

At least 19 recordsLinked to original sources
arXiv2025

A law of large numbers concerning the distribution of critical points of random Fourier series

On the flat torus $\mathbb{T}^m=\mathbb{R}^m/\mathbb{Z}^m$ with angular coordinates $\vecθ$ we consider the random function $F_R=\mathfrak{a}\big(\, R^{-1} \sqrtΔ\,\big) W$, where $R>0$, $Δ$ is the Laplacian on this flat torus, $\mathfrak{a}$ is an even Schwartz function on $\mathbb{R}$ such that $\mathfrak{a}(0)>0$ and $W$ is the Gaussian white noise on $\mathbb{T}^m$ viewed as a random generalized function. For any $f\in C(\mathbb{T}^m)$ we set \[ Z_R(f):=\sum_{\nabla F_R(\vecθ)=0} f(\vecθ) \] We prove that if the support of $f$ is contained in a geodesic ball of $\mathbb{T}^m$, then the variance of $Z_R(f)$ is asymptotic to $const\times R^{m}$ as $R\to\infty$. We use this to prove that if $m\geq 2$, then as $N\to\infty$ the random measures $N^{-m}Z_N(-)$ converge a.s. to an explicit multiple of the volume measure on the flat torus.

math.PR↗
arXiv2024

A law of large numbers concerning the number of critical points of isotropic Gaussian functions

We investigate the distribution of critical points of certain isotropic random functions $Φ$ on $\mathbb{R}^m$. We show that the distribution of critical points of $Φ(Rx)$, suitably normalized, converge a.s. and $L^2$ as random measures to the (deterministic) Lebesgue measure as $R\to\infty$. We achieve this by producing precise asymptotics of the second moments of these distributions as $R\to\infty$.

math.PR↗
arXiv2016

The Gauss-Bonnet-Chern theorem: a probabilistic perspective

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$.

math.DG↗
arXiv2016

Critical points of multidimensional random Fourier series: central limits

We investigate certain families $X^\hbar$, $0<\hbar \ll 1$, of Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^m_\hbar:=\mathbb{R}^m/(\hbar^{-1}\mathbb{Z} )^m$. We show tha,t for any cube $B\subset \mathbb{R}^m$ of size $<1/2$ and centered at the origin, the number of critical points of $X^\hbar$ in the region $\hbar^{-1}B/(\hbar^{-1}\mathbb{Z} )^m\subset\mathbb{T}^m_\hbar$ has mean $\sim c_1\hbar^{-m}$, variance $\sim c_2\hbar^{-m/2}$, $c_1,c_2>0$, and satisfies a central limit theorem as $\hbar\searrow 0$.

math.PR↗
arXiv2015

Critical points of multidimensional random Fourier series: variance estimates

To any positive number $\varepsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function $u^\varepsilon$ on the $m$-torus $T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \sum_{ν\in\mathbb{Z}^m} X_{ν,\varepsilon} \exp\bigl(\; 2π\varepsilon \sqrt{-1} \;(ν\cdot θ)\;\bigr),$$ where $X_{ν,\varepsilon}$ are complex independent Gaussian random variables with variance $w(\varepsilon|ν|)$. Let $N^\varepsilon$ denote the number of critical points of $u^\varepsilon$. We describe explicitly two constants $C, C'$ such that as $\varepsilon$ goes to the zero, the expectation of the random variable $\frac{1}{{\rm vol}\,(T^m_\varepsilon)}N^\varepsilon$ converges to $C$, while its variance is extremely small and behaves like $C'\varepsilon^{m}$.

math.PR↗
arXiv2015

A stochastic Gauss-Bonnet-Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.

math.PR↗
arXiv2014

Pixelations of planar semialgebraic sets and shape recognition

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.

math.DG↗
arXiv2014

Critical sets of random smooth functions on products of spheres

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the expected number of critical points of a random linear combination of a large number eigenfunctions of the Laplacian on the round sphere, tori, or a products of two round spheres. In the case $M=S^1$ we show that the number of critical points of a trigonometric polynomial of degree $\leq ν$ is a random variable $Z_ν$ with expectation $E(Z_ν)\sim 2\sqrt{0.6}\,ν$ and variance $var(Z_ν)\sim cν$ as $ν\to \infty$, $c\approx 0.35$.

math.DG↗
arXiv2014

Critical sets of random smooth functions on compact manifolds

Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\rightarrow \infty$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ and satisfying the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$ as $m\to \infty$.

math.DG↗
arXiv2014

Complexity of random smooth functions on compact manifolds

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.

math.DG↗
arXiv2014

Random Morse functions and spectral geometry

We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $\varepsilon>0$. We first prove that as $\varepsilon\to 0$ the expected distribution of critical values of this random function approaches a universal measure on $\mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1)\times (m+1)$ symmetric matrices. In contrast, we prove that the metric $g$ and its curvature are determined by the statistics of the Hessians of the random function for small $\varepsilon$.

math.DG↗
arXiv2012

The blowup along the diagonal of the spectral function of the Laplacian

We formulate a precise conjecture about the universal behavior near the diagonal of the spectral function of the Laplacian of a smooth compact Riemann manifold. We prove this conjecture when the manifold and the metric are real analytic, and we also present an alternate proof when the manifold is the round sphere.

math.DG↗
arXiv2012

Combinatorial Morse flows are hard to find

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.

math.GT↗
arXiv2010

On the normal cycles of subanalytic sets

We present a short complete proof of the existence of the normal cycle of a compact subanalytic set. The approach is inspired by some old ides of Joseph Fu, uses Morse theoretic techniques and $o$-minimal topology.

math.DG↗
arXiv2010

Statistics of linear families of smooth functions on knots

Given a knot K in an Euclidean space E and a finite dimensional space V of smooth functions on K, we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V. When V consists of the restrictions to K of homogeneous polynomials of degree d on E, this invariant takes the form of the total curvature of a certain immersion of K. In particular, when K is the unit circle in the plane centered at the origin, then the expected number of critical points of the restriction to K of a random homogeneous polynomial of degree d is $2\sqrt{3d-2}$, and the expected number of critical points on K of a trigonometric polynomial of degree d is approximately 1.549d.

math.DG↗
arXiv2009

Dirac operators on cobordisms: degenerations and surgery

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set $C_t$ of a fixed Morse function defining this cobordism. We show that as we approach the critical level set $C_0$ from above and from below these operators converge in the gap topology to (different) selfadjoint operators $D_\pm$ that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators $D_t$ on the complement of $C_0$ and the Kashiwara-Wall index of a triplet of finite dimensional lagrangian spaces canonically determined by $C_0$.

math.DG↗