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Farid Madani

Publications and source records attributed to Farid Madani.

17 recordsLinked to original sources
arXiv2024

Exploring quantum criticality in a 4D quantum disordered system

Phase transitions are prevalent throughout physics, spanning thermal phenomena like water boiling to magnetic transitions in solids. They encompass cosmological phase transitions in the early universe and the transition into a quark-gluon plasma in high-energy collisions. Quantum phase transitions, particularly intriguing, occur at temperatures near absolute zero and are driven by quantum fluctuations rather than thermal ones. The strength of the fluctuations is very sensitive to the dimensionality of the physical systems, which determines the existence and nature of phase transitions. Low-dimensional systems often exhibit suppression of phase transitions, while high-dimensional systems tend to exhibit mean-field-like behavior. The localization-delocalization Anderson transition stands out among quantum phase transitions, as it is thought to retain its non-mean-field character across all dimensions. This work marks the first observation and characterization of the Anderson transition in four dimensions using ultracold atoms as a quantum simulator with synthetic dimensions. We characterize the universal dynamics in the vicinity of the phase transition. We measure the critical exponents describing the scale-invariant properties of the critical dynamics, which are shown to obey Wegner's scaling law. Our work is the first experimental demonstration that the Anderson transition is not mean-field in dimension four.

cond-mat.dis-nn↗
arXiv2019

LcK structures with holomorphic Lee vector field on Vaisman-type manifolds

We give a complete description of all locally conformally Kähler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex manifold, any lcK metric with potential and with holomorphic Lee vector field admits a potential which is positive and invariant along the anti-Lee vector field.

math.DG↗
arXiv2019

On Weyl-reducible conformal manifolds and lcK structures

A recent result of M. Kourganoff states that if $D$ is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold $M$, then the universal covering of $M$, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of $D$, is isometric to $\mathbb{R}^q\times N$ for some irreducible, incomplete Riemannian manifold $N$. Moreover, he characterized the case where the dimension of $N$ is $2$ by showing that $M$ is then a mapping torus of some Anosov diffeomorphism of $T^{q+1}$. We show that in this case one necessarily has $q=1$ or $q=2$.

math.DG↗
arXiv2018

The Equivariant Second Yamabe Constant

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.

math.DG↗
arXiv2016

On toric locally conformally Kähler manifolds

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

math.DG↗
arXiv2016

Conformally related Kähler metrics and the holonomy of lcK manifolds

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of conformal classes on compact manifolds containing two non-homothetic Kähler metrics. The second one is the classification of compact Einstein locally conformally Kähler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compact locally conformally Kähler manifolds. We show that every locally (but not globally) conformally Kähler compact manifold of dimension $2n$ has holonomy $\mathrm{SO}(2n)$, unless it is Vaisman, in which case it has restricted holonomy $\mathrm{SO}(2n-1)$. We also show that the restricted holonomy of a proper globally conformally Kähler compact manifold of dimension $2n$ is either $\mathrm{SO}(2n)$, or $\mathrm{SO}(2n-1)$, or $\mathrm{U}(n)$, and we give the complete description of the possible solutions in the last two cases.

math.DG↗
arXiv2015

S^1-equivariant Yamabe invariant of 3-manifolds

We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold.

math.DG↗
arXiv2014

A tropical characterization of complex analytic varieties to be algebraic

In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the variety is algebraic. This gives a converse of a theorem of Bieri and Groves and generalizes a result proven in \cite{MN2-11}. More precisely, if the dimension of the ambient space is at least twice of the dimension of the generic analytic subvariety, then these properties are equivalent to the volume of the amoeba of the subvariety being finite.

math.AG↗
arXiv2012

On the Volume of Complex Amoebas

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.

math.AG↗
arXiv2012

Generalized logarithmic Gauss map and its relation to (co)amoebas

We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.

math.AG↗
arXiv2011

A construction of conformal-harmonic maps

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky-Viaclovsky and Lamm.

math.DG↗
arXiv2011

Analytic varieties with finite volume amoebas are algebraic

In this paper, we study the amoeba volume of a given $k-$dimensional generic analytic variety $V$ of the complex algebraic torus $(\C^*)^n$. When $n\geq 2k$, we show that $V$ is algebraic if and only if the volume of its amoeba is finite. In this precise case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the $k-$linear spaces will be given.

math.AG↗
arXiv2009

Equivariant Yamabe problem and Hebey-Vaugon conjecture

In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey--Vaugon conjecture in some new cases.

math.DG↗
arXiv2009

Le probléme de Yamabe avec singularités et la conjecture de Hebey-Vaugon

In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not necessarily the usual regularity (it can be $C^1$). To solve this problem, we start the study of the Yamabe type equations. We show that all the known properties in the smooth case are still valid. Under some assumptions, we prove the existence and uniqueness of solutions. The second part is dedicated to the Hebey-Vaugon conjecture, stated in their paper about the equivariant Yamabe problem. We prove that this conjecture is true in some new cases, after we generalize T. Aubin's theorem.

math.DG↗
arXiv2009

The Yamabe problem with singularities

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $φ$ solution of the following Yamabe type equation Δφ+ hφ= \tilde h φ^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde h\in \mathbb R$. We give the regularity of $φ$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature of $g$.

math.AP↗