On the Chromatic Numbers of R2 and R3 with Intervals of Forbidden Distances

In this paper, we discuss new upper bounds for the chromatic numbers of ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ with intervals of forbidden distances.     

A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals

In this paper, we consider the weak solution of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in ${\mathbb{R}}^3$. When the initial data are of small energy and initial density is positive and essentially bounded, we prove the existence of a global weak solution in ${\mathbb{R}}^3$. The large-time behavior of a global weak solution is also established.     

Lower bounds for blow-up in a model of chemotaxis

Many special cases of the classical Keller–Segel system for modeling chemotaxis have been investigated in the literature, and typically the solution of the governing equations will blow up at some finite time. However, the question of establishing lower bounds for this blow-up time has been largely ignored. This paper derives such a lower bound in a parabolic–parabolic model in both ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$.     

BV weak solutions to Gauss–Codazzi system for isometric immersions

The isometric immersion problem for surfaces embedded into ${\mathbb{R}}^3$ is studied via the fluid dynamic framework introduced in Chen et al. (2010) [6] as a system of balance laws of mixed-type. The techniques developed in the theory of weak solutions of bounded variation in continuum physics are employed to deal with the isometric immersions in the setting of differential geometry. The so-called BV framework is formed that establishes convergence of approximate solutions of bounded variation to the Gauss–Codazzi system and yields the $C^{1,1}$ isometric realization of two-dimensional surfaces into ${\mathbb{R}}^3$. Local and global existence results are established for weak solutions of small bounded variation to the Gauss–Codazzi system for negatively curved surfaces that admit equilibrium configurations. As an application, the case of catenoidal shell of revolution is provided.     

Absence de spectre absolument continu pour un opérateur d'Anderson à potentiel d'interaction générique

We present a result of absence of absolutely continuous spectrum in an interval of $\mathbb{R}$, for a matrix-valued random Schrödinger operator, acting on $L^2(\mathbb{R})?{\mathbb{R}}^N$ for an arbitrary $N?1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the N non-negative Lyapunov exponents of the operator. The method, based upon the formalism of Fürstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit construction of the wanted interval of energies.     

G-exceptional vector bundles on P2 and representations of quivers

It is known that the category of homogeneous bundles on ${\mathbb{P}}^2$ is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of G-exceptional bundles on ${\mathbb{P}}^2$ and to prove that they are stable. We also study the G-exceptionality of Fibonacci bundles on ${\mathbb{P}}^2$.     

Global weak solutions to the Vlasov–Poisson–BGK system for initial data in Lp(R3×R3)

We prove the existence of a global nonnegative weak solution to the Cauchy problem of the Vlasov–Poisson–BGK system for initial datum having finite mass and energy and belonging to $L^p({\mathbb{R}}^3\times {\mathbb{R}}^3)$ with $p>3$.     

Emergent dynamics of Cucker–Smale particles under the effects of random communication and incompressible fluids

We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in ${\mathbb{R}}^d$$(d=2,3)$. Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in ${\mathbb{R}}^2$. In a large coupling regime and periodic spatial domain ${\mathbb{T}}^2:={\mathbb{R}}^2/{\mathbb{Z}}^2$, we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different.