Communications in Contemporary Mathematics

This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to $2$-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-representation theory of $2$-groups, as studied by the second author.

In this paper, we are concerned with the existence of multi-bump solutions for the following class of $p(x)$-Laplacian equations:

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a $C^{1,1}$ domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs equilibrium measures. In this paper, we prove that in a certain limiting regime of high temperature the classical KMS condition can be derived from the quantum condition in the simple case of the Bose–Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden–Thompson inequality, Bogoliubov inequality and semiclassical analysis.

In this note, we study the maximal perimeter of a convex set in ${ℝ}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on ${ℝ}^n$, satisfying very mild assumptions, there exists a convex set of $\mu$-perimeter at least $C\frac{\sqrt{n}}{\sqrt[4]{\mathrm{\text{Var}}\left|X\right|}\sqrt{𝔼\left|X\right|}}.$ This implies, in particular, that for any isotropic log-concave measure $\mu$, one may find a convex set of $\mu$-perimeter of order $n^{\frac{1}{8}}$. Secondly, we derive a general upper bound of $Cn\left|\right|f|{|}_{\infty }^{\frac{1}{n}}$ on the maximal perimeter of a convex set with respect to any log-concave measure with density $f$ in an appropriate position.

Our lower bound is attained for a class of distributions including the standard normal distribution. Our upper bound is attained, say, for a uniform measure on the cube.

In addition, for isotropic log-concave measures, we prove an upper bound of order $n^2$ for the maximal $\mu$-perimeter of a convex set.

Let $X\subseteq {\mathbb{P}}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo–Mumford regularity $\mathrm{\text{reg}}({{𝒪}}_X)\le d-e$, as predicted by the Eisenbud–Goto regularity conjecture. Such a bound fails for general projective varieties by a recent result of McCullough–Peeva. The main techniques are Noma’s classification of non-degenerate projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_0^m(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is any bounded, smooth, open subset of ${ℝ}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

We discuss a version of the Chevalley–Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in $S^2\times S^2$ with various rigidities.

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed $v$, traveling wave solutions exist and converge to the zero wave as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L_x^2{H}_y^s({ℝ}^2)$, with $s>\frac{1}{2}$. The critical case $s=\frac{1}{2}$ still stands as an interesting open problem.