Bulletin of Mathematical Sciences

We study a possible Majorana representation $${{\mathcal R}}$$ of the alternating group A6 of degree 6 such that for some involutions s and t in A6, generating a D6-subgroup, the corresponding Majorana axes a s and a t generate a subalgebra of type 3C. We show that there exists at most one such representation $${{\mathcal R}}$$ and that its dimension is at most 70. The representation $${{\mathcal R}}$$ does not correspond to a subalgebra in the Monster algebra generated by a subset of the Majorana axes canonically indexed by the involutions of an A6-subgroup in the Monster.

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan–Kinderlehrer–Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón’s problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. 65–73, 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?

In this survey paper, we consider variational problems involving the Hardy–Schrödinger operator $$L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}$$ on a smooth domain $$\Omega $$ of $$\mathbb {R}^n$$ with $$0\in \overline{\Omega }$$ , and illustrate how the location of the singularity 0, be it in the interior of $$\Omega $$ or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli–Kohn–Nirenberg inequalities. The latter can be stated as: $$\begin{aligned} C\left( \int _{\Omega }\frac{u^{2^*(s)}}{|x|^s}dx\right) ^{\frac{2}{2^*(s)}}\le \int _{\Omega } |\nabla u|^2dx-\gamma \int _{\Omega }\frac{u^2}{|x|^2}dx \quad \hbox {for all } u\in H^1_0(\Omega ), \end{aligned}$$ where $$\gamma <\frac{n^2}{4},\, s\in [0,2)$$ and $$2^\star (s):=\frac{2(n-s)}{n-2}$$ . We address questions regarding the explicit values of the optimal constant $$C:=\mu _{\gamma , s}(\Omega )$$ , as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties often lead to situations where the best constants $$\mu _{\gamma , s}(\Omega )$$ do not depend on the domain, and hence they are not attainable. We consider two different approaches for “breaking the homogeneity” of the problem, and restoring compactness. One approach was initiated by Brezis–Nirenberg, when $$\gamma =0$$ and $$s=0$$ , and by Janelli, when $$\gamma >0$$ and $$s=0$$ . It is suitable for the case where the singularity 0 is in the interior of $$\Omega $$ , and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub–Kang for $$\gamma =0,\, s>0$$ , and by C.S. Lin et al. and Ghoussoub–Robert, when $$\gamma \ne 0, s\ge 0$$ . It consists of considering domains, where the singularity 0 is on the boundary. Both of these approaches are rich in structure and in challenging problems. If $$0\in \Omega $$ , then a negative linear perturbation suffices for higher dimensions, while a positive “Hardy-singular interior mass” theorem for the operator $$L_\gamma $$ is required in lower dimensions. If the singularity 0 belongs to the boundary $$\partial \Omega $$ , then the local geometry around 0 (convexity and mean curvature) plays a crucial role in high dimensions, while a positive “Hardy-singular boundary mass” theorem is needed for the lower dimensions. Each case leads to a distinct notion of critical dimension for the operator $$L_\gamma $$ .

We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, , has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter . The goal is to prove estimates for these operators and, as a consequence, to obtain estimates for the canonical Cauchy–Szegö and Bergman projection operators (which are not of Cauchy–Fantappié type).

We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this second part include absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato’s ultimate Trotter Product Formula.

We indicate how recent work of Figalli–Kim–McCann and Vetois can be used to improve previous results of Trudinger and Wang on classical solvability of the second boundary value problem for Monge–Ampère type partial differential equations arising in optimal transportation together with the global regularity of the associated optimal mappings.

In this paper, we investigate the structure of finite groups that are products of two supersolvable groups and gain a sufficient condition for a group to be supersolvable. Our main theorem is the following: Let the group $$G=HK$$ be the product of the subgroups $$H$$ and $$K$$ . Assume that $$H$$ permutes with every maximal subgroup of $$K$$ and $$K$$ permutes with every maximal subgroup of $$H$$ . If $$H$$ is supersolvable, and $$K$$ is nilpotent and $$K$$ is $$\delta $$ -permutable in $$H$$ , where $$\delta $$ is a complete set of Sylow subgroups of $$H$$ , then $$G$$ is supersolvable. Some known results are generalized.