We obtain novel criteria for the existence of local bifurcation for periodic solutions of Hamiltonian systems by a comparison principle of the spectral flow. Our method allows to find the appearance of new solutions by a simple inspection of the coefficients of the system.

This paper proves a convergence theorem for the push-forward Wiener measures on holonomy groups via stochastic parallel transports along convergent metric connections.

Let [Formula: see text] be a [Formula: see text]-smooth Riemannian surface. A classical theorem in differential geometry states that the Gauss curvature function [Formula: see text] vanishes everywhere if and only if the surface is locally isometric to the Euclidean plane. We give an asymptotically sharp quantitative version of this theorem with respect to an isothermal coordinate chart. Roughly speaking, we show that if [Formula: see text] is a Riemannian disc of radius [Formula: see text] with [Formula: see text] for some [Formula: see text], then there is an isothermal coordinate map from [Formula: see text] onto an Euclidean disc of radius [Formula: see text] which is bi-Lipschitz with constant [Formula: see text].

We deal with weighted Hardy–Sobolev type inequalities for functions on [Formula: see text], [Formula: see text]. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

We show that a uniformly Euclidean metric with isolated singularity on closed [Formula: see text], where [Formula: see text] or [Formula: see text], [Formula: see text] spin, and non-negative scalar curvature on the smooth part is flat and extends smoothly over the singularity. This confirms Schoen’s Conjecture in these cases. The novel approach here, which is the key to the proof, is to show that the space has non-negative synthetic Ricci curvature, i.e. an [Formula: see text] space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.

In this paper, let [Formula: see text] and [Formula: see text] be the potential along the moment curve [Formula: see text] ([Formula: see text]) defined by setting, for any suitable function [Formula: see text], [Formula: see text] The authors obtain some equivalent characterizations via capacities of the embedding [Formula: see text] from Lebesgue spaces [Formula: see text] associated with the [Formula: see text]-dimensional Lebesgue measure to [Formula: see text] endowed with the non-negative Radon measure [Formula: see text] in the form of [Formula: see text] for a weight [Formula: see text]. The main ingredient of this paper lies in the different approach to the estimate of the capacity of the anisotropic cubes [Formula: see text] along [Formula: see text].

In this paper we survey some recent results concerning scattering and non-scattering in the context of the linear Helmholtz equation and inhomogeneities of nontrivial contrast. We examine isotropic as well as anisotropic media. Part of the survey deals with the so-called transmission spectrum, namely those wave numbers at which non-scattering potentially may occur. For wave numbers that are not transmission eigenvalues any incident wave leads to scattering, however, being at a transmission eigenvalue is far from sufficient to guarantee the occurence of non-scattering for even a single incident wave. For instance the inhomogeneity generically has to be smooth for non-scattering to occur. Similarly many smooth geometric shapes will be scattering for natural incident waves even at a transmission eigenvalue. Part of the survey discusses recent results of that nature.

We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and only if the mappings in the corresponding homotopy classes are homotopic on a codimension one skeleton of a triangulation of the domain. When this is the case, the heterotopic energy of a mapping is the sum of its Sobolev energy and its disparity energy, defined as the minimum energy of a bubble to pass between these homotopy classes. At the more technical level, we rely on a framework that works when the target and domain manifolds are not simply connected and there is no canonical isomorphism between homotopy groups with different basepoints.

Mean Field Game (MFG) models implicitly assume “rational expectations”, meaning that the heterogeneous agents being modeled correctly know all relevant transition probabilities for the complex system they inhabit. When there is common noise, it becomes necessary to solve the “Master equation”, in which the infinite-dimensional density of agents is a state variable. The rational expectations assumption and the implication that agents solve Master equations is unrealistic in many applications. We show how to instead formulate MFGs with non-rational expectations. Departing from rational expectations is particularly relevant in “MFGs with a low-dimensional coupling”, i.e. MFGs in which agents’ running reward function depends on the density only through low-dimensional functionals of this density. This happens, for example, in most macroeconomics MFGs in which these low-dimensional functionals have the interpretation of “equilibrium prices.” In MFGs with a low-dimensional coupling, departing from rational expectations allows for completely sidestepping the Master equation and for instead solving much simpler finite-dimensional HJB equations. We introduce an adaptive learning model as a particular example of non-rational expectations and discuss its properties.