Mean curvature self-shrinkers of high genus: Non-compact examples

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in $\mathbb{R}^3$ over a $2\pi/(g+1)$-periodic graph on an equator of the unit sphere $\mathbb{S}^2\subseteq\mathbb{R}^3$, with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted H\"older spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew. Mat

Isometric Immersions and the Waving of Flags

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values into $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. The submanifold result is then used to derive the equations of motion, after equipping the space of isometric immersions with its natural kinetic energy. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.Comment: 25 pages, 1 figur

Job Displacement and Health Outcomes: A Representative Panel Study

We investigate whether job loss as the result of displacement causes ill health. In doing this we use much better data than any previous investigators. Our data are a random 10% sample of the adult population of Denmark for the years 1981-1999. For this large representative panel we have very full records on demographics, health and work status for each person throughout the data period. As well as this we can link every person to a firm (if they are working) and can identify all workers who are displaced in any year, using a variety of definitions of displacement. We focus on one very precise health outcome, hospitalisation for stress related disease, since this is a grave condition and is widely thought to be likely to be associated with unemployment. We use the method of ‘matching on observables’ to estimate the counter-factual of what would have happened to the health of a particular group of displaced workers if they had not in fact been displaced. Our results indicate unequivocally that being displaced in Denmark does not cause hospitalisation for stress related disease. An analysis of the power of our test suggests that even though we are looking for a relatively rare outcome, our data set is large enough to show even quite small an effect if there were any. Supplementary analyses do not show any causal link from displacement or unemployment to our health outcomes for particular groups that might be thought to be more susceptible.unemployment; job displacement; health; matching on observables

A relaxed approach for curve matching with elastic metrics

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of $H^2$-metrics with constant coefficients and scale-invariant $H^2$-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.Comment: 27 page

Exact Calculation of the Product of the Hessian Matrix of Feed-Forward Network Error Functions and a Vector in 0(N) Time

Several methods for training feed-forward neural networks require second order information from the Hessian matrix of the error function. Although it is possible to calculate the Hessian matrix exactly it is often not desirable because of the computation and memory requirements involved. Some learning techniques do, however, only need the Hessian matrix times a vector. This paper presents a method to calculate the Hessian matrix times a vector in O(N) time, where N is the number of variables in the network. This is the same order as the calculation of the gradient to the error function. The usefulness of this algorithm is demonstrated by improvement of existing learning techniques

Non-existence for self-translating solitons

This paper establishes geometric obstructions to the existence of complete, properly embedded, mean curvature flow self-translating solitons $\Sigma^n\subseteq \mathbb{R}^{n+1}$, generalizing previously known non-existence conditions such as cylindrical boundedness.Comment: 15 page

Moduli spaces of 2-stage Postnikov systems

Using the obstruction theory of Blanc-Dwyer-Goerss, we compute the moduli space of realizations of 2-stage Pi-algebras concentrated in dimensions 1 and n or in dimensions n and n+1. The main technical tools are Postnikov truncation and connected covers of Pi-algebras, and their effect on Quillen cohomology.Comment: Version 3: Added conventions in section 1.3. Minor change