Fine properties of functions of bounded deformation - an approach via linear PDES

In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are L1-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems

Spectral inequalities in quantitative form

We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian.Comment: 71 pages, 4 figures, 6 open problems, 76 references. This is a chapter of the forthcoming book "Shape Optimization and Spectral Theory", edited by Antoine Henrot and published by De Gruyte

STEM graduates and secondary school curriculum: does early exposure to science matter?

Increasing the number of Science, Technology, Engineering and Math (STEM) university graduates is considered a key element for long-term productivity and competitiveness in the global economy. Still, little is known about what actually drives and shapes students' choices. This paper focusses on secondary school students at the very top of the ability distribution and explores the effect of more exposure to science on enrolment and persistence in STEM degrees at the university and on the quality of the university attended. The paper overcomes the standard endogeneity problems by exploiting the different timing in the implementation of a reform that induced secondary schools in the UK to offer more science to high ability 14 year-old children. Taking more science in secondary school increases the probability of enrolling in a STEM degree by 1.5 percentage point and the probability of graduating in these degrees by 3 percentage points. The results mask substantial gender heterogeneity: while girls are as willing as boys to take advanced science in secondary school - when offered -, the effect on STEM degrees is entirely driven by boys. Girls are induced to choose more challenging subjects, but still the most female-dominated ones

Essential connectedness and the rigidity problem for Gaussian symmetrization

We provide a geometric characterization of rigidity of equality cases in Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer's definition of indecomposable current.Comment: 38 page

Non-collapsed spaces with Ricci curvature bounded from below

\u2014 We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding\u2019s volume convergence theorem and of Cheeger-Colding dimension gap estimate for RCD spaces. In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence