Invariances in variance estimates

We provide variants and improvements of the Brascamp-Lieb variance inequality which take into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems

The (B) conjecture for uniform measures in the plane

We prove that for any two centrally-symmetric convex shapes $K,L \subset \mathbb{R}^2$, the function $t \mapsto |e^t K \cap L|$ is log-concave. This extends a result of Cordero-Erausquin, Fradelizi and Maurey in the two dimensional case. Possible relaxations of the condition of symmetry are discussed.Comment: 10 page

The openness conjecture and complex Brunn-Minkowski inequalities

We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Koll\'ar.Comment: This is an account of the results in arXiv:1305.5781 together with some background material. It is based on a lecture given at the Abel symposium in Trondheim, June 2013. 13 page

Optimal Concentration of Information Content For Log-Concave Densities

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with more general sufficient conditions for equality in Theorem 2.3. Also some minor corrections and added reference

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Amp\`ere equation

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.Comment: 49 page

A putative relay circuit providing low-threshold mechanoreceptive input to lamina I projection neurons via vertical cells in lamina II of the rat dorsal horn

Background: Lamina I projection neurons respond to painful stimuli, and some are also activated by touch or hair movement. Neuropathic pain resulting from peripheral nerve damage is often associated with tactile allodynia (touch-evoked pain), and this may result from increased responsiveness of lamina I projection neurons to non-noxious mechanical stimuli. It is thought that polysynaptic pathways involving excitatory interneurons can transmit tactile inputs to lamina I projection neurons, but that these are normally suppressed by inhibitory interneurons. Vertical cells in lamina II provide a potential route through which tactile stimuli can activate lamina I projection neurons, since their dendrites extend into the region where tactile afferents terminate, while their axons can innervate the projection cells. The aim of this study was to determine whether vertical cell dendrites were contacted by the central terminals of low-threshold mechanoreceptive primary afferents. Results: We initially demonstrated contacts between dendritic spines of vertical cells that had been recorded in spinal cord slices and axonal boutons containing the vesicular glutamate transporter 1 (VGLUT1), which is expressed by myelinated low-threshold mechanoreceptive afferents. To confirm that the VGLUT1 boutons included primary afferents, we then examined vertical cells recorded in rats that had received injections of cholera toxin B subunit (CTb) into the sciatic nerve. We found that over half of the VGLUT1 boutons contacting the vertical cells were CTb-immunoreactive, indicating that they were of primary afferent origin. Conclusions: These results show that vertical cell dendritic spines are frequently contacted by the central terminals of myelinated low-threshold mechanoreceptive afferents. Since dendritic spines are associated with excitatory synapses, it is likely that most of these contacts were synaptic. Vertical cells in lamina II are therefore a potential route through which tactile afferents can activate lamina I projection neurons, and this pathway could play a role in tactile allodynia

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page