Shot-noise random fields: some geometric properties and some applications for images

Agnès Desolneux
École normale supérieure de Cachan; Member, School of Mathematics
November 10, 2014
Shot-noise random fields can model a lot of different phenomena that can be described as the additive contributions of randomly distributed points. In the first part of the talk, I will give some properties of these random fields. And in a second part, I will discuss an application to images with the texture synthesis problem. The talk is intended for a wide mathematical audience: I will recall the needed notions of probability and show a lot of illustrations.

Asymptotic expansions of the central limit theorem and its applications

Anindya De
Center for Discrete Mathematics and Theoretical Computer Science; Visitor, School of Mathematics
November 11, 2014
In its simplest form, the central limit theorem states that a sum of n independent random variables can be approximated with error \(O(n^{-1/2})\) by a Gaussian with matching mean and second moment (given these variables are not too dissimilar). We prove an extension of this theorem where we achieve error bounds of \(O(n^{-(s-1)/2})\) by matching \(s\) moments (for any \(s > 0\)). While similar results were known in literature before this, the only explicit error bounds were known when the summands are continuous i.i.d. variables.

Zarhin's trick and geometric boundedness results for K3 surfaces

François Charles
Université Paris-Sud
November 11, 2014
Tate's conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin's trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

Universal Chow group of zero-cycles on cubic hypersurfaces

Claire Voisin
Centre national de la recherche scientifique; Distinguished Visiting Professor, School of Mathematics
November 12, 2014
We discuss the universal triviality of the \(\mathrm{CH}_0\)-group of cubic hypersurfaces, or equivalently the existence of a Chow-theoretic decomposition of their diagonal. The motivation is the study of stable irrationality for these varieties. Our main result is that this decomposition exists if and only if it exists on the cohomological level.

Hyperbolic groups, Cannon-Thurston maps, and hydra

Timothy Riley
Cornell University; Member, School of Mathematics
November 17, 2014
Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of "Cannon-Thurston maps" between boundaries. Also, I will give examples where this map between boundaries fails to be defined. This is joint work with O. Baker.

Toric origami manifolds and origami templates

Tara Holm
Cornell University; von Neumann Fellow, School of Mathematics
November 18, 2014
A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope.