Members Seminar

\(P = W\): a strange identity for \(\mathrm{GL}(2,\mathbb C)\)

Mark deCataldo
Stony Brook University; Member, School of Mathematics
November 24, 2014
Start with a compact Riemann surface \(X\) and a complex reductive group \(G\), like \(\mathrm{GL}(n,\mathbb C)\). According to Hitchin-Simpson's ``non abelian Hodge theory", the pair \((X,G)\) comes with two new complex manifolds: the character variety \(\mathcal M_B\) and the Higgs moduli space \(\mathcal{M}_\text{Dolbeault}\). When \(G= \mathbb C^*\), these manifolds are two instances of the usual first cohomology group of \(X\) with coefficients in the abelian \(\mathbb C^*\).

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.

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.

Apery, irrationality proofs and dinner parties

Francis Brown
Centre national de la recherche scientifique, Institut des Hautes Études Scientifiques
October 27, 2014
After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of \(\zeta(3)\). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game. The talk is intended for a general mathematical audience.

Act globally, compute locally: group actions, fixed points and localization

Tara Holm
Cornell University; von Neumann Fellow, School of Mathematics
October 20, 2014
Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero. This often turns topological questions into combinatorial ones and vice versa. I will give an overview of how this technique arises in symplectic geometry.

Hodge theory, coniveau and algebraic cycles

Claire Voisin
Centre national de la recherche scientifique; Distinguished Visiting Professor, School of Mathematics
October 6, 2014
My talk will be a broad introduction to what is the (mostly conjectural) higher dimensional generalization of Abel's theorem on divisors on Riemann surfaces, namely, the relationship between the structure of the group of algebraic cycles on a complex projective variety and the complexity of its so-called Hodge structures.

A Riemann-Roch theorem in Bott-Chern cohomology

Jean-Michel Bismut
Université Paris-Sud
April 21, 2014
If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image.

Toroidal Soap Bubbles: Constant Mean Curvature Tori in \(S^3\) and \(R^3\)

Emma Carberry
University of Sydney
April 14, 2014
Constant mean curvature (CMC) tori in \(S^3\), \(R^3\) or \(H^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes. I will address these periodicity questions for the spherical and Euclidean cases, using Whitham deformations, which I will explain.

Gambling, Computational Information, and Encryption Security

Bruce Kapron
University of Victoria; Member, School of Mathematics
March 24, 2014
We revisit the question, originally posed by Yao (1982), of whether encryption security may be characterized using computational information. Yao provided an affirmative answer, using a compression-based notion of computational information to give a characterization equivalent to the standard computational notion of semantic security. We give two other equivalent characterizations.