Daniel Spielman: Ramanujan graphs of every degree

6th November, 2014

We explain what Ramanujan graphs are, and prove that there exist infinite families of bipartite Ramanujan graphs of every degree. Our proof follows a plan suggested by Bilu and Linial, and exploits a proof of a conjecture of theirs about lifts of graphs. Our proof of their conjecture applies the method of interlacing families of polynomials to Mixed Characteristic Polynomials. A bound on the roots of these polynomials will follow from a bound of Heilmann and Lieb on the roots of the matching polynomials of graphs. We also prove that there exist infinite families of irregular bipartite Ramanujan graphs.

Daniel Spielman: The solution of the Kadison-Singer problem

5th November, 2014

We will explain our recent solution of the Kadison-Singer Problem and the equivalent Bourgain-Tzafriri and Paving Conjectures. We will begin by introducing the method of interlacing families of polynomials and use of barrier function arguments to bound the roots of polynomials. To prove the Paving Conjecture, we introduce the Mixed Characteristic Polynomial of a collection of matrices, and use the theory of Real Stable polynomials and multivariate generalizations of the barrier function arguments to bound their roots.

Daniel Spielman: Sparsification of graphs and matrices

3rd November, 2014

Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be approximated by a sparse graph. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor. Our algorithms follow from the solution of a problem in linear algebra. Given an expression for a rank-n symmetric matrix A as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.

Alessia Mandini: On the Gromov width of polygon spaces

31st October, 2014

After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold (M,ω) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in (M,ω).

I will discuss techniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in ℝ3 with edges of lengths (r1,…,rn). Under some genericity assumptions on lengths ri, the polygon space is a symplectic manifold. After introducing this family of manifolds, I will concentrate on the spaces of 5-gons and calculate their Gromov width.

Edward Belbruno: Designing low-energy capture transfers for spacecraft to the Moon and Mars

28th October, 2014

In 1991 a new type of transfer to the Moon was operationally demonstrated by the Japanese spacecraft, Hiten, using ballistic capture. It was designed by this speaker and James Miller. This is capture about the Moon which is automatic so that no rocket engines are required. It was accomplished due to the existence of regions in phase space called weak stability boundaries, where ballistic capture occurs. These are complex fractal regions of unstable chaotic motion. Until recently it was thought that such a transfer to Mars was not feasible. Recent work by Francesco Topputo and this speaker has shown that a new type of ballistic capture transfer exists to Mars, with interesting implications. It will be described how to design these transfers given realistic constraints and why is so challenging.

James Pascaleff: Equivariant structures in mirror symmetry

17th October, 2014

When a variety X is equipped with the action of an algebraic group G, it is natural to study the G-equivariant vector bundles or coherent sheaves on X. When X furthermore has a mirror partner Y, one can ask for the corresponding notion of equivariance in the symplectic geometry of Y. The infinitesimal notion (equivariance for a single vector field) was introduced by Seidel and Solomon (GAFA 22 no. 2), and it involves identifying a vector field with a particular element in symplectic cohomology. I will describe the analogous situation for a Lie algebra of vector fields, and discuss the application of this theory to mirror symmetry of flag varieties. In this situation, we expect to find a close connection to the canonical bases of Gross-Hacking-Keel.

Laura Starkston: Symplectic fillings and star surgery

25th September, 2014

Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations called star surgery to construct examples of exotic 4-manifolds.