Singular moduli spaces and Nakajima quiver varieties

Giulia Saccà
Member, School of Mathematics
October 28, 2014
The aim of this talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations.

Designing low energy capture transfers for spacecraft to the Moon and Mars

Edward Belbruno
Princeton University and Innovative Orbital Design, Inc.
October 28, 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.

Mirror symmetry & Looijenga's conjecture

Philip Engel
Columbia University
October 29, 2014
A cusp singularity is an isolated surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In the 1980's Looijenga conjectured that a cusp singularity is smoothable if and only if the minimal resolution of the dual cusp is the anticanonical divisor of some rational surface. This conjecture can be related to the existence of certain integral affine-linear structures on a sphere.

Exhibit and Special Event: Mark Podwal

Mark Podwal
October 30, 2014
"All This Has Come Upon Us" is a collection of archival pigment prints by Mark Podwal, a selection of which is on display, which illustrate the tragedies and injustices suffered by the Jewish people. Each work, conceived as a page of a book, illuminates the saying that "Misfortune seldom misses a Jew," but as Podwal's powerful work attests, the Jews sustained their extraordinary faith and persevered. This event features a film about the making of the artworks, followed by a question and answer session with Podwal.

On the Gromov width of polygon spaces

Alessia Mandini
University of Pavia
October 31, 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 \((M, \omega)\) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in \((M, \omega)\).

Information percolation for the Ising model

Eyal Lubetzky
New York University
November 3, 2014
We introduce a new method of obtaining sharp estimates on mixing for Glauber dynamics for the Ising model, which, in particular, establishes cutoff in three dimensions up to criticality. The new framework, which considers ``information percolation'' clusters in the space-time slab, shows that total-variation mixing exhibits cutoff with an \(O(1)\)-window around the time at which the magnetization is the square-root of the volume.

Sparsification of graphs and matrices

Daniel Spielman
Yale University
November 3, 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.

Sign rank, spectral gap and VC dimension

Noga Alon
Tel Aviv University; Visiting Professor, School of Mathematics
November 4, 2014
The signrank of an \(N \times N\) matrix \(A\) of signs is the minimum possible rank of a real matrix \(B\) in which every entry has the same sign as the corresponding entry of \(A\). The VC-dimension of \(A\) is the maximum cardinality of a set of columns \(I\) of \(A\) so that for every subset \(J\) of \(I\) there is a row \(i\) of \(A\) so that \(A_{ij}=+1\) for all \(j\) in \(J\) and \(A_{ij}=-1\) for all \(j\) in \(I-J\). I will describe explicit examples of \(N \times N\) matrices with VC-dimension 2 and signrank \(\Omega(N^{1/4})\).