Beauville's splitting principle for Chow rings of projective hyperkaehler manifolds

Lie Fu
Member, School of Mathematics
November 4, 2014
Being the natural generalization of K3 surfaces, hyperkaehler varieties, also known as irreducible holomorphic symplectic varieties, are one of the building blocks of smooth projective varieties with trivial canonical bundle. One of the guiding conjectures in the study of algebraic cycles of such varieties is Beauville's splitting principle. Concerning the weak form of the splitting principle, I want to report some progress on the closely related Beauville-Voisin conjecture.

Elliptic genera of Pfaffian-Grassmannian double mirrors

Lev Borisov
Rutgers University
November 5, 2014
For an odd integer \(n > 3\) the data of generic n-dimensional subspace of the space of skew bilinear forms on an n-dimensional vector space define two different Calabi-Yau varieties of dimension \(n-4\). Specifically, one is a complete intersection of n hyperplanes in the Grassmannian \(G(2,n)\) and the other is a complete intersection of \(n(n-3)/2\) hyperplanes in the Pfaffian variety of degenerate skew forms. In \(n=7\) case, these have been investigated by Rodland and were (heuristically) found to have the same mirror family.

The solution of the Kadison-Singer problem

Daniel Spielman
Yale University
November 5, 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.

Ramanujan graphs of every degree

Daniel Spielman
Yale University
November 6, 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.

\(C^0\)-characterization of symplectic and contact embeddings

Stefan Müller
University of Illinois at Urbana-Champaign
November 7, 2014
Symplectic and anti-symplectic embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of \(C^0\)-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C.

The Parthenon Sculptures: Decoding Images of Ancient Myths

Joan Breton Connelly
Professor of Classics at New York University
November 7, 2014
Joan Breton Connelly, Hetty Goldman Member (2010­–11) in the School of Historical Studies, explores how the Parthenon sculptures conveyed genealogical myths that answered for the Athenians the basic human question: Where do I come from? She shows how cosmic and epic narratives, and the great boundary catastrophes of war and deluge, established frameworks for understanding the distant past.

Finding Signs of Life on Earth-Like Exoplanets

Hanno Rein
Assistant Professor at the University of Toronto
November 7, 2014
Hanno Rein, former Member (2010–13) in the School of Natural Sciences, discusses using biosignatures to find signs of life, both in the Solar System and beyond. Additionally, Rein explains why chances are good for finding an Earth-like exoplanet around a Sun-like star within the next few years, as well as the challenges faced when that planet is discovered.

This lecture is sponsored by the Association of Members of the Institute for Advanced Study (AMIAS).

Talagrand's convolution conjecture and geometry via coupling

James Lee
University of Washington
November 10, 2014
Consider an image with two colors--black and white--and where only 1% of the pixels are white. If we apply a Gaussian blur, can it be that the non-black pixels of the (now greyscale) image are largely concentrated on a single shade of grey? Sure, if we apply enough blur, the whole picture will be a dull grey. But can this happen for a very light shade of grey? That seems preposterous--certainly the colors should fade gracefully from light to dark. Talagrand conjectured in 1989 that we should see at least 50 shades of grey. We will prove it.