Recent

Symplectic field theory and codimension-2 stable Hamiltonian submanifolds

Richard Siefring
Ruhr-Universität Bochum
April 20, 2017
Motivated by the goal of establishing a "symplectic sum formula" in symplectic field theory, we will discuss the intersection behavior between punctured pseudoholomorphic curves and symplectic hypersurfaces in a symplectization. In particular we will show that the count of such intersections is always bounded from above by a finite, topologically-determined quantity even though the curve, the target manifold, and the symplectic hypersurface in question are all noncompact.

Thermodynamical approach to the Markoff-Hurwitz equation

Michael Magee
Yale University
April 19, 2017
I'll first introduce the Markoff-Hurwitz equation and explain how it plays a fundamental role in different areas of mathematics. The main result I'll discuss is a true asymptotic formula for the number of real points in a fixed orbit of the automorphism group of the Markoff-Hurwitz variety with bounded maximal entry. In particular this establishes an asymptotic count for the number of integer solutions to the Markoff-Hurwitz equation of bounded height.

Billiards and Hodge theory

Simion Filip
Harvard University
April 19, 2017
A polygon with rational angles can be unfolded and glued into a finite genus Riemann surface equipped with a flat metric and some singularities. The moduli space of all such structures carries an action of the group $\mathrm{PSL}(2,\mathbb R)$ and this can be viewed as a renormalization of the billiard flow in the initial polygon. After introducing the basics, I will explain how Hodge theory can give information on the $\mathrm{PSL}(2,\mathbb R)$ dynamics, in particular on the Lyapunov exponents and orbit closures.

Albert Hirschman Award Ceremony and Program

featuring Peter Lange, Duke University; Didier Fassin, Institute for Advanced Study; and Ira Katznelson, Social Science Research Council
April 19, 2017
Albert O. Hirschman Prize Ceremony and Program
Please join the Social Science Research Council and the Institute for Advanced Study for the Albert O. Hirschman Prize Ceremony and Program honoring Amartya Sen.
featuring Peter Lange, Duke University; Didier Fassin, Institute for Advanced Study; and Ira Katznelson, Social Science Research Council
April 19, 2017 at 5 p.m.
Institute for Advanced Study, Wolfensohn Hall

Bounds on roots of polynomials (and applications)

Adam Marcus
Princeton University; von Neumann Fellow, School of Mathematics
April 18, 2017
I will discuss methods for deriving bounds on the roots of polynomials, and how one can use such bounds to assert the existence of combinatorial structures with certain spectral properties. This will include introducing the "method of interlacing polynomials" and showing how one can use it prove the existence of Ramanujan graphs. Lastly, I will show how one can interpret these methods as a finite version of the previous week's results.