String topology coproduct: geometric and algebraic aspects

Manuel Rivera
University of Miami
May 11, 2017
The string topology coproduct is an intersection type operation, originally described by Goresky-Hingston and Sullivan, which considers transverse self-intersections on chains of loops in a smooth manifold and splits loops at these intersection points. The geometric chain level construction of string topology operations involves deforming chains to achieve certain transversality conditions and these deformations introduce higher homotopy terms for algebraic compatibilities and properties.

The $p$-curvature conjecture and monodromy about simple closed loops

Ananth Shankar
Harvard University
May 11, 2017
The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$, for almost all primes $p$. We prove that if the variety is a generic curve, then every simple closed loop has finite monodromy.

Floer theory in spaces of stable pairs over Riemann surfaces

Timothy Perutz
University of Texas, Austin; von Neumann Fellow, School of Mathematics
May 4, 2017
I will report on joint work with Andrew Lee, which explores the notion that spaces of stable pairs over Riemann surfaces (in the sense of Bradlow and Thaddeus) could form a natural home for a “non-abelian” analog of Heegaard Floer homology for 3-manifolds - just as the g-fold symmetric product is the home of Heegaard Floer homology - thereby circumventing the problems with singularities that beset instanton-type theories. In an initial foray into this area, we set up a theory not for Heegaard splittings but for fibered 3-manifolds, based on fixed-point Floer homology.

Lagrangian Floer theory in symplectic fibrations

Douglas Schultz
Rutgers University
April 27, 2017
Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fiber Lagrangians. The primary example that we have in mind is the manifold of full flags in ${\mathbb C}^3$ which fibers as ${\mathbb P}^1 \to {\rm Flag}({\mathbb C}^3) \to {\mathbb P}^2$, and a Lagrangian $T^3$ that fibers over the Clifford torus in ${\mathbb P}^2$.

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.