## Marcus Khuri : On the Penrose Inequality

- Geometry and Topology ( 181 Views )The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.

## Zheng Zhang : On motivic realizations for variations of Hodge structure of Calabi-Yau type over Hermitian symmetric domains

- Geometry and Topology ( 177 Views )Based on the work of Gross and Sheng-Zuo, Friedman and Laza have classified variations of real Hodge structure of Calabi-Yau type over Hermitian symmetric domains. In particular, over every irreducible Hermitian symmetric domain there exists a canonical variation of real Hodge structure of Calabi-Yau type. In this talk, we wil review Friedman and Lazas classification. A natural question to ask is whether the canonical Hermitian variations of Hodge structure of Calabi-Yau type come from families of Calabi-Yau manifolds (geometric realization). In general, this is very difficult and is still open for small dimensional domains. We will discuss an intermediate question, namely does the canonical variations occur in algebraic geometry as sub-variations of Hodge structure of those coming from families of algebraic varieties (motivic realization). In particular, we will give motivic realizations for the canonical variations of Calabi-Yau type over irreducible tube domains of type A using abelian varieties of Weil type.

## Justin Sawon : Lagrangian fibrations by Jacobians of low genus curves

- Geometry and Topology ( 157 Views )The Beauville-Mukai integrable system is a well-known Lagrangian fibration, i.e., a holomorphic symplectic manifold fibred by Lagrangian complex tori. It is constructed by beginning with a complete linear system of curves on a K3 surface, and then taking the compactified relative Jacobian of the family of curves. One may ask whether other families of curves yield Lagrangian fibrations in this way. Markushevich showed that this is not the case in genus two: a Lagrangian fibration by Jacobians of genus two curves must be a Beauville-Mukai system. We generalize his result to genus three curves, and also to non-hyperelliptic curves of genus four and five.

## Jason Parsley : Petal Links

- Geometry and Topology ( 144 Views )A petal diagram of a knot or link consists of a center point surrounded by n non-nested loops; it represents n strands of the link at various heights which all project onto the same center point. Though every knot has a petal diagram, extremely few links have petal diagrams. The goal of this project is to characterize and enumerate which links do. First, we tabulate all petal links of 2-5 components. We then show all petal links arise as circle graphs -- the intersection graph of a set of chords of a circle. This establishes lower bounds on the number of petal links and allows us to conjecture upper bounds. We then discuss using petal diagrams to model certain classes of knots and links.

## Ákos Nagy : From instantons to vortices on spherically symmetric ALF manifolds

- Geometry and Topology ( 125 Views )Yang-Mills theory on Asymptotically Locally Flat (ALF) 4-manifolds has been intensely studied by geometers and physicists since the late 70's. The most important examples are R^3 x S^1, the (multi-)Taub-NUT spaces, and the Euclidean Schwarzschild manifold. In this talk, I will outline the correspondence between spherically symmetric Yang-Mills instantons and planar Abelian vortices (following the ideas of Witten, Taubes, and Garcia-Prada), and then apply this instanton-vortex duality to spherically symmetric ALF 4-manifolds. Finally, I will show how this construction can be used to describe the low energy instanton moduli spaces of the Euclidean Schwarzschild manifold, and its generalizations. This is a joint work with Gonçalo Oliveira (IMPA).