## Kristen Hendricks : Periodic Knots and Heegaard Floer Homology

- Geometry and Topology ( 198 Views )We introduce periodic knots and discuss two classical results concerning their geometry, namely Murasugi's condition on the Alexander polynomial and Edmonds' condition on the genus. We then show how spectral sequences in Heegaard Floer link homology can be used to give a generalization of these two results in the case of doubly-periodic knots.

## Diana Davis : Periodic paths on the pentagon

- Geometry and Topology ( 180 Views )Mathematicians have long understood periodic billiard trajectories on the square table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my work on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior. There will be lots of pictures. This is joint work with Samuel LeliÃ¨vre.

## Ziva Myer : Product Structures for Legendrian Submanifolds with Generating Families

- Geometry and Topology ( 166 Views )In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.

## Fernando Schwartz : On the topology of black holes

- Geometry and Topology ( 159 Views )An important special case of the general construction of black holes translates into a problem in Riemannian geometry, since a totally geodesic slice of spacetime is an asymptotically flat Riemannian manifold with nonnegative scalar curvature, and the restriction of the event horizon to the slice is the apparent horizon in the slice. In this talk we show how to construct new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any $n,m\ge 1$, we construct asymptotically flat, scalar flat Riemannian manifolds with apparent horizon that is a smooth outermost minimal hypersurface with topology $S^n\times S^{m+1}$.

## Graeme Wilkin : Morse theory and stable pairs

- Geometry and Topology ( 151 Views )In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.

## Jesse Madnick : The Mean Curvature of (Co)Associative Submanifolds

- Geometry and Topology ( 144 Views )In flat R^7, two classes of submanifolds stand out: the associative 3-folds and coassociative 4-folds, which enjoy the remarkable property of being area-minimizing in their homology class. In fact, these submanifolds make sense in any 7-manifold with a G2-structure, and it is natural to ask: Under what conditions to do they continue to be minimal? We answer this question by deriving pleasantly simple formulas for their mean curvature. Time permitting, we will explain how these formulas suggest new avenues for the construction of minimal submanifolds of high codimension. This is joint work with Gavin Ball.

## Goncalo Oliveira : Monopoles in Higher Dimensions

- Geometry and Topology ( 130 Views )The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G_2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

## Yi Wang : The Aleksandrov-Fenchel inequalities of k+1 convex domains

- Geometry and Topology ( 120 Views )In this talk, I will report some recent joint work with Sun-Yung Alice Chang in which we partially generalize the Aleksandrov-Fenchel inequalities for quermassintegrals from convex domains in the Euclidean space to a class of non-convex domains.

## Matt Hogancamp : Categorical diagonalization of the full twist.

- Geometry and Topology ( 118 Views )I will discuss recent joint work with Ben Elias in which we introduce a theory of diagonalization of functors. Our main application is the diagonalization of the the Rouquier complex associated to full-twist braid, acting on the category of Soergel bimodules. The ``eigenprojections'' yield categorified Young symmetrizers, which are related to the flag Hilbert scheme by a beautiful recent conjecture of Gorsky-Rasmussen. Finally, I will mention a relationship with stable homology of torus links, which was recently investigated by myself and Michael Abel.

## Chi Li : Construction of rotationally symmetric Kahler-Ricci solitons

- Geometry and Topology ( 117 Views )Using Calabi's method, I will construct rotationally symmetric Kahler- Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.

## Henry Segerman : Connectivity of the set of triangulations of a 3- or 4-manifold

- Geometry and Topology ( 110 Views )This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.

## Carla Cederbaum : The Newtonian Limit of Geometrostatics

- Geometry and Topology ( 110 Views )Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.

## Christopher R. Cornwell : Bindings of open book decompositions and lens spaces

- Geometry and Topology ( 106 Views )We will discuss recent work on Legendrian and transverse links in universally tight contact lens spaces. There is a relationship between such links and the Berge Conjecture. The surgery duals to Berge knots in the corresponding lens space are all bindings of a rational open book decomposition. We will discuss whether these Berge duals support the universally tight contact structure on that lens space.