Georgia Institute of Technology College of Sciences

The braid group embeds in the mapping class group, and so the symplectic representation of the mapping class group gives rise to a symplectic represenation of the braid group. The basic question Tara Brendle and I are trying to answer is: how can we describe the kernel? Hain and Morifuji have conjectured that the kernel is generated by Dehn twists. I will present some progress/evidence towards this conjecture.

Caratheodory's famous conjecture, dating back to 1920's, states that every closed convex surface has at least two umbilics, i.e., points where the principal curvatures are equal, or, equivalently, the surface has contact of order 2 with a sphere. In this talk I report on recent work with Ralph howard where we apply the divergence theorem to obtain integral equalities which establish some weak forms of the conjecture.

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin subgroup quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. This is joint work with Chris Leininger and Johanna Mangahas.

Knotted trivalent graphs (KTGs) along with standard operations defined on them form a finitely presented algebraic structure which includes knots, and in which many topological knot properties are defineable using simple formulas. Thus, a homomorphic invariant of KTGs places knot theory in an algebraic context. In this talk we construct such an invariant: the starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90's using the theory of associators. We present an elementary construction building on Kontsevich's original definition, and discuss the homomorphic properties of the invariant, which, as it turns out, intertwines all the standard KTG operations except for one, called the edge unzip. We prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once, and present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains all the good properties of the original KTGs: it is finitely presented, includes knots, and is closely related to Drinfel'd associators. (Partly joint work with Dror Bar-Natan.)