Georgia Institute of Technology College of Sciences

For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points.

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.

We show that after stabilizations of opposite parity and braid isotopy, any twobraids in the same topological link type cobound embedded annuli. We use this to prove thegeneralized Jones conjecture relating the braid index and algebraic length of closed braidswithin a link type, following a reformulation of the problem by Kawamuro. This is joint workwith Doug Lafountain.

The notion of distance for a Heegaard splitting of athree-dimensional manifold $M$, introduced by John Hempel, has provedto be a very powerful tool for understanding the geometry and topologyof $M$.

Let MCG(g) be the mapping class group of a surface of genus g. For sufficiently large g, the nth homology (and cohomology) group of MCG(g) is independent of g. Hence we say that the family of mapping class groups satisfies homological stability. Symmetric groups and braid groups also satisfy homological stability, as does the family of moduli spaces of certain higher dimensional manifolds. The proofs of homological stability for most families of groups and spaces follow the same basic structure, and

The scissors congruence group of polytopes in $\mathbb{R}^n$ is defined tobe the free abelian group on polytopes in $\mathbb{R}^n$ modulo tworelations: $[P] = [Q]$ if $P\cong Q$, and $[P \cup P'] = [P] + [P']$ if$P\cap P'$ has measure $0$. This group, and various generalizations of it,has been studied extensively through the lens of homology of groups byDupont and Sah.

Contact geometry in three dimensions is a land of two disjoint classes ofcontact structures; overtwisted vs. tight. The former ones are flexible,means their geometry is determined by algebraic topology of underlying twoplane fields. In particular their existence and classification areunderstood completely. Tight contact structure, on the other hand, arerigid. The existence problem of a tight contact structure on a fixed threemanifold is hard and still widely open. The classification problem is evenharder.