Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

Homology surgery and invariants of 3-manifolds

We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of \pi-algebraically-split links in 3-manifolds with fundamental group \pi . Using this class of links, we define a theory of finite type invariants of 3-manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of \pi-decorated trivalent graphs. We also consider an associated notion of surgery equivalence of \pi-algebraically split links and prove a classification theorem using a generalization of Milnor's \mu-invariants to this class of links.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper18.abs.htm

Systems of difference equations as a model for the Lorenz system

We consider systems of difference equations as a model for the Lorenz system of differential equations. Using the power series whose coefficients are the solutions of these systems, we define three real functions, that are approximation for the solutions of the Lorenz system

Noncommutative Geometry

Noncommutative Geometry applies ideas from geometry to mathematical structures determined by noncommuting variables. This meeting emphasized the connections of Noncommutative Geometry to number theory and ergodic theory

Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings

Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any assumption of properness/compactness, keeping in mind the motivating example of $\mathbb H^\infty$, the infinite-dimensional rank-one symmetric space of noncompact type over the reals. The monograph provides a number of examples of groups acting on $\mathbb H^\infty$ which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. Such examples often demonstrate the optimality of our theorems. We introduce a modification of the Poincar\'e exponent, an invariant of a group which gives more information than the usual Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones theorem relating the Hausdorff dimension of the radial limit set to the Poincar\'e exponent of the underlying semigroup. We give some examples based on our results which illustrate the connection between Hausdorff dimension and various notions of discreteness which show up in non-proper settings. We construct Patterson--Sullivan measures for groups of divergence type without any compactness assumption. This is carried out by first constructing such measures on the Samuel--Smirnov compactification of the bordification of the underlying hyperbolic space, and then showing that the measures are supported on the bordification. We study quasiconformal measures of geometrically finite groups in terms of (a) doubling and (b) exact dimensionality. Our analysis characterizes exact dimensionality in terms of Diophantine approximation on the boundary. We demonstrate that some Patterson--Sullivan measures are neither doubling nor exact dimensional, and some are exact dimensional but not doubling, but all doubling measures are exact dimensional.Comment: A previous version of this document included Section 12.5 (Tukia's isomorphism theorem). The results of that subsection have been split off into a new document which is available at arXiv:1508.0696