**50-Minute Talks:**

** Joel Hass**

introduce a simplicial energy that is closely connected to a discretized area associated to a simplicial complex. We will discuss applications to both

mathematical and applied problems. This is joint work with Peter Scott.

conical singularities

these data and a conductance function we construct a canonical pair (S, f ) where S is a special type of a genus (m-1 ) singular flat surface, tiled by rectangles and f is an energy preserving mapping from the edges of the decomposition onto S. In this lecture, we will employ a Dirichlet-Neumann boundary value problem.

embedded pairs of pants and one-holed tori in the surface. This is a joint work with Ser Peow Tan.

Abstract:

**Igor Rivin** (Temple University)

*Title: Conformal matching*

metric leads to a surprisingly rich theory of discrete conformal maps. I will explain some of the salient features of this theory and its connection with

hyperbolic polyhedra. This elucidates the relationship with the theory of circle packings, and it leads to a variant of discrete conformal equivalence that allows

mapping to the hyperbolic plane. This is joint work with Alexander Bobenko, Ulrich Pinkall, and Peter Schröder

**25-Minutes Talks:**

** Richard Bamler**

power of) f extends into the handlebody. The proof rests on an analysis of the accumulation points of a certain sequence of representations from the

fundamental group of S into PSL (2, C). Joint work with Jesse Johnson and Yair Minsky.

**Christine Breiner** (Massachusetts Institute of Technology)

*Title: Symmetries of genus-g helicoids*

short geodesics in hyperbolic 3-manifolds are isotopic into strongly irreducible Heegaard surfaces.

geometry in dimensions 2 and 3. Though this metric is incomplete, its completion is a CAT(0) metric space on which the mapping class group

acts co-compactly, and the quotient of this completion by the mapping class group is the Deligne-Mumford compactification of moduli space

g,n. I will give a brief introduction to Weil-Petersson geometry and discuss joint work with Hugo Parlier that studies the growth of diam

( g,n as g and n) tend to infinity.

a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. The same statement is also true for absolutely area minimizing surfaces, too.

of decompositions reminiscent of the pair of transverse laminations that we'd get in the pseudo-Anosov case. The fundamental group acts on the

orbit space preserving this structure and this can be used to construct an action on a circle at infinity. We use this to translate some properties of the flow to properties the circle action. In particular, we give sufficient conditions for finding closed orbits in the flow. This is part of a conjectural proof that every

quasigeodesic flow on a closed hyperbolic manifold has closed orbits.

complement of τ in M is a genus-2 handlebody. Fifteen years ago, Colin Adams asked a series of questions about how the topological data of an unknotting

tunnel fits into the hyperbolic structure on M. For example: is τ isotopic to a geodesic? Can it be arbitrarily long, relative to a maximal cusp neighborhood?

Does τ appear as an edge in the canonical polyhedral decomposition? Although the most general versions of these questions are still open today, I will

describe fairly complete answers in the case where M is created by a ``generic'' Dehn filling. As an application, there is an explicit family of knots in S3 whose

tunnels are arbitrarily long. This is joint work with Daryl Cooper and Jessica Purcell.

proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in A. Stone (A density function and the structure of singularities of the mean curvature flow.

on functions.

Jacobi operator associated to the minimal immersion plays a particularly crucial role in understanding how the minimal surfaces vary when the ambient quasi-

Fuchsian metric is varied in the deformation space of quasi-Fuchsian metrics.

Mostow Rigidity theorem, Waldhausen's theorem and the result in this paper, we can answer the same question for all prime 3-manifolds.

manifolds arise in the proof of the homotopy rigidity theorem proved by D. Gabai, R. Meyerhoff and N. Thurston. The authors made several conjectures

about the exceptional manifolds most of which have been proved. In my talk I will present an improved version of the conjecture and show that some

exceptional manifolds non-trivially cover manifolds. The proof is based on the results obtained by programs Snap and SnapPy.

Cantor set. Various attempts have been made to make the construction volume preserving. By studying the above extension problem we conclude, in

particular, that the Bowen-type mapping on positive measured Cantor sets can never be volume preservingly extended to the torus. This is joint work with Charles Pugh and Amie Wilkinson.

ideally triangulated punctured surface (S, T) of negative Euler characteristic. If h ≥ 0, the decorated Teichmüller space in the Ψh coordinate becomes an explicit convex polytope P(T) independent of h; and if h < 0, the decorated Teichmüller space becomes an explicit bounded convex polytope Ph(T) so that Ph(T) ⊂ Ph'(T) if h <h'. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichmüller space is reproduced