Christine Breiner (Massachusetts Institute of Technology)
Title: Symmetries of genus-g helicoids
Abstract: Every embedded genus-1 helicoid possesses an orientation preserving isometry. In this talk we outline how to extend this result to genus-g helicoids that have a hyperelliptic underlying conformal structure. The proof relies on the existence of a non-trivial biholomorphic involution as well as an understanding of the weak asymptotic geometry of genus-g helicoids. This is joint work with J. Bernstein.
William Breslin (University of Michigan)
Title: Short geodesics and Heegaard surfaces in hyperbolic 3-manifolds
Abstract: I will discuss how fat Margulis tubes, bounded area sweepouts, the Rubinstein-Scharlemann graphic, and thin position can be used to show that
short geodesics in hyperbolic 3-manifolds are isotopic into strongly irreducible Heegaard surfaces.
Will Cavendish (Princeton University)
Title: On the Growth of the Weil-Peterson Diameter of Moduli Space
Abstract: The Weil-Petersson metric on Teichmuller space is a negatively curved Kähler metric that relates in interesting ways to hyperbolic
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. Baris Coskunuzer (Koc University)
Title: Generic uniqueness of area minimizing disks for extreme curves
Abstract: In this talk, we will give a sketch of the proof of the following statement: For a generic nullhomotopic simple closed curve C in the boundary of
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.
Steven Frankel (California Institute of Technology)
Title: Closed Orbits of Quasigeodesic Flows
Abstract: We discuss quasigeodesic flows on hyperbolic 3-manifolds. Danny Calegari has shown that the orbit space of such a flow comes with a pair
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.
David Futer (Temple University)
Title: The geometry of unknotting tunnels
Abstract: Given a 3-manifold M, with boundary a union of tori, an unknotting tunnel for M is an arc τ from the boundary back to the boundary, such that the
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.
Stephen Kleene (Massachusetts Institute of Technology)
Title: Embedded and immersed MCF self-shrinkers
Abstract: (Joint work with N. Kapouleas and N. M. Moller). We present examples of embedded and Immersed MCF self-shrinkers, and discuss relevant gluing and doubling constructions.
Nam Le (Columbia University)
Title: Blow-up rate of the mean curvature during the mean curvature flow
Abstract: In this talk, we will show that at the first singular time of any compact, Type I mean curvature flow, the mean curvature blows up at the same rate as the second fundamental form. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our
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. Calc. Var. Partial Differential Equations {2} (1994), no. 4, 443--480.) coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow. This talk is based joint work with Natasa Sesum.
Ovidiu Munteanu (Columbia University)
Title: Rigidity theorems for complete noncompact manifolds
Abstract: I will talk about certain characterizations of the hyperbolic and complex hyperbolic spaces by their bottom of spectrum of the Laplace operator
on functions.
Andy Sanders (Maryland)
Title: Closed minimal immersions in quasi-Fuchsian 3-manifolds
Abstract: I will quickly review some of the known facts about closed minimal surfaces in quasi-Fuchsian 3-manifolds and explain how the
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.
Hongbin Sun (Princeton University)
Title: Degree ±1 self-maps and self-homeomorphisms on S3-manifolds
Abstract: We determine which 3-manifolds supporting S3 geometry admit adegree 1 or -1 self-map that does not homotopic to a self-homeomorphism. By
Mostow Rigidity theorem, Waldhausen's theorem and the result in this paper, we can answer the same question for all prime 3-manifolds.
Trnkova, Maria
Title: Hyperbolic Exceptional Manifolds
Abstract: An exceptional manifold is a closed hyperbolic manifold which does not have a shortest geodesic with an embedded tube of radius ln(3)/2. These
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.
Lu Wang (Massachusetts Institute of Technology)
Title: Uniqueness of Self-Shrinkers of Mean Curvature Flow
Abstract: In this talk, we will discuss the uniqueness of selfshrinking ends of mean curvature flow in 3-dimension Euclidean space, given fixed asymptotic behaviours and its relation with the classification problem of non-compact complete self-shrinkers.
Conan Wu (Princeton University)
Title: Volume preserving extensions and ergodicity of Anosov diffeomorphisms
Abstract: Given a C1 self-diffeomorphism of a compact subset in ℝn, from Whitney's extension theorem we know exactly when does it C1 extend to ℝn. How about volume preserving extensions? It is a classical result that any volume preserving Anosov di ffeomorphism of regularity C1+ɛ is ergodic. The question is open for C1. In 1975 Rufus Bowen constructed an (non-volume-preserving) Anosov map on the 2-torus with an invariant positive measured
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.
Tian Yang (Rutgers University)
Title: A Deformation of Penner's Coordinate of the Decorated Teichmüller Space
Abstract: We find a one-parameter family of coordinates {Ψh}hϵℝ which is a deformation of Penner's coordinate of the decorated Teichmüller space of an
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