50-Minute Talks:

Joel Hass (University of California, Davis)
Title: Discretizing area and energy, and applications

Abstract: There have been several approaches to computing discrete, or combinatorial, analogs of the energy of a map between two manifolds. We
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.

Sa'ar Hersonsky (University of Georgia)
Title: Boundary Value Problems on Planar Graphs and Flat Surfaces with integer
conical singularities

Abstract: We are given a cellular decomposition of a planar, bounded domain and its boundary, where each 2-cell is either a triangle or a quadrilateral. From
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.

Feng Luo (Rutgers University)
Title: A dilogarithm identity on the moduli space of curves

Abstract: Given any closed hyperbolic surface of a fixed genus, we establish an identity involving dilogarithm of lengths of simple closed geodesics in all
embedded pairs of pants and one-holed tori in the surface. This is a joint work with Ser Peow Tan.

William Minicozzi (Johns Hopkins University)
Title: Mean Curvature Flow

I will describe joint work with Toby Colding on singularities of mean curvature flow, including the dynamics near a singularity.


Igor Rivin (Temple University)
Title: Conformal matching

Abstract: We will describe some of the mathematics involved in finding the best conformal map between two densities (mostly in low dimensions).

Boris Springborn (Universität Bonn)
Title: Discrete conformal maps and ideal hyperbolic polyhedra

Abstract: A straightforward discretization of the concept of conformal change of
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 (Princeton University)
Title: Stability of symmetric spaces of noncompact type under Ricci flow

Abstract: We establish stability results for symmetric spaces of noncompact type under Ricci flow, i.e. we will show that any small perturbation of the symmetric metric is flown back to the original metric under an appropriately rescaled Ricci flow. It will be important for us which smallness assumptions we have to impose on the initial perturbation. We will find that as long as the symmetric space does not contain any hyperbolic or complex hyperbolic factor, we don't have to assume any decay on the perturbation. Furthermore, in the hyperbolic and complex hyperbolic case, we show stability under a very weak assumption on the initial perturbation. This will generalize a result obtained by Schulze, Schnürer and Simon in the hyperbolic case. The proofs of those results make use of an improved L1 -decay estimate for the heat kernel in vector bundles as well as elementary geometry of negatively curved spaces.

Jacob Bernstein (Stanford University)
Title: A Variational Characterizaton of the Catenoid

Abstract: We show that the catenoid is the unique surface of least area within a geometrically natural class of minimal surfaces. The proof relies on a technique involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli.

Ian Biringer (Yale University)
Title: Extending pseudo-Anosov maps into handlebodies

Abstract: Let f be a pseudo-Anosov homeomorphism of the boundary S of a handlebody. We show how the attracting lamination of f determines whether (a
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

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