Joel Hass - Research Summary


Current CV (PDF file)

My research centers on low-dimensional topology and geometry.

Publications by Date (PDF file)

Publications by Area (PDF file)


Recent Papers and Preprints

2017

The Distribution of Knots in the Petaluma Model.
(with C. Even-Zohar, N. Linial, T. Nowik)
The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of every specific knot decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots.

2016

Isoperimetric Regions in Nonpositively Curved Manifolds.
We show that isoperimetric regions in two and three-dimensional nonpositively curved manifolds are not necessarily balls, and need not even be connected.

2015

Minimal Fibrations of Hyperbolic 3-manifolds.
We give examples of hyperbolic 3-manifolds that fiber over the circle but do not admit minimal fibrations.

Landmark-free geometric methods in biological shape analysis.
(with P. Koehl)
We propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates.

A Metric for genus-zero surfaces.
(with P. Koehl)
We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms give a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.

The number of surfaces of fixed genus in an alternating link complement.
(with A. Thompson and A. Tsvietkova)
Let L be a prime alternating link with n crossings. We show that for each fixed g, the number of genus g incompressible surfaces in the complement of L is bounded by a polynomial in n. Previous bounds were exponential in n.

2014

Invariants of Random Knots and Links.
(with C. Even-Zohar, N. Linial, T. Nowik)
We study random knots and links in using the Petaluma model. We obtain formulae for the distribution of the linking number of a random two-component link and for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3.

How round is a protein? Exploring protein structures for globularity using conformal mapping.
(with P. Koehl)
We present a new approach to measuring the roundness of a genus-zero surface. We then test and apply this to measure the roundness of the platonic solids and the roundness of a collection of 533 proteins.

Automatic Alignment of Genus-Zero Surfaces.
(with P. Koehl)
Gives a new method to automatically find optimal conformal mappings between two triangulated spherical surfaces.

Simplical Energy and Simplicial Harmonic Maps
(with P. Scott)
This paper defines a new type of energy function on maps from a triangulated surface to a manifold. The energy captures many proerties of the Dirchlet energy, but it is much simpler to establish properties such as existence, uniqueness etc.

Topological And Physical Link Theory Are Distinct
(with A. Coward)
We show that the theory of classical knots and links does not coincide with that of knots and links having thickness and fixed length, as in the real world. In particular we prove the existence of a two component link that is split in the classical theory but cannot be split with a physical isotopy.

2013

Globally Optimal Cortical Surface Matching With Exact Landmark Correspondence.
(with A. Tsui, D. Fenton, P. Vuong, P. Koehl, N. Amenta, D. Coeurjolly, C. DeCarli, and O. Carmichael)
A paper from the Information Processing in Medical Imaging conference. This applies hyperbolic orbifolds to align brain cortex surfaces with prescribed landmarks.

What is an Almost Normal Surface?
We explain how almost normal surfaces emerge naturally from the study of geodesics and minimal surfaces.


Many of my recent preprints are available on the math arxives. and on academia.edu.

Research Areas

Three dimensional manifolds
Computational complexity in topology
Minimal surfaces and bubbles
Geometric problems in Biology

The focus of my research is in three dimensional geometry and topology. This subject contains many fascinating problems, with important applications.

Geometric Design

I worked on problems related to Computer Aided Design with Rida Farouki. This work can be seen here.


Minimal Surfaces

Minimal Surfaces and Constant Mean Curvature Surfaces are models for soap films and for optimizing shapes that enclose given volumes.


Click here for information and graphics on Double Bubbles.

Curve Evolution

I have looked at various aspects of the motions of curves and surfaces, with dynamics related to the curvature.


Click here for material on Curve Flows. You can draw a curve and see it flow on your web browser.

Complexity

In work with Jeff Lagarias, Nick Pippenger, Ian Agol and Bill Thurston, I studied the computational complexity of problems associated with unknotting. Tahl Nowik and I developed invariants of knot diagrams which gave quadratic lower bounds on the number of Reidemeister moves needed to connect distinct diagrams of a knot.

Geometric Biology

I am working with Patrice Koehl on geometric problems arising in biology.


The behavior of proteins and other biological molecules is significantly affected by their geometry. Many fascinating geometric problems are closely related to biological applications.

We are working with Nina Amenta, and Owen Carmichael on a project in shape analysis and shape matching. These problems connect to brain imaging and the cortical surface of the brain. It also connects to problens such as facial recognition and understanding the relations between fossils and evolutionary trees.


My Recent Conferences and trips:

FoCM'17
Computational Topology and Geometry workshop.
Barcelona
July 10-19, 2017.

My 60th birthday conference:
Geometry, Topology and Complexity of Manifolds, and applications to Biology
UC Berkeley
May 20-22, 2015.


Geometric Structures on 3-manifolds
School of Mathematics, Institute for Advanced Study
September 1, 2015 to April 30, 2016
I was on sabbatical in 2015-2016, at the above program.

Applied Topology and High-Dimensional Data Analysis
University of Victoria, British Columbia, Canada
August 17-28, 2015

COMCA'15
Congreso Matematica Capricornio
Iquique - Chile
August 5-7, 2015

AMS Sectional Meeting, Las Vegas
Spring Western Sectional Meeting
University of Nevada, Las Vegas, Las Vegas, NV
April 18-19, 2015

In and Around Combinatorics
The 18th Midrasha Mathematicae
Israel Institute for Advanced Studies
January 18-30, 2015

Hyperbolic Geometry and Minimal Surfaces
IMPA, Rio de Janeiro
01/04/15 to 01/10/15

FoCM'14
Computational Topology and Geometry workshop.
Montevideo
Dec. 15-17, 2014.