# Robert B. Kusner

Director of the Center

(This page -- as well as Rob's curriculum vitae (CV) -- is like a cathedral: a bit old-fashioned, always under reconstruction ;-)

Here's a link to Rob's Class Syllabi, Notes, etc.

Here are useful links to GMail, the copyleft LANL Geometry Preprint arXiv and to the UMassAmherstLibrary MathDatabases which enables off-campus links American Mathematical Society's MathSciNet and other copyrighted material at the UMassAmherstLibrary.

One of the most remarkable accomplishments in differential geometery was announced on 28 February 2012: Fernando Coda Marques and Andre Neves have resolved the Willmore Conjecture! This is one of what the late Bob Osserman referred to (in his early 1980s Berkeley Colloquium Lecture) as "Three Obdurate Conjectures in Differential Geometry" and concerns the total bending energy W of closed surfaces in 3-space (he referred to it as "the sleeper" among the three - the other two were the Caratheodory Conjecture about umbilic points on ovaloids and Lawson's Conjecture about the the uniqueness of the Clifford torus among all embedded minimal surfaces of genus 1 in the 3-sphere, also recently resolved by Simon Brendle). The round sphere achieves the absolute minimum W=4\pi and it's been known since work of Leon Simon from the 1980s that for higher genus surfaces the absolute minimum is strictly greater; indeed Willmore conjectured in the 1960s that W is at least the area 2\pi^2 of the Clifford minimal torus in the 3-sphere, and this is part of what Fernando and Andre have proven. Their work actually shows that any embedded minimal surface in the 3-sphere (other than a great 2-sphere or Clifford torus) has area strictly greater than 2\pi^2. Their methods depend on the orientability of the surface (through a very clever degree argument) but don't seem suited resolve a related conjecture of mine for closed surfaces in n-space: W exceeds 6\pi for any surface not diffeomorphic to the 2-sphere, with equality if and only if the surface is a Veronese minimal projective-plane (which is non-orientable) in the 4-sphere.

Most of Rob's work is available electronically in preprint or published form at the above locations, but there are a few things not (yet) posted there:

For example, in April 2003, Rob and two of his students went to hear Grisha Perelman lecture at MIT about geometrization of three-manifolds via Ricci Flow -- here is our triptych, Three-manifolds according to Grisha Perelman (.ps) (.pdf), of the event. Versions of our article also appear in the 2003 UMass Math Dept Newsletter, and in the popular Japanese math journal _Sugaku_ (v. 42, no. 10, pp. 4-6, September 2003).

Four years later (April 2007) our former department head, Eduardo Cattani, wanted to update this topic (now that Perelman was awarded - and has declined! - the 2006 Fields Medal for his work in this area), so we each wrote short articles for the 2007 UMass Math Dept Newsletter about the Poincare and Geometrization Conjectures. Here's an .html version of Rob's article A Bit of (Cosmic) Background which includes some links to the classical literature mentioned therein.

Here's where you can get the (semifinal: April 2003) Postscript and (final: May 2004) PDF versions of Rob's Clay/MSRI Summer 2001 note on Conformal Structures and Necksizes of CMC Surfaces (Postscript) (PDF) which supersede the arXiv version from July 2002. This volume was finally published by the AMS late in 2005.

And here is where you can read the October 1998 Science News cover story by Ivars Petersen of how Kusner's Minimax Sphere Eversion was computed and animated. (Compare with Rob's original account [ PDF version], written in 1995 when Ken Brakke and John Sullivan were working on the first animation with him at GANG.)

Bio

Rob Kusner is Professor of Mathematics and Co-Director of the Center for Geometry, Analysis, Numerics and Graphics (GANG). His research focuses on variational problems in low dimensional geometry and topology, with applications in the natural sciences. Most of his work has focused on minimal and constant mean curvature surfaces, as well as on the the geometry of energy minimizing knots and links. He has also begun working with students on curvature flows, and with physicists on 3-dimensional contact geometry (as a setting for the theory of chiral liquid crystals).

Kusner has been involved with GANG since its inception in the 1980's, and became its Co-Director in 1993, after a year at the Institute for Advanced Study in Princeton on an NSF Fellowship. Rob returned to the IAS, and also participated in the special geometric topology year at MSRI in Berkeley, during the 1996-97 academic year. During his 2003-04 sabbatical, Rob was again in the Bay Area, participating in the MSRI special year in geometry, and working with colleagues at Berkeley, Stanford and (just over the mountains, a few more hours to the east) Utah. He will be a Research Professor at MSRI in their Spring 2016 program on differential geometry.

Rob spent his most recent sabbaticals (Spring 2008 and 2011, and Fall 2015) at the University of Pennsylvania where he remains a long-term visitor in their Mathematics and Physics Departments. During June-July 2008, Rob also ran a workshop on the Geometry of Condensed Matter at the Aspen Center for Physics, where he also spent part of the summer of 2004. Rob organized a workshop on geometric knots last summer (June-July 2011) at CRM Ennio de Giorgi in Pisa, and in 2012 he ran a summer school on knotted fields at KITP (the Kavli Institute for Theoretical Physics) in Santa Barbara. During fall 2012 Rob was a visiting fellow at the Isaac Newton Institute in Cambridge, England, participating in the Topological Dynamics program there, and during the spring of 2015 he commuted between Amherst and Providence to participate in and help organize a semester-long program at ICERM, Brown Univerity's new experimental mathematics institute.

During the spring of 1996, Rob was visiting professor at the University of Minnesota in Minneapolis, where he was also a member of the Institute for Mathematics and its Applications. He has also served as a visting faculty member at Stanford (1987-88), at the University of California, Santa Barbara (1988-89), at Rice (1992), at the Universite de Tours, France (2003-), and at Penn (2008-). Rob organized the NSF-supported Five Colleges Geometry Institute, serving as its first research director (1990-91). (More on all this is documented in Rob's CV.)

In 1995, Kusner joined the editorial board of the journal Experimental Mathematics. He invites mathematicians working in all areas of geometry and topology to consider publishing their highest quality experimental work there.

Rob is often working with honors students; for example, James Lawrence has written programs identifying symmetries of and for viewing high-dimensional objects; Elyse Fosse investigated optimally packed helical tubes; Lillian Carasquillo worked on the geometry of knotted bands; Evan Innis finished a senior thesis on knotted ribbons and raceways (two versions of "flattened rope") in 2007; and in 2010, Amherst College senior Michael Kreisel worked with Rob on Riemann surfaces with Euclidean cone metrics (Mike's honors thesis was awarded the College's geometry prize)!

Here is a compendium of Kusner's research papers in the GANG Preprint Series. Some of Rob's recent work can also be found in Kusner's Mathematical Giftshop, or in his curriculum vitae. Rob also enjoys sharing some of his more amusing inventions with the public.

An alternative transportation advocate, Kusner has served for many years on the Town of Amherst Public Transportation and Bicycle Committee, which he chaired from 1998-2000. In 2007, Rob became the Chair of the Massachuestts Department of Conservation and Recreation's Norwottuck Rail Trail Advisory Committee. After many years of work to get a design that's both environmentally and economically optimal, the the Norwottuck has finally (summer 2015) been reconstructed! He is also a founding board member of the Norwottuck Network (a non-profit Massachusetts corporation dedicated to helping acquire cycling and hiking trail easements and to preserving their environs), and was a member of the Amherst Conservation Commission (2000-03).

In the Spring of 2005 Rob's neighbors asked him to run for one of the seats on the Town of Amherst's five-member Select Board to which Rob was elected while simultaneously leading an effort to maintain and improve Amherst's "direct democracy" Town Meeting. The Town is still grappling with the form of government it will ultimately adopt, but for now it appears that Town Meeting and the Select Board have been reenergized, despite (or, perhaps, because of) the controversies of the "Kusner (2005-2008) era" in Amherst. Cognizant of the fate of another bearded American leader during an earlier constitutionally divisive time, Rob decided to not seek re-election in 2008. He remains, nevertheless! For example, since late in 2006, Rob has been a host of the public affairs program Focus, broadcast each Sunday at noon on WMUA (91.1 MHz FM) in the Amherst area.

(By the way, if the above photo looks unfamiliar, occasionally Rob shaves :-)

        1435G Lederle Graduate Research Tower
Department of Mathematics
University of Massachusetts at Amherst
Amherst MA 01003
USA


Phone 413 545 6022

Lab 413 545 4605

Secretary 413 545 0510

Fax 413 545 1801

Electronic Mail kusner@math.umass.edu

All material on this website is Copyleft* by Rob Kusner.