*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* ;-)

**Useful links**

Here are links to Rob's UMass Math Department homepage and to his 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 to 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=4π 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
2π^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 2π^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 6π 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. Rob was invited to write
the cover
article for the Spring 2016 MSRI newsletter *Emissary* about
this topic.

Recently (summer 2019, in joint work with Peng Wang) we have proven
a number of related theorems, including the W-stability (in fact, the
locally W-minizing property) of the Clifford torus in all dimensions,
with similar W-minimizing and rigitdity results for Lawson's family of
embedded minimal surfaces of every genus - more on this at the arXiv
soon! Even more recently (spring 2021, in joint
work with Peter McGrath)
we have solved the *existence* part of the Canham Problem, posed
around 1970 in the context of explaining the equilbrium shapes of
erythrocytes (red blood cells) variationally: we prove that there is a
W-minimizing surface of given genus and isoperimetric ratio by
constructing comparison surfaces (resembling concentric spheres
perurbed by singular solutions to the Willmore-Jacobi equation joined
by tiny catenoidal necks at the singularities) of arbitrarily small
isoperimetric ratio with W<8π.

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). [Please note: in two places the text mentions Victor Kac as the person introducing and later thanking Perelman; the person was actually David Vogan, the chair of the MIT math department at the time.]

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 has led 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, with physicists on 3-dimensional contact geometry (as a setting for the theory of chiral liquid crystals), and most recently on the geometry of critical configurations on spheres.

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 was a Research Professor at MSRI in their Spring 2016 program on differential geometry, and wrote the cover article for their Emissary newsletter then about recent work on the Willmore Bending Energy for surfaces.

During June 2017, Rob led a working group on Packing Hard Flexible Structures in 2D & 3D in conjunction with the Packing of Continua workshop at the Aspen Center for Physics, where he also organized a 2008 workshop on the Geometry of Condensed Matter; he participated in a multi-week ACP workshop during the summer of 2004 as well. Rob organized a workshop on geometric knots for 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. Rob was a visiting fellow at ICERM again in Spring 2018.

Rob spent part of 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, and where he has occasionally led seminars or taught courses.

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 (2003-07), 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, he decided to not seek re-election in 2008. Rob remains a Precinct 3 Town Meeting rep and otherwise engaged in public affairs. Beginning in 2006, Rob was among regular the hosts - informally know as the "Friends of Ken [Mosakowski - a long-time former host who passed away late that November]" -] of Focus, broadcast each Sunday at noon on WMUA (91.1 MHz FM) in the Amherst area. Unfortunately, a 2016 "reorganization" at WMUA ended the nearly-5-decade run of Focus to detriment of the entire community. And even more recently, a new Amherst Charter Commission was seated and proposed a city charter which passed by a slim majority in March 2018. So in the fall of 2018, the city-known-as-the-Town of Amherst held its first unitary (combined executive-legislative authority) Council election - guess who was nominated and is now the ballot as a candidate for one of the three "at-large" Council seats?! Though I made it through the September primary, the 2144 votes garnered in November means my voice will come over the public microphone if and when I address those who were elected.

Coming full circle, in 2015 Rob was elected by the Association of Members of the Institute for Advanced Study to its board of trustees; he was re-elected in fall 2018 and currently serves as AMIAS treasurer. That's a completely different electorate and a very different duty, so it's been a pleasure to help the institution which has helped so many of us fortunate former members advance our research and understanding.

Finally, in 2020 — to enhance his teaching, to producee his math videos, to participate in virtual seminars, and to carry out collaborative research with colleagues around the world during the pandemic from a remote virtual place — Rob founded Coronavirus University, where the faculty are strong, the administrators are good looking, and all the students are above average (with a whimsical wink at Garrison Keillor, even though Coronavirus University is perfectly real in a virtual way ;-)!

*This website was one of the first in the world, and this webpage may be the first at UMass, dating from the time of the above photo (1994±1, so if the person depicted looks unfamiliar, it's because time passes - and because occasionally Rob shaves :-)

**Address**

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 2762

**Fax** 413 545 1801

**Electronic Mail** profkusner "at" gmail "dot" com, kusner "at" math "dot" umass "dot" edu

**Department Webpage** www.math.umass.edu/directory/faculty/rob-kusner

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

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