[Last updated February 2, 2012.]

Lee Rudolph

Professor of Mathematics, Clark University, Worcester, MA, 01610.
Recent interests: mathematical psychology, mathematical aspects of robotics and motion planning, computational geometry. Long-standing, still very active interests: knot theory, low-dimensional topology, algebraic geometry, visualization.
Office: Room 335, Mathematics, Computer Science, and Physics Building, (508-793-)7345.


I attended the Second Abel Conference, A Mathematical Celebration of John Milnor on January 30 and 31, and February 1.


The annotated list of my mathematical publications (1970-2011) includes links to on-line versions, reviews, illustrations, co-authors' homepages, and so on. The list of citations of my work (updated April 15, 2011) contains over 500 books and papers (with links to on-line versions when available) by nearly 360 other researchers from Abe and Abhyankar to Zoladek and Zuddas (with links to homepages or other relevant information when available); maintaining it has been a good way for me to keep up to date with developments.
On April 21, 2007, soprano Karen Guderian and pianist Sima Kustanovich gave a workshop performance of Stephen Dembski's setting of my poem Little Prayer in November at Razzo Recital Hall in Clark University's Center for the Arts. Many thanks to Matt Malsky for making this event happen, and to the performers for being so patient with the writer and composer.

In November 2006, the Academy of American Poets included that poem in a list of ``poems appropriate to Thankgiving'' (where it is nestled in between ``He who binds himself to joy'' by William Blake and ``To a Friend Whose Work Has Come to Nothing'' by William Butler Yeats!), whence it found its way to USA Today's "On Deadline" blog. Garrison Keillor read it on The Writer's Almanac.

front cover of the book The poem appears in my poetry collection A Woman and a Man, Ice-Fishing (published in January 2006), which won the Texas Review Press's 2004 competition for the X. J. Kennedy Poetry Prize, judged by John Hollander. Support your local bookstore by buying a copy there--if your bookstore doesn't carry it yet, you (or your bookstore) can order it directly from TRP or through The Writer's Almanac.

I love these strange, witty, passionate poems, so rare in their range of far and near, here and there, light and dark. ``A singing lamp,'' ``lamp skull.'' ---Jean Valentine (National Book Award winner for poetry, 2004)
Lee Rudolph's poems are elegant--in both the mathematical and stylistic senses of the word--and witty, as if John Donne and Henri Poincaré had been channeled to create the superb lyrics of A Woman and a Man, Ice-Fishing. Some poems are like knots that loop around themselves, resist untangling, but then resolve through form, the poetic equivalent of an equation. Other poems proceed in a seemingly-more narrative way, but surprise through their collision of lexicons and events: ``You wrote me, I thought, in another language. / Today I recognize it as my own.'' Enter Rudolph's thought-world, back yard, attic; solve his word problems; read these poems and ``open who knows what unexpected door.'' ---Sue Standing
Lee Rudolph is among the least boring poets I know. His bold, inventive work keeps handing us fresh surprises, from the surreal hilarity of the title poem to the moving `` Little Prayers.'' Master of many instruments, Rudolph can deliver fat-free free verse as well as song lyrics (``Lullaby''), experimental forms (``Escape Reading,'' ``Scraps from the Dream Newspaper''), and tightly rhymed lyrics (``Beauty'', ``Weather Report''). You never know what to expect from him, except that each poem will be powerful, arresting, and original. ---X. J. Kennedy

You can find most of my previously published poetry (including the contents of my two previous collections, Curses and The Country Changes) starting at this page. Some new poems are here (updated October 2, 2007).

cover of the book The Handbook of Knot Theory, edited by William W. Menasco and Morwen Thistlethwaite, is now in print. Individual chapters are all available on-line free of charge at the ArXiv, where their authors intend to keep them updated as the field continues to develop. My contribution to the volume is a survey of ``Knot theory of complex plane curves''. Here's what Joan Birman has to say about it in Mathematical Reviews.

The focus of this article is the knots that arise, in various ways, as subsets of complex curves in C². The author's stated goal is to view the subject "through a narrow lens, that of quasipositivity". To this reviewer that seems like an excellent decision, because the author is surely the world expert on quasipositivity. Section headings include braids and braided surfaces, transverse C-links, complex plane curves in the small and the large, totally tangential C-links, and relations to other research areas.

The author is meticulous and thorough, the result being that his 79-page review includes 15 pages of preliminaries, an 8-page index for those who need to recall a definition that has been forgotten, and an 8-page bibliography. This has a downside because his emphasis on scholarship could lose readers who like to browse before deciding whether to spend more time and read in detail. For example, the term quasipositivity is not even mentioned until we are 30 pages into this 79-page article, and then it is defined with such a wealth of symbols and notation that had been set up earlier that it makes even the definition of this `guiding principle' inaccessible to this impatient reader. Still, it's good to know that it's all there, for a day when one just might be motivated to do the very hard work needed to get started.

Despite its name, this chapter neither contains all, nor supersedes most, of the content of my old survey article, ``Some knot theory of complex plane curves'' (originally published in the monograph Noeuds, Tresses, et Singularités, 1983).


Robotics and Motion Planning


Between June 27 and July 1, 2009, I attended 2009 Robotics: Science and Systems (RSS 2009) at the University of Washington, in Seattle, to present two papers written with my colleague Li Han: Explicit Parametrizations of the Configuration Spaces of Anthropomorphic Multi-Linkage Systems and Explicit parametrizations and regression in robotics, the latter as part of the Workshop on Regression in Robotics - Approaches and Applications.

On May 29, 2009, members of the Clark University Multibody Systems Research Group will attend the Fifth Annual New England Manipulation Symposium (NEMS) at Worcester Polytechnic Institute.

The paper Bending and Kissing: Computing Non-Self-Colliding Configurations of Planar Loops with Revolute Joints by Li Han, Lee Rudolph, Sam Dorsey-Gordon, Dylan Glotzer, Dan Menard, Jon Moran, and James R. Wilson has been presented (by Jon) at International Conference on Robotics and Automation, being held in Kobe, Japan, May 12 - 17, 2009. Sam, Dylan, Dan, Jon, and James are Clark undergraduates; this research was partially supported by NSF award RI0713335 and an NSF supplementary Research Experience for Undergraduates award.

Li Han and I co-organized a Special Session on Topological Robotics at the Spring Eastern Sectional Meeting of the American Mathematical Society at Worcester Polytechnic Institute, April 25, 2009.


Between June 25 and 28, 2008, I attended 2008 Robotics: Science and Systems (RSS 2008) at ETH in Zurich, Switzerland, where I presented the paper, Simplex-Tree Based Kinematics of Foldable Objects as Multi-body Systems Involving Loops (written with Li Han).

On May 30, 2008, members of the Clark University Multibody Systems Research Group attended the Fourth Annual New England Manipulation Symposium (NEMS) at Brown University.


On August 18, 2007, the National Science Foundation announced Award 0713335, for research on Practical Parametrization and Efficient Motion Planning of Linkage Systems. Li Han is Principal Investigator and I am co-PI. An important feature of our successful proposal is our plan to continue ``training and involvement of undergraduate students in mathematical and algorithmic aspects of robotics research''. For more details, consult the Clark University Multibody Systems Research Group homepage.

On April 13, 2007, Li Han presented our paper A unified geometric approach to inverse kinematics of a spatial chain with spherical joints at ICRA (the IEEE International Conference on Robotics and Automation) in Rome.


In 2006, Li Han and I attended three robotics conferences and presented two papers.

Mathematical Psychology

The debate is not about whether the numbers are right but whether it is right to have numbers. (Adam Gopnik, The New Yorker, September 6, 2004)

On November 3, 2007, I spoke at the 2007 Purdue Winer Memorial Lectures in West Lafayette, Indiana. The theme of the lecture series was ``Mathematical Psychology as Applied Mathematics''; my talk was titled ``The Hole in Emotion Space''. I had a great time listening to a wide variety of interesting talks.

Some of my ideas about qualitative topological models for developmental psychology were presented in a session at JPS 2007, the 37th annual meeting of the Jean Piaget Society, Amsterdam, May 31-June 2, 2007.

During calendar year 2004, a National Science Foundation Interdisciplinary Grant in the Mathematical Sciences let me spend all my time in Clark's psychology department. In 2005, I returned to the mathematics department, while continuing my work with colleagues in psychology, graduate and undergraduate students in psychology, mathematics, and computer science, and Li Han. The abstract "Mathematical Psychology: Geometry, Mapping and Dynamics in Emotion Space" describes what I thought I would be doing when I wrote the proposal in November 2002; it's all still true, but more is going on--including some "Research on the stimuli in tactile, auditory, and visual domains that elicit emotional responses" performed with physiological monitoring equipment purchased with the aid of another NSF grant. To date, I've published three papers sketching some ideas for new kinds of mathematical modeling in psychology, and have several more in the works--watch this spot for more developments!

The point ... is not that man is the measure of all things; it is that man can, like all things, be measured. (Adam Gopnik, The New Yorker, January 17, 2005)

Knot theory, low-dimensional topology, and algebraic geometry

Between May 20 and May 26, 2007, I attended the conference BRAIDS AND THEIR RAMIFICATIONS: Configuration Spaces, Arrangements, Mapping-Class Groups, 3-Manifolds in Cortona, Italy. During the first week of June, I attended a small meeting on quasipositivity and low-dimensional topology hosted by the Forschungsinstitut Mathematik at ETH--Zurich.

Some of the themes of my work are sketched in abstracts of research proposals funded by the National Science Foundation, and in an article by Ruth Auerbach giving proofs of two theorems I stated in a talk to the participants in the 1995 Research Experience for Undergraduates on algebraic singularities, directed by Alan Durfee at Mt. Holyoke College.

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