Alfred Noel, Mathematics

As a mathematician, I am used to working five, six years until I have something that I could say that was interesting.

ALFRED G. NOEL - Professor, UMass Boston

MLK Visiting Professor 2000-2001
Hosted by the Department of Mathematics

Alfred Noel is a Research Mathematician and Professor of Mathematics at the University of Massachusetts Boston. At the time of his MIT appointment, he was a Lecturer in Mathematics, Computer Science and Bio-Statistics at Northeastern University. Research interests: Representation Theory of Lie Groups using a combination of classical and computational techniques, mathematics education.

2000-2001 Scholars

Kazhdan-Lusztig-Vogan Polynomials for E8

The atlas software written by Fokko du Cloux can compute (among other things) Kazhdan-Lusztig-Vogan polynomials for real groups. The input is a real reductive group G (specified by a real form of a complex reductive group) and a block B of representations of G with regular integral infinitesimal character. The block B is a Z-module, with two bases: one consisting of irreducible representations, and the other of standard representations. The number of basis elements is called the block size. The Kazhdan-Lusztig-Vogan polynomials (evaluated at 1) give the change of basis matrix; it is a square matrix of size the block size.

The software computes Kazhdan-Lusztig-Vogan polynomials for all groups up to rank 7 quite quickly. See the atlas tables of structure and representation theory. The complex group E8 has three real forms: the compact form, the quaternionic form (K=A1×E7) and the split form (K=D8). Each of these gives rise to a different kind of real group of type E8. Their blocks, and their sizes, are given by the blocksizes command of atlas:

 

  compact quaternionic split
compact 0 0 1
quaternionic 0 3,150 73,410
split 1 73,410 453,060

The last row means that the split group has 3 blocks (at infinitesimal character rho), of sizes 1,  73,410 and 453,060 respectively. (The columns give the group on which the Vogan-dual block lives.)

Kazhdan-Lusztig-Vogan polynomials for the compact and quaternionic groups of type E8 are easily computed, as well as those for the blocks of sizes 1 and 73,410 of the split group. That leaves the block of size 453,060 of the split group.

Computation of Kazhdan-Lusztig-Vogan Polynomials for the large block of split real E8

In principle the atlas software can compute Kazhdan-Lusztig-Vogan polynomials for the block B of size 453,060. However this would require about 200 gigabytes of virtual memory (and therefore certainly a 64-bit processor; 32-bit processors as are found in most PCs cannot possibly address more than 4 gigabytes of virtual memory); moreover this virtual memory should preferably be actual RAM, since once the program starts assigning part of its memory to disk (swap space) its speed becomes orders of magnitude slower. In fact Dan Barbasch successfully completed this calculation on a machine with 128 gigabytes of RAM and 100 gigabytes of swap space. It took 2 days to use up the RAM, and ran for another 12 days using swap space. Dan was then unable to save the answer in a usable form, since the usual textual output format of the atlas would have required more space than would be available on even very large hard disks.

David Vogan started working on this computation over the summer with Fokko (until his death from ALS in November), and in the fall also with Marc van Leeuwen. Birne Binegar did some experiments to get estimates on the size of the computation, and it became clear that even with improved efficiency it would not be possible to do this on a machine with less than 128 gigabytes of RAM.

Following a suggestion of Noam Elkies, David and Marc rewrote the code to compute the polynomials mod n for several values of n, and then obtain the answer using the Chinese remainder theorem. In the end it was necessary to compute four moduli: 251, 253, 255 and 256, which together give the answer modulo N=4,145,475,840. While it is not a-priori possible to prove this is sufficient, to fact that all coefficients modulo N were found to lie in the interval from 0 to 11,808,808 allows us to prove that these coefficients are in fact the correct ones in Z.

The computation were carried out on sage, which was kindly made available to us by William Stein. It has 64 gigabytes of RAM (and 75 gigabytes of swap, which were however not needed) and 16 AMD opteron 64-bit processors. It is physically located at the University of Washington in Seattle, but was operated for this calculation exclusively via the internet. Manual intervention in Seattle was needed several times however, to reboot the computer after crashes (which were unrelated to the atlas computation).

The calculation took place in several steps, between Friday 22 December 2006 and Monday 8 January 2007: this included four runs of the atlas software that were identical except for the modulus used, and finally several post-processing steps of the binary files written, to perform the lifting by the Chinese remainder theorem of 13,721,641,221 polynomial coefficients. The computation took about 77 hours total, if one excludes the runs that had to be aborted due to a crash or that produced useless output due to subtle bugs that were initially present in the I/O procedures. David wrote a more detailed narrative of the process of computing these polynomials.

The final answer is contained in a pair of binary files of respective sizes 14 gigabytes and 60 gigabytes. (By way of comparison, the size of the latter file would allow storing 45 days of continuous music in MP3-format.) There is a utility to print any particular Kazhdan-Lusztig-Vogan polynomial. Our next step is to make the answer available in a useful way. It is not practical to give the answer in the same form as used in the tables referred to above, for example the one for the Kazhdan-Lusztig-Vogan polynomials for the large block of Spin(5,4) (which has type B4).

The Kazhdan-Lusztig-Vogan polynomials are polynomials in an indeterminate q. The matrix alluded to above is given by evaluating at q=1. According to the table below, there is a standard representation which contains a certain irreducible representation with multiplicity 60,779,787.


Here is some information on the Kazhdan-Lusztig-Vogan polynomials for the block B.

Size of the block: 453,060

Number of distinct polynomials: 1,181,642,979

Maximal coefficient: 11,808,808

Polynomial with the maximal coefficient: 152q22 + 3,472q21 + 38,791q20 + 293,021q19 + 1,370,892q18 + 4,067,059q17 + 7,964,012q16 + 11,159,003q15 + 11,808,808q14 + 9,859,915q13 + 6,778,956q12 + 3,964,369q11+ 2,015,441q10 + 906,567q9 + 363,611q8 + 129,820q7 + 41,239q6 + 11,426q5 + 2,677q4 + 492q3 + 61q2 + 3q

Value of this polynomial at q=1: 60,779,787

Size of the matrix: 205,263,363,600=453,0602

Number of coefficients in distinct polynomials: 13,721,641,221

 

Tables of structure theory and representations   Atlas software   Atlas home page

 

Alfred G. Noel is a Research Mathematician and Professor of Mathematics at the University of Massachusetts Boston. At the time of his MIT appointment, he was a Lecturer in Mathematics, Computer Science and Bio-Statistics at Northeastern University. His research interests are: Representation Theory of Lie Groups using a combination of classical and computational techniques, mathematics education. 

Dr. Noel holds a BET in Electrical Engineering (1980) from the Centre Pilote de Formation Technique in Port-au-Prince, Haiti. He earned a Mathematics High School Teacher certificate at the Institut Pédagogique National in Port-au-Prince, Haiti. After a year of undergraduate courses in computer science at Northeastern University in Boston, MA, he enrolled as a graduate student there. Dr. Noel went on to earn an MSc in Applied Mathematics (1986) and a PhD in Puer Mathematics (1997). His dissertation was titled  "Nilpotent Orbits and Theta-Stable Parabolic Subalgebras" and directed by Donald R. King.

He worked as a senior software engineer at Computervision and a research engineer at Peritus Software before joining the faculty at Northeastern in 1984. For the next 15 years Dr. Noel taught most of the undergraduate courses in the Mathematics Department and lectured on Data Structures and Algorithms in the Computer Science Department. He has also regularly taught at Bouve College

In 2005, Dr. Noel became Associate Professor of Mathematics in the College of Science and Mathematics at the University of Massachusetts Boston, where he has remained for the last 16 years. He is a two-time recipient of the Outstanding Achievement Award for Research and Scholarship (2003-04 / 2004-05) academic years.

Dr. Noel gives lectures at national and international conferences worldwide (France, Spain, Poland, Russia, China, Japan, Singapore). He is a member of the American Mathematical Society and the National Association of Mathematicians. Since 1996 he has organized a seminar series for the Haitian Scientific Society, of which he is Founding Member. Dr. Noel has held two posts as Professeur Invité in the Département de Mathématique at the Université de Poitiers, France.

As a 2000-2001 MLK Scholar at MIT, Dr. Noel was hosted by the Department of Mathematics, where he taught a course in Linear Algebra and conducted research. Two papers resulting from the visit were: 1) "Classification of Admissible Nilpotent Orbits in Simple Exceptional Lie Algebras of Type E6(6) and E6(-26)" [American Mathematical Society Journal of Representation Theory, Vol 5 (494-502) 2001] and 2) "Classification of Admissible Nilpotent Orbits in Simple Exceptional Real Lie Algebras of Inner Type" [American Mathematical Society Journal of Representation Theory, Vol 5 (455-493) 2001].

In 2006, Dr. Noel tail-ended a Visiting Scholar position at Harvard with a return to the MIT Mathematics Department as a Visiting Professor. From over a year he worked with David Vogan on development and implementation of modules for computing the Unitary Dual of a given Lie group. This is still one of the most important unsolved problems in the field. This work was done within the "Atlas of Lie Groups and Representations", the team of mathematicians that computed the character table of E8 in 2007, a result that was wildly covered in the US and abroad.

Refereed Papers

  • Progress and Process in Math Excel at Northeastern University. Co authors Maurice P. Gilmore, Thomas P. Stephens,Jr. The Center for Innovation in Urban Education Studies, Volume II, 1996 edited by J. Frazer, Northeastern University.
  • Classification of Nilpotent Orbits in Symmetric Spaces. Proceedings of the American Mathematical Society DIMACS African Americans in Mathematical Sciences volume 34 (123-127) 1997. 
  • Nilpotent Orbits and Theta-stable Parabolic Subalgebras. American Mathematical Society Journal of Representation Theory, Volume 2 (1-32) 1998. Click here to get postscript version 
  • Component Groups of Centralizers of Nilpotents in Symmetric Space. Co-author Donald R. King . Journal of Algebra Vol 232 (94-125) 2000. Click here to get postscript version 
  • Classification of Admissible Nilpotent Orbits in Simple Exceptional Lie Algebras of Inner Type. American Mathematical Society Journal of Representation Theory, Volume 5 (455-493) 2001.
  • Classification of Admissible Nilpotent Orbits in Simple Exceptional Lie Algebras of Type E6(6) and E6(-26) American Mathematical Society Journal of Representation Theory, Volume 5 (494-502) 2001.
  • Computing maximal tori using LiE and Mathematica. Lectures Notes in Computer Science, Springer-Verlag.   Vol 2657  (728-736) 2003.
  • Computing theta-stable parabolic subalgebras using LiE. Lectures Notes in Computer Science, Springer-Verlag Volume 3039 (335-342) 2004.
  • A LiE subroutine for computing prehomogeneous spaces associated with nilpotent Complex Orbits Co-author Steven Glenn Jackson Lecture Notes in Computer Science, Springer-Verlag  Vol 3516,  (611—618),  2005.
  • A LiE subroutine for computing prehomogeneous spaces associated to real  nilpotent orbits Co-author Steven Glenn Jackson Lecture Notes in Computer Science, Springer-Verlag Vol 3482, (512—521),  2005.
  • Prehomogeneous spaces associated with complex nilpotent orbits Co-author Steven Glenn Jackson Journal of Algebra Volume 289, Issue 2, (515-557) 2005.
  • Prehomogeneous spaces associated to nilpotent orbits in simple real Lie algebras E6(6) and E6(-26) Co-author Steven Glenn Jackson. Experimental Mathematics Volume 15, Isuue 4, (455-469) 2006.
  • Some remarks on Richardson orbits in complex symmetric spaces  American Mathematical Society Mathematics of Computation. Volume 75 (395-417), 2006.
  • Prehomogeneous spaces associated with real nilpotent orbits, with Steven Glenn Jackson. Journal of Algebra Volume 305, Issue 1, (194-269) 2006.
  • Polarizable theta-stable parabolic subalgebras and $K_C$-saturation in the non-compact real forms of $G_2$ and $F_4$, with Steven Glenn Jackson. Lecture Notes in Computer Science, Springer-Verlag Vol 3992, (422—429), 2006.
  • A general computational scheme for testing admissibility of nilpotent orbits of real Lie groups of inner type. Lecture Notes in Computer Science, Springer-Verlag Vol 4151, (1—11), 2006.
  • Maximal Tori of Reductive Centralizers of Nilpotents in Complex Symmetric Space  American Mathematical Society Contemporary Mathematics, Volume 467 (103 – 139)   2008.
  • The Atlas of Lie groups and representations: scope and successes American Mathematical Society Contemporary Mathematics, Volume 467 (85 – 101) 2008.
  • A new approach for computing generators for U(g)^K with  Steven Glenn Jackson Journal of Algebra Volume 322, Issue 8, (2607-2620) 2009.
  • Nilpotent Orbits attached to Coxeter Cells with  Steven Glenn Jackson (Submitted) 2009.

Memoirs and Books

  • Ph. D. Thesis, Northeastern University, March 1997.
  • Council for African American Researchers in the Mathematical Sciences: Volume III. Fifth Conference for African American Researchers in the Mathematical Sciences, June 22-25, 1999, University of Michigan, Ann Arbor, Michigan. Alfred G. Noël, Earl Barnes, Sonya A. F. Stephens Editors. American Mathematical Society Contemporary Mathematics, Volume 275. (2001)
  • Council for African American Researchers in the Mathematical Sciences: Volume V. Thirteenth Conference for African American Researchers in the Mathematical Sciences, June 19-22, 2007, Northeastern University and the University of Massachusetts Boston. Alfred G. Noël, Donald R. King, Gaston M. N’Guérékata, Edray H. Goins  Editors. American Mathematical Society Contemporary Mathematics, Volume 467. (2008)

Problems and Abstracts

A canonical basis for maximal tori of the reduc tive centralizer of a nilpotent element Proceedings of the NATO  Advanced Study Institute  Computational Noncommutative Algebra  and Applications July 6--19, 2003 Il Ciocco Resort Hotel, Italy. Edited by Jim Byrnes, Nato Science Series II. Mathematics, Physics and Chemistry – Vol. 136.

A Short Note on Nipotent Orbits Associated to Coxeter Cells (with Steven Jackson) ACM Communications in Computer Algebra, Issue 168, Vol. 43, N0. 2, (52-53), June 2009.

Expository Papers and Selected Talks

   

Preprints, Tables and Work in Progress

Math research team maps E8
Calculation on paper would cover Manhattan

 

Elizabeth A. Thomson, MIT News Office
March 18, 2007

 

An international team of 18 mathematicians, including two from MIT, has mapped one of the largest and most complicated structures in mathematics. If written out on paper, the calculation describing this structure, known as E8, would cover an area the size of Manhattan.

The work is important because it could lead to new discoveries in mathematics, physics and other fields. In addition, the innovative large-scale computing that was key to the work likely spells the future for how longstanding math problems will be solved in the 21st century.

MIT's David Vogan, a professor in the Department of Mathematics and member of the research team, presented the work Monday, March 19 to a standing-room-only crowd in Room 1-190. His talk, "The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness," was peppered with jokes and laughter.

E8 (pronounced "E eight") is an example of a Lie (pronounced "Lee") group. Lie groups were invented by the 19th-century Norwegian mathematician Sophus Lie to study symmetry. Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects.

Mathematicians study symmetries in higher dimensions. E8 has 248 dimensions.

"What's attractive about studying E8 is that it's as complicated as
symmetry can get. Mathematics can almost always offer another example that's harder than the one you're looking at now, but for Lie groups E8 is the hardest one," Vogan said.

"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Jeffrey Adams, project leader and a mathematics professor at the University of Maryland. "This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large-scale computing to solve complicated mathematical problems."

The mapping of E8may well have unforeseen implications in mathematics and physics that won't be evident for years to come.

"There are lots of ways that E8 appears in abstract mathematics, and it's going to be fun to try to find interpretations of our work in some of those appearances," said Vogan. "The uniqueness of E8 makes me hope that it should have a role to play in theoretical physics as well. So far the work in that direction is pretty speculative, but I'll stay hopeful."

"This is an exciting breakthrough," said Peter Sarnak, a professor of mathematics at Princeton University and chair of the scientific board at the American Institute of Mathematics (AIM)."Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists," said Sarnak, who was not involved in the work.

The magnitude and nature of the E8 calculation invite comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E8 calculation, which contains all the information about E8and its representations, is 60 gigabytes. This is enough to store 45 days of continuous music in MP3-format.

The mapping of E8 is also unusual because it involved a large team of mathematicians, who are typically known for their solitary style. "People will look back on this project as a significant landmark and because of this breakthrough, mathematics will now be viewed as a team sport," said Brian Conrey, executive director of AIM.

The E8 calculation is part of an ambitious project sponsored by AIM and the National Science Foundation known as the Atlas of Lie Groups and Representations. The goal of the Atlas project is to determine the unitary representations-roughly speaking, symmetries of a quantum mechanical system-of all the Lie groups (E8 is the largest of the exceptional Lie groups). This is one of the most important unsolved problems of mathematics. The E8 calculation is a major step and suggests that the Atlas team is well on the way to solving this problem.

The Atlas team consists of 18 researchers from around the globe. The core group consists of Adams and Vogan, plus Dan Barbasch (Cornell), John Stembridge (University of Michigan), Peter Trapa (University of Utah), Marc van Leeuwen (University of Poitiers) and (until his death in 2006) Fokko du Cloux (University of Lyon). Additional team members include Dan Ciubotaru, the CLE Moore Instructor in MIT's Department of Mathematics, and Alfred Noel, a professor at the University of Massachusetts at Boston and an MIT visiting scholar.

For more information on E8 visit aimath.org/E8/.

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