Curriculum Vitae

 Alex Jay Feingold

Department of Mathematical Sciences, SUNY-Binghamton, Binghamton, NY 13902-6000

Personal:

Born: April 1, 1950, Baltimore, Maryland, USA
Permanent Address: 45 Matthews Street, Binghamton, NY 13905-4038
Phone Numbers: (607) 729-3637 (home), (607) 777-2465 (office)
Spouse: Nancy Tittler
Married: December 18, 1977
Children: Emily Ruth (Born Aug. 27, 1985), Judith Marian (Born Mar. 10, 1988)

Education:

Dissertation:

``Tensor products of modules for Lie algebras''
Director:  Professor George B. Seligman

Academic Honors:

Grants:

  1. 1994-96    National Security Agency Grant, Mathematical Sciences Program (2 years), for ``Vertex Operator Algebras and Representation Theory''
  2. 1987-88    Grant-in-Aid from the Institute for Advanced Study, Princeton, NJ
  3. 1987        National Science Foundation Grant (2 years) for ``Affine and Hyperbolic Kac-Moody Algebras''
  4. 1985        National Science Foundation Grant (2 years) for ``Affine and Hyperbolic Kac-Moody Algebras''
  5. 1985        National Science Foundation Grant, Mathematical Sciences Research Equipment, (1 year), jointly with other Department members
  6. 1985        Dean's Research Semester Award
  7. 1984        SUNY Faculty Research Fellowship Award
  8. 1982        SUNY Faculty Research Fellowship Award
  9. 1980-81    National Science Foundation Grant (2 years) for ``Generalized Cartan Matrix Lie Algebras''
  10. 1980        SUNY Faculty Research Fellowship Award (declined because of NSF grant)
  11. 1978-79    National Science Foundation Grant for ``Generalized Cartan Matrix Lie Algebras and Power Series Identities''

Professional Societies:

Professional Experience:

Professional Activities:

    Transactions of the American Mathematical Society,
    Journal of Algebra,
    Journal fur die Reine und Angewandte Mathematik,
    Journal of Number Theory,
    Conference Proceedings on Lie Algebras and Related Topics at Madison, Wisconsin, 1988,
    Duke Mathematical Journal,
    Journal of Pure and Applied Algebra, Advances in Mathematics,
    Communications in Mathematical Physics,
    Proceedings of the Conference Moonshine, The Monster,and Related Topics, 1994.

Invited Lectures:

Attended Conferences:

Teaching Interests:

Algebra (Groups, Rings, Fields, etc.), Linear Algebra, Lie Algebras, Vertex Operators,
Modular Forms, Siegel Modular Forms, Conformal Field Theory, Fusion Algebras.

Present Research Interests:

My area of special interest is the theory of Lie algebras, their representations, connections to other parts of mathematics and applications to physics. My thesis concerned the decomposition of tensor products of finite-dimensional modules for complex semisimple Lie algebras. While still a graduate student at Yale, strongly influenced by my teacher, Jim Lepowsky, I extended my research into the infinite-dimensional Kac-Moody Lie algebras, independently introduced in 1968 by V.G. Kac (M.I.T.) and R.V. Moody (University of Saskatchewan). This has been an exciting and fruitful area of research because of its remarkable connections with physics (e.g., solitons, quantum field theory, string theory) and other areas of mathematics (e.g., combinatorics, group theory, modular forms, singularities, differential equations, knot theory).

During the period from 1981 to 1991 I had several collaborations with Igor Frenkel (Yale University). Our first paper studied hyperbolic Kac-Moody algebras, showing one such algebra to be closely connected with the theory of Siegel modular forms of genus two and with the related problem of lifting elliptic modular forms (the Saito-Kurokawa conjecture). We also gave a construction which provided closed formulas for an infinite number of root multiplicities (on levels 0, 1 and 2). Our second paper studied affine Kac-Moody algebras, providing a unified approach to constructing certain representations of all the classical affine algebras. These were based on underlying associative algebras of commutation or anticommutation relations whose bosonic or fermionic representations are important in quantum field theory. Another paper, with J. F. X. Ries, studied representations of hyperbolic Kac-Moody algebras, constructing all irreducible highest weight standard modules and providing closed formulas for an infinite number of weight multiplicities (on levels 0, 1 and 2). Other collaborations, also with Ries, studied the vertex operator algebras known in physics as chiral algebras. These algebras play a central role in string theory, conformal field theory, and in the Frenkel-Lepowsky-Meurman construction of the ``Monster'' group. Our main objectives were to obtain independent vertex and spinor constructions of chiral algebras, the isomorphism between the two viewpoints, known as a ``boson-fermion correspondence'', and constructions of the exceptional affine algebra E8(1) based on D4(1) spinor constructions and the principle of triality. Such representations of E8(1) are essential in the anomaly-free heterotic superstring theory of particle physics. Some of these results were announced at the 1988 Conference on Lie Algebras and Related Topics, Madison, Wisconsin. Those results which only involve the spinor constructions are in our Contemporary Mathematics monograph ([10]). A sequel (with Ries only) was planned to give the vertex picture and the boson-fermion correspondence, but the untimely death of Ries has substantially delayed the completion of that project.

In recent years, I have studied vertex operator algebras, their modules and intertwining operators constructed from the discrete series of $c<1$ Virasoro modules or from higher level representations of affine Kac-Moody algebras. These constructions, known in physics as minimal models and Wess-Zumino-Witten models, are deeply connected with braid groups and quantum groups, topics of great current interest. The structure of the intertwining operators is governed by the fusion rules, and some of my most recent work ([14]-[16]) and current projects involve new combinatorial interpretations and constructions of fusion algebras. I have also returned to the study of hyperbolic algebras [17] in collaboration with Hermann Nicolai (Albert Einstein Institute, Potsdam, Germany), and, with Nicolai and Axel Kleinschmidt, I am extending the first work I did with Frenkel to higher rank hyperbolic algebras. With Stefan Fredenhagen (Albert Einstein Institute, Potsdam, Germany), I am completing a proof of a conjecture of Mark Walton which shows that the Frenkel-Zhu fusion rule theorem can be restated as a beautiful generalization of the classical tensor product formula of Parthasarathy, Ranga Rao and Varadarajan.

Invited Addresses:

  1. The Special Session on Lie algebras, organized by Maria Wonenberger at the 775th Meeting of the American Mathematical Society at Bloomington, IN, April 11-12, 1980.
  2. The Special Session on Kac-Moody Lie Theory, organized by Howard Garland and James Hurley at the 789th Meeting of the American Mathematical Society at the University of Massachusetts, Amherst, October 16-18, 1981.
  3. The workshop on Vertex Operators in Mathematics and Physics, organized by James Lepowsky at the Mathematical Sciences Research Institute in Berkeley, CA, November 10-17, 1983.
  4. The Lie Algebras and Related Topics Conference at the University of Wisconsin, Madison, organized by J. Marshall Osborn and Georgia Benkart, May 22-June 1, 1988.
  5. The 1991 American Mathematical Society Summer Research Institute, on Algebraic Groups and Their Generalizations, Pennsylvania State University, University Park, PA, July 8-26, 1991.
  6. The Special Session on Rings and Representations, organized by Martin Lorenz and Shari A. Prevost at the 868th Meeting of the American Mathematical Society, Temple University, Philadelphia, PA, October 12-13, 1991.
  7. The Structure and Representation Theory of Lie Algebras conference in honor of George Seligman, April 10-12, 1992, Yale University, New Haven, CT.
  8. The 884th Meeting of the AMS, Special Session on Lie Theoretical Methods in Mathematical Physics, Sept. 18-19, 1993, Syracuse University, Syracuse, NY.
  9. The Joint Summer Research Conference in the Mathematical Sciences, Moonshine, The Monster, and Related Topics, Mount Holyoke College, South Hadley, MA, June 18-24, 1994.
  10. The 906th Meeting of the AMS, Special Session on Quantum Kac-Moody Lie Algebras and Related Topics, Nov. 17-18, 1995, Greensboro, North Carolina.
  11. The 922nd Meeting of the AMS, Special Session on  VOA's, Monstrous Moonshine and Related Topics, May 2-4, 1997, Detroit, Michigan.
  12. The Conference on Generalized Kac-Moody Algebras at the Mathematical Research Institute at Oberwolfach, Germany, organized by Richard Borcherds and Peter Slodowy, July 19-25, 1998.
  13. The 943rd Meeting of the AMS, Special Session on Representations of Lie Algebras, April 24-25, 1999, State University of New York at Buffalo, NY.
  14. Infinite Dimensional Lie Theory and It's Applications Program, Workshop on Vertex Operator Algebras in Mathematics and Physics, Oct. 23 - 27, 2000, The Fields Institute, Toronto, Ontario, Canada.
  15. Ramanujan International Symposium on Kac-Moody Lie Algebras and Applications January 28--31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India.
  16. The 1024th Meeting of the AMS, Special Session on Geometric and Combinatorial Methods in Representation Theory, March 3-4, 2007, Davidson College, Davidson, NC.

Publications:

  1. ``Zones of uniform decomposition in tensor products'', Proceedings of the American Mathematical Society, Vol. 70, No. 2, July 1978, 109-113.
  2. ``Tensor products of finite dimensional modules for complex semisimple Lie algebras'', Lie Theories and Their Applications, Proceedings of the 1977 Annual Seminar of the Canadian Mathematical Congress, Queen's Papers in Pure and Applied Mathematics No. 48, Editors: A. J. Coleman and P. Ribenboim, Queen's University, Kingston, Ontario, 1978, 394-397.
  3. ``The Weyl-Kac character formula and power series identities'', Advances in Mathematics 29, No. 3, September 1978, 271-309 (with J. Lepowsky).
  4. ``A hyperbolic GCM Lie algebra and the Fibonacci numbers'', Proceedings of the American Mathematical Society, Vol. 80, No. 3, November 1980, 379-385.
  5. ``Tensor products of certain modules for the Generalized Cartan Matrix Lie Algebra A_1^(1)'', Communications in Algebra, Vol. 9, No. 12, 1981, 1323-1341.
  6. ``A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2'', Mathematische Annalen 263, 1983, 87-144 (with I. Frenkel).
  7. ``Classical affine algebras'', Advances in Mathematics, Vol. 56, No. 2, May 1985, 117-172 (with I. Frenkel).
  8. ``Some applications of vertex operators to Kac-Moody algebras'', Vertex Operators in Mathematics and Physics. Proceedings of a conference Nov. 10-17, 1983. Edited by J. Lepowsky, S. Mandelstam, I. M. Singer. Publications of the Mathematical Sciences Research Institute #3, Springer-Verlag, 1985, 185-206.
  9. ``The exceptional affine algebra E8(1), triality and chiral algebras'', Lie Algebras and Related Topics, Proceedings of a Conference held in Madison, Wisconsin, May 22 to June 1, 1988; Editors: G. Benkart and J. M. Osborn; Contemporary Mathematics, Vol. 110, American Mathematical Society, Providence, RI, 1989, (with Igor Frenkel and John F. X. Ries).
  10. ``Spinor Construction of Vertex Operator Algebras, Triality and E8(1)'', Contemporary Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 1991, 146 pp. monograph (with Igor Frenkel and John F. X. Ries).
  11. ``Representations of hyperbolic Kac-Moody algebras'', Journal of Algebra, Vol. 156, No. 2, April 1993, 433-453 (with Igor Frenkel and John F. X. Ries).
  12. ``Constructions of vertex operator algebras'', Proceedings of Symposia in Pure Mathematics, Vol. 56, Algebraic Groups and Their Generalizations, William J. Haboush and Brian J. Parshall, Editors, American Mathematical Society, Providence, RI, Part 2, 317-336, April 1994.
  13. ``Spinor construction of the c = 1/2 minimal model'', Moonshine, The Monster, and Related Topics, Contemporary Mathematics, Vol. 193, Chongying Dong and Geoffrey Mason, editors, American Mathematical Society, Providence, RI, 1995 (with John F. X. Ries and Michael Weiner), 45-92.
  14. ``Minimal model fusion rules from 2-groups'', Letters in Mathematical Physics, Vol. 40, No. 2 (1997), 159-169, (with Fusun Akman and Michael Weiner).
  15. ``Type A Fusion Rules From Elementary Group Theory'', Comtemporary Mathematics, Vol. 297, Proceedings of the Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, Charlottesville, VA, American Mathematical Society, Providence, RI, 2002 (with Michael Weiner), 97--115.
  16. ``Fusion Rules for Affine Kac-Moody Algebras'', Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, Vol. 343, American Mathematical Society, Providence, RI, 2004, 53--96.
  17. ``Subalgebras of hyperbolic Kac-Moody Algebras'', Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, Vol. 343, American Mathematical Society, Providence, RI, 2004, 97--114, (with Hermann Nicolai).
  18. ``A New Perspective on the Frenkel-Zhu Fusion Rule Theorem'', (with Stefan Fredenhagen), to be submitted to the Duke Journal of Mathematics.

Work in Progress:

  1. ``Hyperbolic Kac-Moody algebras and the four normed division algebras'', (with Hermann Nicolai and Axel Kleinschmidt).
  2. ``Tensor product decompositions and the Rogers-Ramanujan power series, (with Antun Milas).
  3. ``Tensor and fusion products, Berenstein-Zelevinsky triangles, and lattice-point enumeration in polytopes'', (with Matthias Beck).
  4. ``Vertex Operator Algebras, Triality and E8(1)'', preprint, (with the late John F. X. Ries).

Graduate Students:

  1. Michael D. Weiner, Ph.D. 1994, Thesis: ``Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras'', accepted for publication in the Memoirs of the American Mathematical Society. (Currently Associate Professor at the Altoona Campus of Penn State University.)
  2. Omar Saldarriaga, Ph.D. 2004, Thesis: Fusion Algebras, Symmetric Polynomials, Orbits of $N$-Groups, and Rank-Level Duality. (Currently Assistant Professor at William Patterson University, Wayne, New Jersey.)
  3. Currently working with Quincy Loney and Darryl Daugherty.

Links back to:
Webpage of Alex Feingold
Department of Mathematical Sciences
Binghamton University

This page last modified on 9/26/2007.