 Office: WH 115
 Office hours:
 Courses:
 Ph. D. Students:

Diego Penta,
Spring,
2016
(Coadvisors Fernando Guzman and Marcin Mazur)
Thesis: Decomposition of the Rank 3 KACMoody Lie Algebra F with Respect to the Rank 2 Hyperbolic Subalgebra Fib

Christopher Mauriello,
Spring,
2013
Thesis: Branching Rule Decomposition of Irreducible Level1 E_6^(1)modules with respect to F_4^(1)

Quincy Loney,
Summer,
2012
Thesis: Decomposition of Level1 Representations of D_4^(1) With Respect to its Subalgebra G_2^(1) in the Spinor Construction

Omar Saldarriaga,
Summer,
2004
Thesis: Fusion Algebras, Symmetric Polynomials, Orbits of Elementary NGroups, and RankLevel Duality

Mike Weiner,
Spring,
1994
Thesis: Bosonic Construction of Vertex Operator ParaAlgebras from Symplectic Affine KacMoody Algebras
Alex Feingold
Welcome to my website
Information about my current courses, Spring 2016
For information and the syllabus of my Spring 2016 course, Math 226/227: Calculus 2, link to
Math 226/227.
For information and the syllabus of my Spring 2016 course, Math 404: Advanced Linear Algebra, link to
Math 404: Advanced Linear Algebra.
Squirrel Mail
Here is a link to the Department of Mathematical Sciences
Squirrel Mail login.
Here is a link to my page containing a number of interesting animated gifs,
some just fun, but some with real mathematical contents:
My animated GIFs Gallery.
Mathematical Sculptures by Alex Feingold
I have a strong interest in sculptures, especially those related to mathematics, and
I have made some in stone, bronze and wood. Here are links to webpages showing my work,
and links to the artists who have inspired and influenced me. You can also follow the
following link to an
Artist's Statement
which I have written.
From December 3  30, 2015, two of my sculptures were on exhibition at the
Art Mission, Prospect Street in downtown Binghamton.
From September 8  October 29, 2015, some of my sculptures are on exhibition at the
Broome County Public Library, on Court Street in downtown Binghamton.
Starting in Fall 2014 I have been working with a 3D graphics program, Rhino for Mac, to make a computer
model of a figure 8 knot whose cross section is a triangle with concave sides (close to a hypocycloid) where
the triangle rotates slightly as it goes around the curve so that when it comes back to the start, it has
rotated just 120 degrees. That means there is only one edge and one surface, like a Mobius strip. I have made
such sculptures in bronze before in a large size, most recently in Spring 2014, but my goal was to use 3D
printing to make it accurately in a small size for jewelry. Pictures of the results, and of other 3D printed
sculptures I designed, can be seen on my webpages:
Sculptures Fall 2014
and
Sculptures Spring 2015.
In summer 2014 I donated one of my bronze cast sculptures, a Mobius Trefoil Knot, to the Albert Einstein
Institute (AEI), also called the Max Planck Institute for Gravitational Physics, in Potsdam/Golm, Germany.
During my visit, June 116, 2015, there was a dedication ceremony, showing how it had been mounted on a
specially made wooden pedestal, with a plaque showing the title of the piece, the donor, and a few words.
There are some pictures on my webpage
Sculptures Spring 2015
showing the mounted sculpture, and some of the dedication ceremony, conducted by Prof. Hermann Nicolai,
my good friend, collaborator, and a director of AEI.
In Spring 2014 I made a bronze sculpture based on the logo of the Institut Hautes des Etudes Scientifique (IHES),
BuresSurYvette, France, and donated it to IHES in graditude for the research support they gave me the last three
summers. It was made by bending a 1/4" diameter bronze rod 96" long. A picture of the sculpture and a graphic of
the logo on which it was based are shown on my webpage:
Sculptures Spring 2014.
From December 3, 2014  January 29, 2015, two of my sculptures were on exhibition at the
Art Mission, Prospect Street in downtown Binghamton.
From September 4  October 31, 2014, some of my sculptures were on exhibition at the
Broome County Public Library on Court Street in downtown Binghamton.
From January 8  February 7, 2014, two of my sculptures were on exhibition at the
Art Mission, Prospect Street in downtown Binghamton.
From December 3  27, 2012, some of my sculptures were on exhibition at the
Art Mission, Prospect Street in downtown Binghamton.
From June 1  July 27, 2012, some of my sculptures were on exhibition at the
Broome County Public Library on Court Street in downtown Binghamton.
One of my bronze sculptures from Fall 2010, a 12" diameter bronze torus (3,5) knot mounted on a bluestone
base over a spherical mirror, was selected for the
2011 Roberson Regional Art Exhibition,
and will be on display at the
Roberson Museum and Science Center,
located at 30 Front Street, Binghamton, NY, 13905, during
July 9  October 16, 2011.
From September 130, 2010, some of my sculptures were on exhibition at the
Broome County Public Library on Court Street in downtown Binghamton.
From June 927, 2010, eleven of my kinetic sound sculptures will be on exhibition
in the lobby of the Know Theater, located at
74 Carroll Street, Binghamton, NY. This exhibition was selected to compliment their
current production, ``Black Comedy" by Peter Shaffer.
On September 4, 2009, 7  9 PM, in the lobby of the Downtown Campus of Binghamton
University, I exhibited 19 of my sculptures as part of the First Friday Art Walk. The
sculptures shown included many of my kinetic sound sculptures, some in the style of
Harry Bertoia, both bronze and steel examples, and some others based on knot theory. This was
my first solo exhibition, and I want to thank Jennifer Keegin, Associate Director of
Campus Life for Campus Activities, for her help in arranging this exhibition, and Prof.
Jim Stark for the use of his pedestals for the display of the artwork.
The 11th annual Bridges conference took place in the birthplace of M. C. Escher,
Leeuwarden, The Netherlands, July 2428, 2008, with a Mathematical Art Public Activity Day
on July 29. The Bridges conferences are a literal bridge between mathematics, music, art,
architecture and culture, and include an exhibition for 2D and 3D mathematical art.
Four of my sculptures were submitted and accepted for the exhibition, but could not
attending conference so I am sent pictures of them. This link takes you to the list
of artists who participated in the exhibition, and includes links to pictures of their
work:
Bridges 2008 Artists.
The following link to the conference website has many interesting links to previous conferences,
and to mathematical art on the internet:
Bridges Conference 2008.
Two of my kinetic sound sculptures were in the Members Exhibition Dec. 728, 2007 at the
Art Mission in Binghamton, NY. Another bronze sculpture
of mine is part of their Members Exhibition Dec. 528, 2008.
One of my sculptures was exhibited November 1  4, 2007, at the Center Gallery at the
University of South Florida, Tampa, FL, in an exhibition ``Rhythm of Structure: Beyond the
Mathematics", curated by John Sims as part of the conference
Knotting Mathematics and Art:
A Conference in Low Dimensional Topology and Mathematical Art. You can see pictures I took
of the pieces in the exhibition, and pictures I took at the conference, by following the link:
Knot Art Conference Pictures.
Four of my sculptures were exhibited May 3  24, 2007, at the
Altered Esthetics Gallery
in Minneapolis, MN, during their show ``Art For The Other Senses". A writeup about the
exhibition which mentions my kinetic sound sculptures can be found at the following link:
Altered Esthetics Exhibition in vita.mn story.
Kinetic Sound Sculpture
inspired by the work of Harry Bertoia.
Stone and wood sculpture projects
inspired by Helaman Ferguson.
Bronze metal sculptures
made as castings of earlier works in other media and by welding.
Steel and plastic sculptures
inspired by hyperbolic surfaces and the tensegrity sculptures of
Kenneth Snelson
and
John Robinson.
I have also collected some beautiful examples of
netsuke.
Sculptures from the Spring 2006 semester can be
seen on the following webpage:
Spring 2006 Sculptures.
Sculptures from the Summer of 2006 are shown on the following webpage:
Summer 2006 Sculptures.
Sculptures from the Fall of 2006 are shown on the following webpage:
Fall 2006 Sculptures.
Sculptures from the Spring 2007 semester can be
seen on the following webpage:
Spring 2007 Sculptures.
Sculptures from the Summer of 2007 are shown on the following webpage:
Summer 2007 Sculptures.
Sculptures from the Spring 2008 semester can be
seen on the following webpage:
Spring 2008 Sculptures.
Sculptures from the Summer of 2008 are shown on the following webpage:
Summer 2008 Sculptures.
Sculptures from the Fall of 2008 are shown on the following webpage:
Fall 2008 Sculptures.
Sculptures from the Summer of 2009 are shown on the following webpage:
Summer 2009 Sculptures.
Sculptures from the Fall of 2009 are shown on the following webpage:
Fall 2009 Sculptures.
Sculptures from the Spring of 2010 are shown on the following webpage:
Spring 2010 Sculptures.
Sculptures from the Fall of 2010 are shown on the following webpage:
Fall 2010 Sculptures.
Sculptures from the Spring and Summer of 2011 are shown on the following webpage:
Spring and Summer 2011 Sculptures.
Sculptures from the Spring and Summer of 2012 are shown on the following webpage:
Spring 2012 Sculptures.
Sculptures from the Fall 2012 and Spring 2013 are shown on the following webpage:
Fall 2012 and Spring 2013 Sculptures.
Sculptures from the Fall 2013 are shown on the following webpage:
Fall 2013 Sculptures.
Sculptures from the Spring and Summer of 2014 are shown on the following webpage:
Spring 2014 Sculptures.
Sculptures from the Fall 2014 are shown on the following webpage:
Fall 2014 Sculptures.
Sculptures from the Spring 2015 are shown on the following webpage:
Spring 2015 Sculptures.
Sculptures from the Fall 2015 are shown on the following webpage:
Fall 2015 Sculptures.
Sculptures from the Spring 2016 are shown on the following webpage:
Spring 2016 Sculptures.
I am president of our local chapter of Phi Beta Kappa, Psi of New York.
Phi Beta Kappa is the oldest and most prestigious liberal arts honor society in the country.
The link above will take
you to our webpage which has information about the history of the society,
our selection procedures, and lists of students who qualified in recent years.
I am the Permanent Faculty Correspondent of our local chapter of Pi Mu Epsilon,
the Alpha Omicron of New York, the National Mathematical Honor Society. For information
about our newly formed chapter, follow the link above.
I have been a serious ice skater (figure skating, not hockey!) since 1968. I am a member
of the United States Figure Skating Association, was a member of the Professional Skaters Association, and
I am currently a member and President of the
Binghamton Figure Skating Club.
The link in the heading above this paragraph will take you to a webpage where I have put my
skating resume and some pictures.
Curriculum Vitae:
Follow the indicated link to my
Curriculum Vitae
Research Interests:
KacMoody Lie Algebras and their Representations, Vertex Operator Algebras,
Conformal Field Theory, Fusion Rules, Hyperbolic Weyl Groups.

Weyl Groups of Some Hyperbolic KacMoody Algebras, A paper written in collaboration
with Daniel Vallières, Jan. 15, 2016. The proof of Theorem 6.5 in this paper refers
to some calculations done with PARI. The PARI code used can be found at the following link:
PARI code file and some explanation of the code can be found in the following text file:
readme.txt file. This paper can also be found on the arXiv through the following link:
arXiv link to this paper.
 Decomposition of Level1 Representations of
D_{4}^{(1)} With Respect to its Subalgebra G_{2}^{(1)} in the Spinor Construction
, A Ph.D. dissertation written by Quincy Loney under my direction.
 Branching Rule Decomposition of Irreducible Level1
E_{6}^{(1)}modules with respect to F_{4}^{(1)}
, A Ph.D. dissertation written by Christopher Mauriello under my direction.

The 3State Potts Model and RogersRamanujan Series, A paper written in collaboration
with Antun Milas (SUNYAlbany), the Central European Journal of Mathematics, Jan. 2013, Vol. 11,
Issue 1, pp. 116, (DOI) 10.2478/s1153301200867. A link to the online journal is
here.

Matrix Realizations of Hyperbolic Triangle Groups, A paper written in collaboration
with Elizabeth Dwornik (UnionEndicott High School, Endicott, NY) as the culmination of
a oneyear mentoring project. This paper is looking for a home. If you know of a nice
journal to which it should be submitted, please contact Alex Feingold.
 Hyperbolic Weyl Groups and the
Four Normed Division Algebras, A paper written in collaboration with Axel
Kleinschmidt (Universite Libre de Bruxelles & International Solvay Institutes,
Bruxelles, Belgium) and Hermann Nicolai (Max Planck Institute for Gravitational
Physics, Albert Einstein Institute, Golm, Germany),
Journal of Algebra 322 (2009), pp. 12951339. Another paper with the same title
and authors was accepted for publication in the proceedings of an international
conference on vertex operators and related areas in honor of Geoffrey Mason. That
conference took place at Illinois State University in July, 2008. That shorter paper is an
introduction to and announcement of the results in this paper, and will appear in
Vertex operator algebras and related areas,
Contemporary Mathematics, Amer. Math. Soc., Providence, RI.

A New Perspective on the FrenkelZhu Fusion Rule Theorem,
A paper written in collaboration with Stefan Fredenhagen (Max Planck Institute
for Gravitational Physics, Albert Einstein Institute, Golm, Germany), submitted
for publication in October 2007, accepted in June 2008,
to appear in the Journal of Algebra 320 (2008), pp. 20792100.

Fusion Algebras, Symmetric Polynomials, Orbits of NGroups, and RankLevel Duality,
The Ph.D. thesis of Omar Saldarriaga, done under the direction of Alex Feingold.
Also available as a pdf file:
pdf file of Omar's thesis.

Subalgebras of Hyperbolic KacMoody Algebras,
KacMoody Lie Algebras and Related Topics, Ramanujan International Symposium
on KacMoody Algebras and Applications, Jan. 2831, 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, 97114, (with Hermann Nicolai,
Max Planck Institute for Gravitational Physics, Potsdam, Germany).

Fusion Rules For Affine KacMoody Algebras,
KacMoody Lie Algebras and Related Topics, Ramanujan International Symposium
on KacMoody Algebras and Applications, Jan. 2831, 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, 5396.

Valedictory Address by Alex J. Feingold to the
Ramanujan International Symposium on
KacMoody Lie Algebras and Applications, Jan. 2831, 2002,
Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai, India.

Type A Fusion Rules From Elementary Group Theory,
Comtemporary Mathematics, Vol. 297, Proceedings of the
Conference on InfiniteDimensional Lie Theory and Conformal Field Theory,
Editors: S. Berman, P. Fendley, Y. Huang, K. Misra, and B. Parshall,
Charlottesville, VA, American Mathematical Society, Providence, RI, 2002
(with Michael Weiner), 97115.

Minimal Model Fusion Rules From 2Groups,
Letters in Mathematical Physics, Vol. 40, No. 2, April (II), 1997, 159169,
(with Fusun Akman and Michael D. Weiner).

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, pp. 4592 (with John F. X. Ries and
Michael D. Weiner).

Constructions of vertex operator algebras, Proceedings of Symposia in Pure
Mathematics, Vol. 56, Algebraic Groups and Their Generalizations, Editors
William J. Haboush and Brian J. Parshall, American Mathematical Society,
Providence, RI, April 1994, Part 2, 317336.

Representations of hyperbolic KacMoody algebras, Journal of Algebra, Vol.
156, No. 2, April 1993, 433453 (with Igor B. Frenkel and John F. X. Ries).

Spinor Construction of Vertex Operator Algebras, Triality and E_8^(1),
Contemporary Mathematics, Vol. 121, American Mathematical Society,
Providence, RI, 1991, 146 pp. monograph (with Igor B. Frenkel and John F. X.
Ries).

Classical Affine Algebras (1),
Classical Affine Algebras (2),
Classical Affine Algebras (3),
Classical Affine Algebras (4),
Classical Affine Algebras (5),
Advances in Mathematics, Vol. 56, No. 2, May 1985, 117172 (with Igor B. Frenkel).
I divided the paper into five pdf files rather than one big file.

Some applications of vertex operators to KacMoody algebras,
Vertex Operators in Mathematics and Physics. Proceedings of a conference Nov. 1017, 1983.
Edited by J. Lepowsky, S. Mandelstam, I. M. Singer. Publications of the Mathematical Sciences
Research Institute #3, SpringerVerlag, 1985, 185206.

A Hyperbolic KacMoody Algebra and the Theory of Siegel Modular Forms of Genus 2,
Mathematische Annalen 263 (1983), 87144, (with Igor B. Frenkel). Appearing here with
the permission of Niedersachsische Staatsund Universitatsbibliothek, which has the
volume at
this link.
Interesting Sites on the Web:
 MSRI Berkeley
 Public Broadcasting Service
 American Mathematical Society
 Max Planck Institut für
Mathematik, Bonn, Germany
 Max Planck Institut für
Gravitationsphysik, Albert Einstein Institute, Potsdam, Germany.
Interesting Mathematical Pictures From The Minnesota Geometry Center
 Picture 1, Jekyll and Hyde
 Picture 2, Red Eyes
 Picture 3, Crabs
 Picture 4, Frosted Pane
 Picture 5, Henon Ring
 Picture 6, Henon Isthmus
 Picture 7, Henon Crucified
 Picture 8, Hyperbolic Space Tiled with
Dodecahedra
 Picture 9, Inversion
 Picture 10, Penrose by Quasitiler
 Picture 11, Escher Fish
Solomon Mandelkern was a famous biblical scholar, known for writing a concordance of the
bible (Old Testiment). He is a distant relative of mine, and my father, Bernard Feingold,
did a lot of research on Mandelkern's life and wrote a short biography which was published
in an issue of the Jewish Historical Society of Maryland. A scaned copy of that article is
available through the link above.
My father, Bernard Feingold, a retired Brig. General in the Maryland National Guard, was also
very interested in military history, and helped to develop a museum at the Fifth Regiment Armory
in Baltimore, Maryland. He served as a director of the museum, as mentioned on the webpage whose
link is above. The museum has considerably expanded since his passing in 1999.
This page was last modified on 1/21/2016.
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