Welcome to the Research Pages of William Sit
Professor Emeritus
Department of Mathematics
The City College of The City University of New York
160 Convent Avenue, New York, NY 10031
Office: R6/291D (North Academic Center)
Office Tel: 212-650-5179
Office Fax: 212-862-0004
Email: wsit@ccny.cuny.edu

Home Page: wsit.ccny.cuny.edu  Last Updated on: January 31, 2016

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Professor Sit was trained as a differential algebraist. Under the supervision of Ellis R. Kolchin at Columbia University, Sit's 1972 doctoral thesis was on differential algebraic subgroups of the special linear group SL(2), with an application to the classification of certain strongly normal extensions in differential Galois theory. He then worked on the differential dimension polynomial of a (differentially) finitely generated extenion by which the dependence of the generators on arbitrary functions can be measured when the generators are viewed as a generic solution to a system of partial (or ordinary) differential equations. Since a differential algebraic group satisfies a system of differential polynomial equations, it has an associated differential dimension polynomial. One of his main results is a theorem relating certain coefficients of the differential dimension polynomials of two linear differential algebraic subgroups of GL(n) to corresponding coefficients of their intersection and product. Another one is the Well-Ordering Theorem of differential dimension polynomials under a natural ordering.

In the second half of 1970's, he earned a master degree in computer science from The City College of New York, and also attended graduate courses in operations research at Columbia University. Under his mentor Donald Goldfarb, Sit's master degree thesis proves that the worst case behaviour of the steepest edge pivoting rule of Goldfarb and Reid for the simplex algorithm for linear programming is exponential in the size of the problem.

Sit spent a sabbatical year (1987–88) at IBM's T. J. Watson Research Center with the group headed by Dr. Richard Jenks, and was fascinated by the computer algebra system Scratchpad, the predecessor of the now open-sourced CAS Axiom. Sit applied computer algebraic techniques to differential algebra and developed prototype packages in Scratchpad to automate the search for first integrals of parametric autonomous systems, based on his own adaptation of an algorithm originally due to Larry Goldman. As a by-product, he discovered an algorithm for solving parametric linear equations which is vastly superior to the usual Gaussian elimination method. Parametric linear equations are the key to many other problems, including solutions of differential equations. This new method was based on algebraic geometry and the then emerging Gröbner basis techniques. By the turn of the century, Sit had developed other packages for differential algebra in Axiom. These included packages for quasi-algebraic sets (1998) and initial value domains for implicit first order systems of ordinary differential equations (2000–2001).

During 1991–93, Sit worked with Professor Gilbert Baumslag on computational group theory using C and C++. From 1991–94, he was the project director for an NSF-ILI equipment grant and was mainly responsible for setting up a computer laboratory-classroom facility with 24 IBM compatible PC's running Mathematica and Maple. He developed many tutorial style Mathematica notebooks for differential equations.

In 1998, Sit was invited as a visiting scholar to Wolfram Research, Inc. from January 5 through January 30. He experimented in new techniques to program the Mathematica front end to improve user interface and in 1999 was awarded a CCNY Teach21 Presidential Award to update the Mathematica notebooks for differential equations. Almost all of the notebooks were completely rewritten to take advantage of these new techniques. The new version was demonstrated at the August, 1999 International Symposium on Symbolic and Algebraic Computation, held in Vancouver, Canada. Sit developed additional notebooks and devised new programming techniques to further enhance students' learning experience when using these notebooks. The packages were freely available for installation by students on their home computers.

In Fall, 1998 he was a general member at the Mathematical Science Research Institute at Berkeley. He participated in two workshops on Symbolic Computations and was invited to present two talks, one being an introduction to differential algebra, and the other, a research report on initial value problems for differential algebraic equations.

Since his sabbatical at IBM, Sit's research interest has been in computational differential algebra. He is active locally and internationally in organizing conferences. He and Professor Raymond Hoobler organized a special year on Computational Differential Algebra and Algebraic Geometry in 1995–1996. He regularly organized a monthly weekend seminar on Differential Algebra, then held at CCNY and later at Hunter College. In 1997, he received a private grant from Mrs. Kate Kolchin to promote research in differential algebra. In 2000, he co-founded and co-organized with Li Guo, William Keigher of Rutgers University (and others) the first International Workshop on Differential Algebra and Related Topics (DART), which was held on November 2–3, 2000 at the Newark Campus of Rutgers University. This workshop was followed by a Special Session of the AMS at Columbia on November 4–5, which he also co-organized. For the year 2001–2002, he received an award from the CUNY Faculty Development Program to organize a seminar series on Differential Algebra and Symbolic Computations. In 2002, he was principal director and co-organizer of the East Coast Computer Algebra Day 2002 (ECCAD2002), which was (still is) an annual major regional conference of its kind in the United States. ECCAD2002 was held on May 18, 2002 at LaGuardia Community College (Prof. Jerry Ianni of LGCC was the local organizer). Sit co-chaired (with Ziming Li from the Chinese Academy of Sciences) the Program Committee for the Asian Symposium in Computer Mathematics, 2003 (ASCM'03), which was held in Beijing October 23–25, 2003 and co-edited its proceedings.

Before his retirement in September 2006, Sit had been a regular reviewer for the Journal of Symbolic Computation, and at the invitation of the Journal, he completed editorial work as a guest editor (with co-editor Manuel Bronstein) for a special issue on Differential Algebra and Differential Equations, a two-volume issue that appeared in November, 1999. In 2002, he co-edited (with Li Guo, William Keigher, and Phyllis Cassidy) on the proceedings of DART (see above), which appeared in June, 2002, in which he contributed also as an author to a lead tutorial article (70 pages) in differential algebra.

Since 2005, the Kolchin Seminar in Differential Algebra (KSDA) has moved to the Graduate Center and meets weekly on Fridays, often with both a formal morning and an informal afternoon session. After retirement, Sit continued to run KSDA and has been the webmaster of the site. He continued to promote the International Workshop on Differential Algebra and Related Topics (DART), and was on the program committees for four more DART workshops (2007 and 2008 at Rutgers University, Newark, N.J.; 2010 at the Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences at Beijing, China; 2013 at Engineering School Polytech'Lille, Lille, France). He is again a program committee member for the upcoming DART VI, to be held August 10--14, 2015, again in Beijing, China.

On February 6, 2009, Sit was awarded Distinguished Asian Role Model of the Year by the Asian Alumni Group of the Alumni Association of CCNY.

During the last decade, Sit turned his attention to Rota-Baxter algebras, which are algebras with a linear operator that satisfies the Rota-Baxter axiom (an abstraction of the integration by parts formula). With Professor Li Guo of Rutgers University at Newark, he pioneered the use of computer algebra to attack the classification of all linear operators on associative algebras, a problem first posed by Rota. They have identified 6 differential type algebras and 14 Rota-Baxter type algebras. As a side benefit, the enumerations of Rota-Baxter words and differential Rota-Baxter words (which are kind of monomial basis for the various types of free algebras with a linear unary operator) provide new combinatorial results, yielding new interpretation of some well-known integer sequences as well as providing new families of sequences in the well-known database On-Line Encyclopedia of Integer Sequences initated by Neil Sloane. The theory developed has now been applied by other researchers (notably Markus Rosenkranz, Georg Regensburger, and Li Guo) to define axiomatic integral-differential algebras, where initial and boundary value problems can be formulated abstractly and algebraically, potentially leading to symbolic algorithms to solve integro-differential equations.

He was recently invited to give two conference talks, the first at the Regional Meeting of the Canadian Mathematical Society at Winnepeg, in June, 2014 (Session: Gröbner Bases and Computer Algebra), and second at the Applications of Computer Algebra conference, at Fordham University, July 2014 (Session: Algorithmic and Algebraic Aspects of Differential and Integral Operators).