Reading Class 290-84: Hecke Algebras and Orthogonal Polynomials

Winter 2005

Meeting:	The reading class meets alternately Mondays at noon (or 12:05 or 12:10?) (room TBA) Thursdays at 3:10 (room TBA) during the Winter Quarter.
Instructor:	Monica Vazirani, Kerr Hall 579, phone: 752-2218, vazirani@math.ucdavis.edu
CRN:	80761
Grading:	Students who plan to attend regularly (missing no more than 3 classes) and participate in the discussion can register for one credit. Students who will present a paper in class can register for three credits.
Web:	http://www.math.ucdavis.edu/~vazirani/W05/290.html

Course description

This reading seminar runs in the context of the Research Focus Group on "Hecke Algebras and Orthogonal Polynomials". We will read and discuss topics of mutual interest. Suggested topics are:

Review of other root systems, in particular type BC, and the Askey-Wilson polynomials
other papers of Eric Rains
More on the double affine Hecke algebra. Related work of Cherednik, Opdam, Gordon, etc.
Other viewpoints of Macdonald polynomials, as in Haiman's work. Connection to diagonal harmonics. A combinatorial description.
special functions
a classical look at orthogonal polynomials of several real variables.
Gauss cubature (the analogue of quadrature)
generalized multivariate Hermite and Laguerre polynomials
generalized classical orthogonal polynomials on the ball and simplex, etc.

Suggested texts/papers are:

Macdonald ``Symmetric functions and orthogonal polynomials,'' QA212.M33
Koornwinder ``Askey-Wilson polynomials for root systems of type $BC$,'' QA353.H9 H97
Hypergeometric functions on domains of positivity, Jack polynomials, and applications
(Tampa, FL, 1991), 189--204, Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992.
Kirillov ``Lectures on affine Hecke algebras and Macdonald's conjectures,''
Charles Dunkl and Yuan Xu "Orthogonal polynomials of several variables." QA404.5.D86
Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press, Cambridge, 2001.
Szegő ``Orthogonal Polynomials,'' ( Colloquium publications (American Mathematical Society) ; v. 23)
Shields Library QA404.5 .S9
Andrews, Askey, Roy ``Special functions," ( Encyclopedia of mathematics and its applications ; v. 71)
Shields Library QA351 .A74

Further suggestions are welcome!

Schedule Winter

January 7: Organizatorial meeting at 12:30 in 693 Kerr
January 31: MONDAY noon. 593 Kerr. Gus Wiseman on Umbral Calculus.
reference: "The Umbral Calculus" by Steven Roman (1984). [this is out of print, but the library carries it]
reference: Steven M. Roman and Gian-Carlo Rota, The Umbral Calculus, Advances in Mathematics 27 (1978), 95-188 [this is available online]
February 10: THURSDAY 3pm. 593 Kerr. Eric Rains on BC_n-symmetric polynomials and/or related topics
Feb 14:MONDAY noon. Philip Sternberg 593 Kerr
Feb 24: THURSDAY 3pm. John Millson of UMD in 693 Kerr
Generalized Kostant convexity theorems, the constant term map on spherical Hecke algebras and branching to Levi subgroups
March 3: THURSDAY 3pm. 593 Kerr.(?) Eric Rains on BC_n-symmetric polynomials and/or related topics
Mar 7 or March 10

Organizatorial meeting is on Friday, Jan 7 at 12:30 in 693 Kerr
For winter quarter-- the possible times are Monday noon, or Thursday 3-4pm (or a mixture of each). (Room TBA).
The reading seminar will only be held if there is sufficient interest.
Email list: the people who now receive emails about this course are listed here. If you are NOT on this list and would like to be (or if you ARE on the list and would like to be removed), please email monica.
vazirani@math anne@math rains@math sternberg@math issy@math deka@math tmcal@math gus@nafindix.com endelman@math capnjim@math wilmarth@math svis@math jpsteinb@math ksharding@ucdavis.edu caputo@math gaudard@math ksharding@math xqwang@math lesyoung@math quillbone@math rdanner@math hallowell@math haws@math lai@math marion@math mrathbun@math dan_rutherford@math xqwang@math