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32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme. rial on Yang-Baxter graphs and Macdonald polynomials is In particular, for superintegrable systems in 3D, the poly- based on joint work with J.-G. Luque. nomial representations of quadratic algebras are given in terms of two-variable polynomials and the two-variable Charles F. Dunkl analog of the Askey scheme, including the quadratic Racah University of Virginia algebra, will be discussed along with the limiting processes [email protected] within the scheme. Sarah Post IP2 University of Hawaii at Manoa Two Variable q-Polynomials Department of Mathematics [email protected] Abstract not available at this time. Mourad Ismail IP7 University of Central Florida A New Look at Classical Orthogonal Polynomials [email protected] There are two possible definitions of classical orthogo- nal polynomials:(i) they satisfy a second order differential IP3 or difference equation;(ii) (generalized) derivative of them On the Asymptotic Behavior of a Log Gas in the gives again orthogonal polynomials. Both definitions are Bulk Scaling Limit in the Presence of a Varying related with concrete forms of corresponding operators. We External Potential propose a new approach dealing with some abstract um- bral operators. This gives a wide generalization of a notion Abstract not available at time of publication. of classical orthogonal polynomials. Percy Deift Alexei Zhedanov Courant Institute Donetsk Institute for Physics and Engineering, Ukraine New York University [email protected] [email protected] IP4 IP8 Title Not Available at Time of Publication Asymptotic and Numerical Aspects of Special Functions Abstract not available at this time. For the numerical evaluation of special functions, asymp- Olga Holtz totic expansions are an important tool. The standard ex- University of California, Berkeley pansions can be used rather straightforwardly. The so- Technische Universitaet Berlin called uniform expansions need more attention, especially [email protected] for critical values of secondary parameters in the asymp- totic problem. For example, the Airy-type expansion of the Bessel function Jν (z) can be used for large domains of the IP5 argument and order, but for the transition value z = ν spe- Integrable Probability and the Role of Painlev cial methods are needed for computing the coefficients. We Functions mention several methods for handling this type of problem. We start with a few examples for which Maple and Mathe- We will review various models in probability that are inte- matica have problems in the evaluation of well-known spe- grable in the sense that various distribution functions can cial functions, like the Kummer U-function, for medium- be explicitly evaluated in terms of Painleve functions and sized values of the parameters. We discuss recent activities their generalizations. We develop in more detail a class in the Santander-Amsterdam project on the evaluation of of stochastic growth models that belong to the Kardar- special functions, in particular for certain cumulative dis- Parisis-Zhang (KPZ) uni- versality class such as the asym- tribution functions. We start with the incomplete gamma FA15 Abstracts 33 functions, and we give recent results for the non central chi- polynomials, which are multivariate orthogonal polynomi- squared or the non central gamma distribution, also called als. Recently Eric Rains defined moments of Koornwinder Marcum Q−function in radar detection and communica- polynomials at q=t, which appear to be polynomials with tion problems. This is joint work with Amparo Gil and positive coefficients when written appropriately in the pa- Javier Segura (University of Cantabria, Santander, Spain). rameters of the ASEP. I’ll explain joint work with Sylvie Corteel in which we show that Koornwinder moments at q=t are related to the 2-species ASEP, an exclusion process Nico M. Temme involving two different types of particles. I’ll also describe Centrum Wiskunde & Informatica complementary work of Olga Mandelshtam and Xavier Vi- The Netherlands ennot providing a combinatorial description of the station- [email protected] ary distribution of the 2-species ASEP. Lauren Williams IP9 University of California Berkeley Multivariate Orthogonal Polynomials and Modified [email protected] Moment Functionals Multivariate orthogonal polynomials can be defined by SP1 d means a measure defined on a domain on R . A very impor- SIAG/OPSF Gbor Szeg Prize Announcement and tant class of multivariate orthogonal polynomials is called Lecture classical because the measure satisfies a matrix analogue of the Pearson differential equation as well as the orthogo- Abstract not available at time of publication.. nal polynomials are the eigenfunctions of a partial second order differential operator. In this talk, we present old and Karl Liechty new results on classical multivariate orthogonal polynomi- DePaul University als. In particular, some classical multivariate orthogonal Department of Mathematical Sciences polynomials and some useful modifications will be studied, [email protected] as well as their impact into the useful properties of the orthogonal polynomials. We study the so–called Uvarov modification obtained by adding to the measure one or a CP1 finite set of mass points. Recently, Christoffel modifica- Killip-Simon Problem and Jacobi Flow on Gsmp tion in several variables, that is, the modification obtained Matrices by multiplying the measure times a polynomial, has been studied in the frame of linear relations. D. Damanik, R. Killip and B. Simon completely described the spectral properties of Jacobi matrices J+,whicharein 2 Teresa E. P´erez asence perturbations of the isospectral torus of periodic Universidad de Granada Jacobi matrices. The spectrum of a periodic Jacobi matrix Departamento de Matem´atica Aplicada is a system of intervals of a very specific nature. Jointly [email protected] with P. Yuditskii, we generalize this result to spectral sets, which are arbitrary finite system of intervals. IP10 Benjamin Eichinger Hypergeometric Series: On Number Theory’s Se- Institute of Analysis, Johannes Kepler University Linz cret Service [email protected] A natural outcome of the theory of generalized hypergeo- CP1 metric functions are rational approximations to the values of Riemann’s zeta functions and alike mathematical con- A Matrix Approach for the Semiclassical and Co- stants. In my talk I plan to outline the way it goes (hyper- herent Orthogonal Polynomials geometric series, hypergeometric Barnes- and Euler-type integrals) and stress on some recent achievements — the We obtain a matrix characterization of semiclassical or- current best irrationality measures of π (due to Salikhov), thogonal polynomials in terms of the Jacobi matrix as- of log 2 (due to Marcovecchio) and of ζ(2). The final part sociated with the multiplication operator in the basis of of the talk will discuss some linear and algebraic indepen- orthogonal polynomials, and the lower triangular matrix dence results that make use of generalized hypergeometric that represents the orthogonal polynomials in terms of the functions. monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals. Wadim Zudilin University of Newcastle Lino G. Garza [email protected] Universidad Carlos III de Madrid [email protected] IP11 Francisco Marcell´an Orthogonal Polynomials and the 2-Species ASEP Universidad Carlos III de Madrid Instituto de Ciencias Matem´aticas (ICMAT) The asymmetric exclusion process (ASEP) is a model of [email protected] particles hopping on a 1-dimensional lattice with open boundaries. The partition function of this model is related Luis E. Garza to moments of Askey-Wilson polynomials. Askey-Wilson Departamento Matematicas,Universidad Nacional polynomials are at the top of the hierarchy of orthogonal Colombia polynomials, and are also
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