[Turkmath:7158] seminer duyurusu

[Songul Esin]

DO�U� UNIVERSITY

DEPARTMENT OF MATHEMATICS

 

SEMINAR

Prof.Dr.Murad ÖZAYDIN

 

OKLAHOMA UNIVERSITY

 

 

q+

 

The limit of a q-analog as q approaches 1 yields its classical counterpart. For instance the q-analog of a natural number n is 1+q +q2 +. . .+qn−1, with the q-factorial and the q-binomial coefficients defined analogously. We can think of q as a deformation parameter, but depending on the context q may more naturally be:

(i) A prime power (when working over finite fields);

(ii) A real number with 0 << q < 1 (in quantum calculus);

(iii) A complex number of modulus 1 (in a Fourier series or a character formula);

(iv) The variable of a generating function (for the growth of a group or a Poincare-Hilbert series).

For instance the q-binomial coefficient, which is a monic palindromic, unimodal polynomial in q of degree k(n − k), is the number of k-dimensional subspaces of an n-dimensional vector space over a finite field with q elements, as well as the Poincare series of the cohomology of the complex Grassmannian Gr(n, k). This is not a coincidence, an explanation is given by a uniform Schubert-Bruhat decomposition of Gr(n, k) over any field. We can get finer versions of many classical notions (q = 1 yielding the classical case) like the q-binomial theorem, the q-Euclidean algorithm, the q-derivative, the q-exponential, etc. For example if A and B q-commute, that is, BA = qAB (e.g., the shift and the modulation operators in signal processing), where q is centralized by both A and B, then the expansion of (A + B)n is in terms of the q-binomial coefficients. Sometimes there is more than one q-analog (what’s amazing of course is that there are so few), such as for the q-Catalan numbers or the q-exponential (the inverse of one q-exponential is given in terms of the other). There are other interesting expressions, let’s call them r-analogs, which yield a classical one when the parameter r specialises to 1. They come up in connection with chromatic polynomials, hyperplane arrangements, Poincare polynomials of (the cohomology of) configuration spaces. They somehow don’t have the “quantum� flavor of q-analogs, but are related to those (via the Berry-Robbins question and the Atiyah conjecture for configuration spaces), usually in connection with a Weyl group. The values obtained when q is specialised to roots of unity (other than 1) has also been of interest in several areas (representation theory, invariants in low dimensional topology, the cyclic sieving phenomena, etc.). Perhaps the most intriguing focus of current interest is the search for the “field with one element�

with high expectations for the consequences.

 

Date : 13 August 2010 at 11:00

Location: H 303 ( with air-condition)

 

 

  

 

 
Asst. Prof. Songül ESİN
Dogus University, Acibadem, Kadikoy, 34722
Istanbul, TURKEY
Tel:  +90 216 3271104 / 1345
Fax: +90 216 5445533
http://www3.dogus.edu.tr/sesin
 
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