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class=MsoNormal align=center><SPAN style="FONT-SIZE: 16pt"><FONT
face="Times New Roman"><FONT size=5><STRONG>q<SPAN
class=049584611-11082010>+</SPAN></STRONG></FONT></FONT></SPAN></P>
<P style="TEXT-ALIGN: center; MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal align=center><SPAN style="FONT-SIZE: 16pt"><FONT
face="Times New Roman"><o:p></o:p></FONT></SPAN> </P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">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:<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">(i) A
prime power (when working over finite fields);<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">(ii)
A real number with 0 << q < 1 (in quantum
calculus);<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">(iii)
A complex number of modulus 1 (in a Fourier series or a character
formula);<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">(iv)
The variable of a generating function (for the growth of a group or a
Poincare-Hilbert series).<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">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 <SPAN class=049584611-11082010>d</SPAN>imensional topology, the
cyclic sieving phenomena, etc.). Perhaps the most <SPAN
class=049584611-11082010>i</SPAN>ntriguing focus of current interest is the
search for the “field with one element”<o:p></o:p></FONT></SPAN></P>
<P style="MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none"
class=MsoNormal><SPAN style="FONT-SIZE: 11pt"><FONT face="Times New Roman">with
high expectations for the consequences.<o:p></o:p></FONT></SPAN></P>
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style="mso-bidi-font-weight: normal"><SPAN
style="FONT-SIZE: 18pt; mso-ansi-language: EN-US" lang=EN-US>Date :
</SPAN></B><SPAN style="FONT-SIZE: 18pt; mso-ansi-language: EN-US" lang=EN-US>13
August 2010 at 11:00</SPAN><SPAN
style="FONT-SIZE: 20pt; mso-ansi-language: EN-US"
lang=EN-US><o:p></o:p></SPAN></FONT></P>
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air-condition)<o:p></o:p></SPAN></FONT></P>
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<DIV dir=ltr id=idSignature82245>
<DIV align=left><FONT size=2 face=System>Asst. Prof. Songül ESİN</FONT></DIV>
<DIV><FONT size=2 face=System>Dogus University, Acibadem, Kadikoy,
34722</FONT></DIV>
<DIV><FONT size=2 face=System>Istanbul, TURKEY</FONT></DIV>
<DIV><FONT size=2 face=System>Tel: +90 216 3271104 / 1345</FONT></DIV>
<DIV><FONT size=2 face=System>Fax: +90 216 5445533</FONT></DIV>
<DIV><FONT size=2 face=System><A
href="http://www3.dogus.edu.tr/sesin">http://www3.dogus.edu.tr/sesin</A></FONT></DIV></DIV></FONT></DIV></DIV>
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