Department
of Mathematics
Arizona
State University
Analysis
Seminar
Fall 97
- Spring 98
Organizers: Joaquin
Bustoz and Sergei
Suslov
SEMINARS:
October
1997: 8,
15,
22,
29.
November
1997: 5,
12,
18-21, 19.
December
1997: 17.
January
1998:
23.
February
1998: 19-26,
23
March
1998:
9, 30.
April
1998:
13, 20
October
8, 1997
John McDonald, Department of
Mathematics
"Representing non-negative polynomials
as sums of squares"
October
15, 1997
John McDonald, Department of
Mathematics
"Representing non-negative polynomials
as sums of squares"(continued)
October
22, 1997
Sergey Nikitin, Department of
Mathematics
"Boundary Value Problem"
October
29, 1997
Sergey Nikitin, Department of
Mathematics
"Boundary Value Problem"(continued)
November
5, 1997
Sergey Nikitin, Department of
Mathematics
"Boundary Value Problem"(continued)
November
18-21, 1997
Mourad Ismail, University of
South Florida
"Three lectures on differential
equations for orthogonal polynomials"
November
12, 1997
Alwin Swimmer, Department of
Mathematics
"On golden spirals"
ABSTRACT: Golden spirals are
those associated with the golden ratio.
In particular some fascinating
properties of the eyes of the spirals
considered will be discussed.The
salient features of Grassmann Algebra,
the main tool used to discover
these properties, will be given so that
the talk will be intelligible
to everyone in the department, all of whom
are cordially invited to attend.
November
19, 1997
Alwin Swimmer, Department of
Mathematics
"On golden spirals"{continued)
December
17, 1997
George Gasper, Department of
Mathematics,
Northwestern University
Evanston, Illinois
"q-Extensions of Erdelyi's Fractional
Integrals and Applications".
January
23, 1998
Analysis Seminar and Colloquium
Dennis Stanton, Department of
Mathematics
University of Minnesota
"Three Combinatorial Positivity
Theorems"
ABSTRACT:
I will present three examples
of combinatorial
positivity theorems from analysis,
algebra, and topology. First
a trigonometric inequality will
be proven combinatorially. Next,
unimodality of a difference
of specialized Schur functions is
proven by representation theory.
Finally, the combinatorial
Laplacian on the simplicial
complex of independent sets in a
matroid is shown to have non-negative
integral eigenvalues.
The final result implies that
one can "hear" the characteristic
polynomial of a matroid. A new Tutte polynomial identity also
follows.
February
19-26, 1998
Mourad Ismail, University of
South Florida
"Three lectures on 'addition'
theorems for basic expotential
functions and related topics"
February
23, 1998
Kevin Kadell, Department of
Mathematics
"Aomoto's machine and the Dyson
constant term identity"
ABSTRACT:
Aomoto has used the fundamental
theorem of calculus
to give an elegant proof of
an extension of Selberg's integral.
A constant term formulation
of Aomoto's argument is based upon
the fact that for $1\le s \le
n$ the constant term in
$t_s{\partial}/{\partialt_s}f(t_1,
..., t_n)$
is zero provided that $f(t_1,
..., t_n)$ has a Laurent expansion at
$t_1=...=t_n=0$. We use this
as the engine for a simple proof of
an Aomoto type extension of
the Dyson constant term identity.
March
9, 1998
Sergei Suslov, Department of
Mathematics
"Basic analog of Fourier series
on a q-quadratic grid"
ABSTRACT:
In our joint paper with Joaquin
Bustoz we prove orthogonality relations
for some analogs of trigonometric
functions on a q-quadratic grid and
introduce the corresponding
q-Fourier series. We discuss several other
properties of this basic trigonometric
system and our q-Fourier series.
Explicit examples of these series
naturally lead to a new class of
formulas never investigated
before from an analytical and/or numerical
viewpoint. The beauty of just
a few known examples makes the study of
basic Fourier series a fascinating
problem in the area of classical
analysis and applied mathematics.
March
30, 1998
Sergei Suslov, Department of
Mathematics
"Basic analog of Fourier series
on a q-quadratic grid"(continued)
April
13, 1998
Monday, 2:40-3:30 pm, PSA
107
Kevin Kadell, Department of
Mathematics
"The Schur functions for partitions with complex parts"
ABSTRACT
We use the classical ratio of of alternants to define the complex
Schur function
$s_{\lambda}(t_1, t_2, ..., t_n)$ where the partition $\lanbda$
has complex parts
and $t_1, ..., t_n$ are in the complex plane cut from zero to infinity.
We use subset
horizontal strips and tournament tableaux to give the branching
rule and the combinatorial
representation for $s_{\lambda}(t_1, t_2, ..., t_n)$ where $t_1,
..., t_n$ are distinct.
We show that $s_{\lambda}(t_1, t_2, ..., t_n)$ satisfies the Pieri
formula and, extending
results of Kadell and Hua, two Selberg q-integration formulas.
We show that if
$t_1, ..., t_n$ are distinct, then $s_{\lambda}(t_1, t_2,
..., t_n)$ is an entire function
of each $\lambda_1,..., \lambda_n$. We extend Pieri induction to
complex partitions
$\lanbda$, thus giving alternative proofs of our results.
April
20, 1998
Monday, 2:40-3:30 pm, PSA
107
Mike Brilleslyper, Department
of Mathematics
"An extension
of the Dirichlet problem"
ABSTRACT
Given
a map from the boundary of the unit disk into the
2-sphere,
we seek extensions that minimize a particular energy
functional.
For highly symmetric boundary data, there is a
classification
of relative homotopy classes that contain energy
minimizing
extensions (harmonic maps). We prove a uniqueness
result
for extensions that are rotationally invariant and give a geometric
characterization
of a harmonic map that is not an absolute minimum for
the energy
functional (i.e., a harmonic map that is not conformal or
anti-conformal).
In some sense, this is the simplest generalization
of the
classical Dirichlet problem to a target space with positive
curvature,
yet some of the most basic uniqueness questions remain open.