Possible
projects for interested students:
Project
1: Theory of multiple orthogonal polynomials on the unit circle
Required knowledge: Linear
Algebra, Complex Analysis
Useful (not
required) knowledge: Measure Theory; Functional Analysis; Operator Theory
Level:
Bachelor, Master, PhD
Details: Multiple orthogonal polynomials are polynomials orthogonal with respect to the several measures.
When the orthogonality measures are on the real line, the theory is very well-developed. When the measures are on the complex unit circle though,
there are many interesting questions that remain open. There has been a very interesting progress in this direction and the project is
to explore these new developments and extend the theory further. The level of the project can be tailored to the ambition of the student.
Project
2: Gaussian quadrature on the real line and on the unit circle
Required
knowledge: Calculus
Useful
knowledge: Real and complex analysis
Level: Bachelor, Master, PhD
Details: Gaussian quadrature is a family of numerical methods for evaluating integrals of f(x) by expresing it as a weighted sums
of f evaluated at appropiately chosen points. To maximize precision the chosen points must be zeros of the corresponding orthogonal polynomials if the integration is over the real line. For integration on the unit circle, one needs zeros of the so-called para-orthogonal polynomials. Depending on the level and ambition of the student, one can even study simultaneous Gaussian quadrature rules involving several integrals at the same time by using theory of multiple orthogonal polynomials.
Project
3: Non-classical finite rank
perturbation of Jacobi and CMV operators
Required
knowledge: Linear Algebra; Complex Analysis; Measure Theory
Useful knowledge: Functional
Analysis; Operator Theory
Level:
Master, PhD
Details: This
project investigates spectral properties of non-Hermitian
perturbations of real Jacobi operators and non-unitary perturbations
of CMV operators. We will restrict ourselves to the case of finite
rank perturbations and compare the situation with the classical
perturbations.
Project
4:
Matrix-valued
multiple orthogonal polynomials
Required
knowledge: Linear Algebra; Complex Analysis; Measure Theory
Useful knowledge: Functional
Analysis; Operator Theory
Level:
Master, PhD
Details: The
project consists of surveying known theory of multiple orthogonality
and matrix-valued orthogonality and the task is to develop theory for
the combination of them: matrix-valued multiple orthogonal
polynomials.
Project
5: Recurrence relations for
orthogonal polynomials: on the real line, the unit circle, and beyond
Required knowledge: Linear
Algebra
Useful (not
required) knowledge: Complex Analysis; Matlab/Mathematica;
Probability
Level:
Bachelor
Details: The
project is to survey the recurrence relations satisfied by the
orthogonal polynomials in various settings: on the real line, on the
unit circle, and for their multivariate and multiple generalizations.
Using Matlab/Mathematica, one could use these relations to
numerically observe asymptotics of the orthogonal polynomials and
their zeros. Depending on the student's ambition and background, one
could include the case of orthogonal polynomials with random
recurrence coefficients.
Rostyslav Kozhan, kozhan [at]
math [dot] uu [dot] se