Here is a description of topics for projects in
analysis. The projects can be done both individually
on in small groups, the level and precise
formulation of the problem will depend on the
student's background and interests.
Are you interested or just want more info feel free
to contact me
office: 540, SBII
ph. 73550257
email: eugenia(at)math.ntnu.no
- Biot-Savart operator and its applications
- In a short note by Gauss (1833) an integral formula for the
linking number of two disjoint closed curves in
three-dimensional space was proved. More than a hundred
years later a topological meaning of the linking number of a
curve and itself was understood, it is so-called the
writhing number of the curve and is an analog of the
helicity of a vector field.
The project stadies integral formulas for mentioned
topological characteristics. Here the Biot-Savart operator
comes into play. Different levels of the project may include
the study of the Hodge decomposition for Biot-Savart
operator, spectral properties of the operator, applications
of writhing number in DNA research.
Literature:
1. J. White, Self-linking and the Gauss integral in higher dimensions,
Amer. Journal of Mathematics 91 (1996), 693--728.
2. W.F. Pohl, DNA and differential geometry,
Math. Intelligencer 3 (1980/81), no. 1, 20--27.
3. De W. Sumners, Untangling DNA,
Math. Intelligencer 12 (1990), no. 3, 71--80
4. J. Cantarella, D. De Turck, H. Gluck The Biot-Savart
operator for application to knot theorey, fluid dynamics and
plasma physics, Jour. of Math. Physics, 42 (2001),
n0.2, 876--904.
- Zero sets of harmonic polynomials
- Local structure of the zero sets of harmonic polynomials of
two variables is well understood, at each critical point it
is a union of finitely many curves that pass through the
point and form equal angles at this point.
The situation changes when we go to dimension three. How the
zero surfaces look like now? It is one of the questions that
is interesting to "look" at (you can get amazing pictures!).
These zero sets, both their local and global structure are
of great interest. One of the important mathematical
questions in geophysics can be formulated in the following
way:
Given two harmonic functions whose level
surfaces are orthogonal at each point. What can be said
about the functions?
Other situations when level (zero) sets of harmonic polynomials
appear include uniqueness sets for the Radon
transform and nodal geometry on Riemannian surfaces.
To work on that project one needs some curiosity, knowledge
of MatLab, Maple, or Mathematica could be useful. It is an
advantage to have some background in Complex analysis, like
TMA4175.
Literature:
1. T. Sheil-Small, Complex polynomials, Cambridge
Studies in Adv. Math. 75, Cambridge Univ. Press, 2002.
2. Sh. Axler, P. Bourdon, W. Ramey, Harmonic function
theory.
Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 1992.
3. H. Groemer, Geometric applications of Fourier series and spherical harmonics. Encyclopedia of Mathematics and its Applications, 61. Cambridge University Press, Cambridge, 1996.
- Clifford algebras and Maxwell equations
- Quaternions appeared in the middle of 19th century in
works by both Hamilton and Maxwell in the search for an
algebra that could substitute the complex numbers in three
dimensional (i.e. natural) physical applications.
For a long period this theme remained exotic. But, since
the beginning of the 1960s, Clifford algebras have been
studied intensively by physicists and
mathematicians. Nowadays Clifford algebras represent a
unified language for mathematical description of
non-relativistic and relativistic classical and quantum
mechanics.
One of the ways to start the study of this field is to
look at monogenic functions with values in a Clifford
algebra as at a generalization of complex valued analytic
function. This way assumes that one is familiar with
complex analysis. Many questions appear when one tries to extend
the results of complex analysis to this setting.
Another way to the field lies through description of
physical laws using the language of Clifford
analysis. As
it always happens with a new way to write down the things
we get a new understanding of the nature of the
laws. Project may start with writing Maxwell equations in
differential forms or Dirac operator using spinors.
Literature:
1. R. Brackx, R. Delange, F. Sommen, Clifford
Analysis, Research Notes in Mathematics 76, Pitman,
London, 1982.
2. P. Lounesto, Clifford algebras and Spinors,
London Math. Soc. Lecture Notes Series 239,
Cambridge University Press, Cambridge, 1996.
Last modified: Mon Nov 14 16:15:36 MET 2005
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