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
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Biot-Savart operator and its applications (Analysis meets Topology)
- 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 studies integral formulas for mentioned
topological characteristics. Here the Biot-Savart operator
comes into play. The study of this operator has been a popular topic
of research the last decade. Work on the project combines basic knowledge of topology
and analysis in a fun way.
Literature:
1. V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998.
2. DeTurck, Dennis; Gluck, Herman Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space. J. Math. Phys. 49 (2008), no. 2, article
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),text"
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.
-
Uncertainty principle: modern view-point
- The Heisenberg uncertainty principle has a mathematics formulation, which roughly says that a function and its Fourier transform can not be highly localized simultaneously (or a signal can not be localized both in time and frequency). Many versions of this statement are known.
Literature:
Last modified: Mon Nov 14 16:15:36 MET 2005
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