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.
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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.
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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. The project consist of studying recent results that can be viewed as discrete versions of the uncertainty principles and their applications.
Literature:
1. D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl.
Math., vol. 49, no. 3, pp. 906–931, 1989.
2. M. Elad and A. M. Bruckstein, A generalized uncertainty principle and sparse representation in
pairs of bases, IEEE Trans. Inf. Theory, vol. 48, no. 9, pp. 2558–2567, Sep. 2002.
3. E.J.Candès, J.Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6 (2006), no. 2, 227--254.
4. E.J.Candès, J.Romberg, T.Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 (2006), no. 2, 489--509.
-
Spherical wavelets
- Traditionally harmonic functions in a ball are represented by series of harmonic polynomials. This technique is
very well studied and many properties of harmonic polynomilas or spherical harmonics are known. In modern calculations it turns out that another bases are more appropriate for different problems. one of the popular approaches is construction of a wavelet basis on the sphere. The method is used in geomathematics and imageprocessing and is interesting from the theoretical point of view. The project will give an overview of the known bases and their applications for different problems.
Literature:
1. P.Schroder, W.Sweldens Spherical wavelets: Efficiently Representing Functions on the Sphere, Proc. Intern. conference on computer graphics and interactive techniques, 1995
2. V.Michel applied and Comp. harmonic analysis, 12, 77-99,2002
3. M.Holschneider, I.Iglewska-Nowak Poisson Wavelets on the sphere, Journ. Fourier Analysis and appl., 13, no.4, 405-419,2007.
M.Hayn, M.HolschniederDirectional spherical multipole wavelets Journ. of Math. Physics, 2009
-
Rudin-Shapiro polynomials
These polynomilas were introduced to solve a minimization problem in 1950s. Rudin-Spapiro polynomilas have coefficients 1 and -1 and their uniform norm on the unit circle is closed to L2-norm, the sequence of the coefficients is called Rudin-Shapiro sequence. The sequences turn out to be connected to various
problems of mathematics and a number of remarkable applications were found.
Literature
J.-P. Allouche, M. Mendès France, On an extremal property of the Rudin-Shapiro sequence. Mathematika 32 (1985), no. 1, 33--38.
2. A.Rodenhausen, Paperfolding, generalized Rudin-Shapiro sequences, and the Thue-Morse sequence. Fractals 3 (1995), no. 4, 679--688
3.
G. BenkeGeneralized Rudin-Shapiro systems J. Fourier Anal. Appl. 1 (1994), 87–101.
4. J. S. Byrnes Quadrature mirror filters, low crest factor arrays, functions achieving optimal
uncertainty principle bounds, and complete orthonormal sequences—a unified approach, Appl.
Comput. Harmon. Anal. 1 (1994), 261–266
5. R. Coifman, F.Geshwind, Y.Meyer, Noiselets. Appl. Comput. Harmon. Anal. 10 (2001), no. 1, 27--44.
Last modified: Mon Nov 14 16:15:36 MET 2005
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