ANALYSIS ON FRACTALS |
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We are looking for a small group of undergraduate students to work on projects
related to ANALYSIS ON FRACTALS. The projects mostly will run in the Summer 2010, but
may begin before it, and spill into subsequent semesters, depending on the desire of
the participants. For the summer, the students are expected to be supported from REU,
SURF and faculty grants. The aim of the projects will be exploration of differential
equations and various operators on fractal domains. Previous undergraduate work
includes published papers on the eigenmodes (vibration modes) of the Laplacian (2nd
derivative) of functions that live on Sierpinski gasket type fractals, and the
electrical resistance of fractal networks, as well as work on Laplacians on
projective limit spaces. For more details and other previous projects and pictures,
see the website www.math.uconn.edu/~teplyaev/fractals/.
The exact choice of
the project will depend on the students' background and interests. Besides being
interesting, taking part in a research project like this may be very useful in the
future (for instance, when applying to graduate schools).
Students in the project are supposed to have the usual background in linear algebra and differential equations. Knowledge of Matlab, Mathematica, other computer algebra systems, or programming, as well as proof writing, mathematical analysis, and probability may be helpful but are not required. Please contact us if you are interested and/or have any questions. For the Summer 2009 REU project applications are no longer accepted, but a new search is expected to start in January 2010 (early applications are encouraged). |
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math.uconn.edu math.uconn.edu
Applet that generates random Sierpinski Gaskets
Applet that computes Green's function of the random Sierpinski Gaskets
Previous completed works with undergraduate students:
S. Makisumi, G. Stadnyk, B. Steinhurst, Modified Hanoi Towers Groups and Limit Spaces preprintarXiv:0909.3520 http://arxiv.org/abs/0909.3520
K. Romeo and B. Steinhurst, Eigenmodes of a Laplacian on Laakso Space, Complex Variables and Elliptic Equations, 54:6, pp. 623-637 (2009). preprint http://arxiv.org/abs/0903.4661
D. Ford and B. Steinhurst, Vibration Spectra of the m-Tree Fractal, preprint http://arxiv.org/abs/0812.2867
Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil, Poorak Mody, Benjamin Steinhurst, Alexander Teplyaev
Vibration modes of 3n-gaskets and other fractals
J. Phys. A: Math. Theor. 41 (2008) 015101 (21pp). pdf file
Vibration Spectra of Finitely Ramified, Symmetric Fractals
Fractals 16 (2008), 243--258. pdf file
project web page
older preprint
in the
Isaac Newton Institute Preprint Series
Mathematica notebooks for the project
B. Boyle, K. Cekala, D. Ferrone,
N. Rifkin and A. Teplyaev Electrical Resistance of N-gasket Fractal Networks.
Pacific Journal of Mathematics 233 (2007), 15--40.
pdf file.
D. Fontaine, T. Smith and A. Teplyaev Resistance of random Sierpinski gaskets.
Quantum Graphs and Their Applications, Contemporary Mathematics 415 (2006), AMS, Providence, RI.
pdf file The project web page is here.
R. Meyers, R. Strichartz and A. Teplyaev Dirichlet forms on the Sierpinski gasket. Pacific Journal of Mathematics 217 (2004), 149-174. pdf file
J. Needleman, R. Strichartz, A. Teplyaev and P.-L. Yung Calculus on the Sierpinski gasket: polynomials exponentials and power series. Journal of Functional Analysis 215 (2004), 290--340. pdf files and from arXiv.org
E.J. Bird, S.-M. Ngai and A. Teplyaev Fractal Laplacians on the Unit Interval. Ann. Sci. Math. Quebec 27 (2003), 135--168. pdf file
B. Adams, S.A. Smith, R. Strichartz and A. Teplyaev The spectrum of the Laplacian on the pentagasket (with), Fractals in Graz 2001 -- Analysis -- Dynamics -- Geometry -- Stochastics, Trends Math., Birkhauser Basel (2003), 1--24. (pdf file)
J. Stanley,R. Strichartz and A. Teplyaev Energy partition on fractals. Indiana University Mathematics Journal 52 (2003), 133--156. pdf file
O. Ben-Bassat,R. Strichartz and A. Teplyaev What is not in the domain of the Laplacian on a Sierpinski gasket type fractal. Journal of Functional Analysis 166 (1999), 197--217. pdf file
