Analysis, Probability and Mathematical Physics on Fractals
Fractals Summer 2011
Each year we are looking for a group of undergraduate students to work on Analysis, Probability and Mathematical Physics on Fractals. The projects run either in the summer or during the academic year. 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.
The exact choice of
the topics to study 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 is not required. Please write if you are interested and/or have any questions.
To apply, please e-mail to Daniel Kelleher the following:
UConn applicants are expected to apply for SURF.
All US non-UConn applicants will be considered for REU stipends.
Applicants who have their own funding are welcome to apply, including foreign applicants (some travel expenses may be covered, but no stipend will be available for foreign students).
Alexander (Sasha) Teplyaev,
Daniel Kelleher, kellehermath.uconn.edu www.math.uconn.edu/~kelleher/
Matt Begue at the
Link to Analysis, Probability and Mathematical Physics on Fractals October 2-3, 2010, Syracuse NY
Chuen Ming Mike Wong (Princeton University),
Christopher Kauffman (University of Rochester),
Amanda Parshall (York College of Pennsylvania),
Evelyn Stamey (Ithaca College),
Robert M Kesler (Princeton University),
REU 2009: Matthew Begue (UConn),
Shotaro Makisumi (Princeton University),
Grace Stadnyk (Hamilton College),
Levi deValve (UConn),
David Miller (Salve Regina University),
UConn Frontiers 2008: posters of Kevin Romeo, Alon Dagan, Michael Khalil
Applet that generates random Sierpinski Gaskets
Applet that computes Green's function of the random Sierpinski Gaskets
Previous completed works with undergraduate students:
M. Begue, D. J. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and Strichartz hexacarpet, arXiv:1106.5567
D. Kelleher, B. Steinhurst and C-M.M. Wong, From Self-Similar Structures to Self-Similar Groups, arXiv:1011.1817
Christopher J. Kauffman, Robert M. Kesler, Amanda G. Parshall, Evelyn A. Stamey and Benjamin A. Steinhurst,
Quantum mechanics on Laakso spaces, preprint
see also B. Steinhurst, Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension, arXiv:0811.1378
M. Begue, L. deValve, D. Miller, B. Steinhurst, Spectrum and Heat Kernel Asymptotics on General Laakso Spaces, arXiv:0912.2176
S. Makisumi, G. Stadnyk, B. Steinhurst, Modified Hanoi Towers Groups and Limit Spaces, to appear in International Journal of Algebra and Computation (IJAC) arXiv:0909.3520
D. Ford and B. Steinhurst, Vibration Spectra of the m-Tree Fractal, Fractals 18 (2010) 157--169. arXiv:0812.2867
K. Romeo and B. Steinhurst, Eigenmodes of a Laplacian on Laakso Space, Complex Variables and Elliptic Equations, 54:6, pp. 623-637 (2009). arXiv:0903.4661
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.
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