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Probability, Analysis and Mathematical Physics on Fractals

Each year we are looking for a group of undergraduate students to work on Probability, Analysis 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 NSF, REU, SURF and other 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, you can e-mail the following to Daniel Kelleher:

  • letter of application, describing your background and interests and your expectations for the program;
  • resume or CV;
  • a list of upper level mathematics courses taken, or unofficial copy of undergraduate transcripts;
  • arrange for one or two recommendation letters from faculty at your home college/university.
Alternatively, you can submit an online application to our departmental REU program. In this case please indicate in which topic(s) you are interested.

Applicants who have their own funding are welcome to apply, including foreign applicants who already have a valid US visa (some travel expenses and/or housing cost may be covered, but no stipend will be available for foreign students). All US non-UConn applicants will be considered for REU stipends.

Alexander (Sasha) Teplyaev,
Daniel Kelleher,

Research supported in part by NSF grant DMS-0505622, and by the University of Connecticut Department of Mathematics

links to

  • Fractals Summer 2011
  • Fractals Summer 2012
    departmental page
  • Matt Begue at the Frontiers 2011

    Link to Probability, Analysis and Mathematical Physics on Fractals October 2-3, 2010, Syracuse NY

    REU 2010: Daniel Kelleher, 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), Ben Steinhurst

    REU 2009: Matthew Begue (UConn), Shotaro Makisumi (Princeton University), Grace Stadnyk (Hamilton College), Levi deValve (UConn), David Miller (Salve Regina University), Ben Steinhurst

    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:

  • Dan Kelleher, Michael J. Ignatowich, Catherine E. Maloney, David J. Miller, Khrystyna Nechyporenko. Computation of Resistance Scaling of the Pillowcase and Fractalina Fractals, arXiv:1204.5815

  • Dan Kelleher, Tyler Reese, Dylan Yott, Toni Brzoska, Analysing properties of the C. Elegans neural network: mathematically modeling a biological system, arXiv:1109.3888 PLoS ONE 7(10): e40483

  • 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 Experimental Mathematics, 21(4):402417, 2012

  • D. Kelleher, B. Steinhurst and C-M.M. Wong, From Self-Similar Structures to Self-Similar Groups, arXiv:1011.1817 to appear in International Journal of Algebra and Computation (IJAC)

  • R. Kesler, B. Steinhurst Casimir Effect on Laakso Spaces (submitted)

  • Christopher J. Kauffman, Robert M. Kesler, Amanda G. Parshall, Evelyn A. Stamey and Benjamin A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys. 53, 042102 (2012). arXiv:1011.3567
    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 Fractals 20:2 (2012)

  • S. Makisumi, G. Stadnyk, B. Steinhurst, Modified Hanoi Towers Groups and Limit Spaces, International Journal of Algebra and Computation (IJAC) 21:6 (2011) arXiv:0909.3520

  • D. Ford and B. Steinhurst, Vibration Spectra of the m-Tree Fractal, Fractals 18:2 (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.
    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

  • 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