## Analysis, Probability and Mathematical Physics on Fractals |
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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
- 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.
UConn applicants are expected to apply for 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). |
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Alexander (Sasha) Teplyaev,
**teplyaev**math.uconn.edu **www.math.uconn.edu/~teplyaev/**

Daniel Kelleher,
**kelleher**math.uconn.edu
**www.math.uconn.edu/~kelleher/**

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

Matt Begue at the
Frontiers 2011

Link to **Analysis, Probability 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:**

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
arXiv:1011.3567

see also B. Steinhurst,

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

D. Ford and B. Steinhurst, ** Vibration Spectra of the m-Tree Fractal**,

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.

older

B. Boyle, K. Cekala, D. Ferrone,
N. Rifkin and A. Teplyaev ** Electrical Resistance of N-gasket Fractal Networks**.
Pacific Journal of Mathematics

D. Fontaine, T. Smith and A. Teplyaev ** Resistance of random Sierpinski gaskets**.
Quantum Graphs and Their Applications, Contemporary Mathematics

R. Meyers, R. Strichartz and A. Teplyaev ** Dirichlet forms on the Sierpinski gasket**.
Pacific Journal of Mathematics

J. Needleman,
R. Strichartz, A. Teplyaev and P.-L. Yung ** Calculus on the Sierpinski gasket:
polynomials exponentials and power series**.
Journal of Functional Analysis

E.J. Bird, S.-M. Ngai and A. Teplyaev ** Fractal Laplacians on the Unit Interval**.
Ann. Sci. Math. Quebec

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. (

J. Stanley,R. Strichartz and A. Teplyaev ** Energy partition on fractals**. Indiana University Mathematics Journal

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

Some directly or indirectly related pictures: