University of Connecticut, August 3rd, 2017
Organizers: Luke Rogers, Phanuel Mariano, and Gamal Mograby
Participating Schools: Amherst, Smith, UConn and UMass
5th Mini-Symposium full program (2017)
Uncategorized
Gradients on Higher Dimensional Sierpinski Gaskets
Group Members
Luke Brown, Giovanni E Ferrer Suarez, Karuna Sangam.
Supervisors
Gamal Mograby, Dan Kelleher, Luke Rogers, Sasha Teplyaev.
Overview
Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.
Publication: arXiv:1908.10539 Fractals Vol. 28, No. 06, 2050108 (2020)
doi.org/10.1142/S0218348X2050108X
Presentation
Poster
Multiplicative LLN and CLT and their Applications
Group Members
Lowen Peng, Anthony Sisti, Rajeshwari Majumdar
Supervisors
Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo
Overview
We study the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturb the random matrices with a parameter and we look to find the dependence of the the Lyapunov exponent on this parameter. We also study the variance related to the multiplicative CLT. We prove and conjecture asymptotics of various parameter dependent plots.
Publication: “Lyapunov exponent and variance in the CLT for products of random matrices related to random
Fibonacci sequences” — arXiv:1809.02294, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), pp 21
Presentations:
Raji Majumdar and Anthony Sisti, will present posters Applications of Multiplicative LLN and CLT for Random Matrices and Black Scholes using the Central Limit Theorem on Friday, January 12 at the MAA Student Poster Session, and give talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.
Spectrum of the Magnetic Laplacian on the Diamond Fractal
Group Members
Stephen Loew, Madeline Hansalik, Aubrey Coffey
Supervisors
Overview
The diamond fractal is a fractal that is obtained in the following manner. Start with a graph with two vertices and an edge and replace the edge with two new vertices connected to our original vertices to obtain a diamond shaped graph. The diamond fractal is defined to be the limiting object after continuing with the edge replacement indefinitely. In the project, the spectrum of magnetic Laplacian operators on graph approximations to the diamond fractal was computed.
Given a level n approximation to the fractal with known magnetic field strengths through cells and holes, it is possible to determine the net magnetic field through the cells and holes of the preceding graph approximations. The spectral similarity relation between the operators on successive graph approximations was worked out, with the corresponding spectral decimation polynomial depending on the magnetic field strengths. A poster and talk on this work was presented at the REU Mini-Symposium at UConn.
Publication: , ,
arXiv:1704.01609
Presentation
3rd Northeast Mathematics Undergraduate Research Mini-Symposium
Organizers
Participating Schools: Amherst, Colgate, Smith, UConn, UMass, Williams
3rd Mini-Symposium full program (2015)
(photo courtesy Megan Brunner)
Mini-Symposium poster