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.
Alexander Teplyaev – The Spectral Dimension of the Universe
May 29, 2015
Professor Alexander Teplyaev will explain some ideas behind the notion of spectral dimension and how they are related to research being done in our department.
Masha Gordina – Random thoughts on Brownian motion
June 5, 2015
Professor Masha Gordina will talk about the fascinating history of the Brownian motion and its applications in the real world.
Keith Conrad – Continued Fractions
June 12, 2015
Professor Keith Conrad will talk about continued fractions, how to compute them, some of their properties, and how to answer seemingly unanswerable questions like this: if an unknown fraction is roughly 2.32558, what is it? (The answer is not 232558/100000.)
Thomas Laetsch – From Brownian motion cometh
June 19, 2015
Following Dr. Gordina’s talk developing Brownian motion, Thomas Laetsch will take us on a short drunkard’s walk through several theories stemming from or related to Brownian motion. R(E)U ready?
Joe Chen – Drunkard, Octopus, and Electrical Networks
June 26, 2015
Joe Chen will summarize the main ideas behind electrical networks and describe two unexpected applications to probability.
We investigated systems of complex-valued ordinary differential equations (ODEs) that blows up in finite time, which we refer to as explosive systems. The goal is to understand for what initial conditions does the system explode and will the addition of noise stabilize it; that is, if we were to perturb the system with an additive Brownian motion, will the system of stochastic differential equation (SDE) still be explosive? In fact, we were able to prove a toy model of the stochastic Burgers’ equation to be ergodic; that is, the SDE is nonexplosive and it has a unique limiting distribution.