## Event

AMS Sectional Meeting @ Stony Brook

Loren Anderson, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew

Alexander Teplyaev, Luke Rogers, Antoni Brzoska, Ulysses Andrews

In this project, Feynman’s analysis of an infinite ladder circuit is extended to fractal circuits, in particular, a Sierpinski Ladder circuit and two variants of a Hanoi circuit. These circuits are constructed by taking the limit of graph approximations with inductors and capacitors placed along edges in a well-defined manner. Inductors, capacitors and resistors all contribute to the impedance within a circuit; but whereas a resistor imparts a real impedance, inductors and capacitors impart a purely imaginary impedance.

For each circuit, the following was accomplished. First, the net impedance between certain boundary points of the circuit was computed. Second, the filter conditions for each circuit were found. A circuit becomes a filter when the net impedance between boundary vertices has a positive real part! Third, it was proved that these impedances can be obtained by placing a small positive resistance epsilon on each edge of the graph approximations, finding the limiting impedance between the boundary vertices, and then taking epsilon to zero. Finally, the construction of harmonic functions on these circuits was outlined.

“*Stabilization by noise of a C^2-valued coupled system.*”Joe P. Chen, Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’Connell, and Fan Ny Shum. *To appear in* Stoch. Dyn.

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.

June 5, 2015

Professor Masha Gordina will talk about the fascinating history of the Brownian motion and its applications in the real world.

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.)

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?

June 26, 2015

Joe Chen will summarize the main ideas behind electrical networks and describe two unexpected applications to probability.

Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’Connell

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.

Stephen Loew, Madeline Hansalik, Aubrey Coffey

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.