Luke Brown, Giovanni E Ferrer Suarez, Karuna Sangam.
Gamal Mograby, Dan Kelleher, Luke Rogers, Sasha Teplyaev.
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.
Courtney George, Samantha Jarvis.
Dan Kelleher, Luke Rogers, Sasha Teplyaev.
We analyze the spectra of a sequence of graphs constructed from the Schreier graphs of the Basilica group. Our analysis differs from earlier work of Grigorchuk and Zuk in that it is based on a macroscopic decomposition of the graphs. This method gives precise information about the multiplicities of eigenvalues and, consequently, good information about the spectral measures of large graphs. It also permits a proof of the existence of gaps in the spectrum of limiting graphs.
Phanuel Mariano – The volume of the unit ball in n dimensions
July 28, 2017
Phanuel Mariano from the University of Connecticut will be giving a talk computing the volume of the unit ball in arbitrary dimension.
Michelle Rabideau – Continued Fractions and the Fibonacci Sequence
July 21, 2017
Michelle Rabideau from the University of Connecticut will be giving a talk related to Continued Fractions.
Hugo Panzo – Laplace’s method and applications to probability
July 14, 2017
Hugo Panzo from the University of Connecticut will be giving a talk related to Laplace’s method.
Patricia Alonso Ruiz – Resistance metric – an electric interpretation of measuring distances
July 7, 2017
Any weighted graph can be seen as an electric linear network where the current flows between nodes (vertices) connected by resistors (weighted edges). This electric interpretation provides a special way to measure distances in a graph via the so-called effective resistance metric. What does this metric actually do, how it is related to energy minimizers and why it is so helpful when graphs become infinite are some of the questions we will address in this talk.
Keith Conrad – An algebraic characterization of differentiation
July 30, 2017
The derivative is defined using limits while the basic rules of differentiation (sum rule, product rule, chain rule) have an algebraic flavor. We will see how differentiation, and more generally differential operators, can be characterized purely algebraically by putting all the analytic conditions into the functions that we want to differentiate.
Ambar Sengupta – Random Matrices: Pictures From Traces and Products
July 23, 2017
Professor Ambar Sengupta will give a talk based on Random Matrices.
Zihui Zhao – Harmonic Measure
June 16, 2017
Zihui Zhao form the University of Washington will be an introductory talk in harmonic measure.
Luke Rogers – Length and volume
June 9, 2017
I will talk a little about Euclidean length and volume, the lengths of curves, Peano curves, the positive area curves of Osgood, and a delightful theorem of Hajlasz and Strzelecki that shows one can measure volume with a string.
Daniel Kelleher – The metric space of metric spaces
June 2, 2017
Metric spaces are sets which have a notion of distance. We will compare two different metric spaces, and see that this comparison makes the set of metric spaces into a metric space (Don’t worry, after that it’s turtles the rest of the way down). The focus will be put on length spaces — metric spaces where distance is given as the length of the shortest curve connecting two points. For these spaces we discover a sense of curvature
Lowen Peng, Anthony Sisti, Rajeshwari Majumdar
Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo
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.