Bitcoin and other cryptocurrencies allow for peer-to-peer transfers of digital tokens. In this talk, we will introduce the Bitcoin network and protocol, providing as we do so a survey of mathematical ideas necessary for Bitcoin's success. Topics to be discussed will include private/public key cryptography, elliptic curve digital signatures, cryptographic hash functions and the feasibility of network attacks.

(Real) Hyperbolic lattices have been heavily studied for their interesting number theoretic and dynamical properties. Their complex hyperbolic analogs, however, are quite a bit less understood and we know of relatively few examples. In this talk, I'll briefly introduce hyperbolic lattices, discuss some of the known results, and introduce my research into methods for constructing complex hyperbolic lattices.

Coherent structures are ubiquitous in environmental and geophysical flows. These structures are transport barriers that dictate the trajectories of deterministic passive scalars. Due to the structure type (shear region or vortex) the transport can either be enhanced or reduced; leading to anomalous transport and diffusion. In this talk we will explore various aspects of modeling anomalous diffusion and LCS detection methods.

The Kuramoto model is a celebrated model of coupled oscillators renowned for the complex self-organizational synchronization behaviour it exhibits with relatively simple dynamics. Much work has been done in the dynamical systems and control communities on predicting which model parameters lead to synchronization and which do not. A popular variation on the model is to embed the phase oscillators into a network and restrict the coupling to only be between oscillators connected by an edge; doing so greatly increases the difficulty in predicting how and when the bifurcation to synchronization will occur. I will be presenting an established method due to Dorfler for identifying the bifurcation, my own variation on the method, and connect the questions we have about the Kuramoto model to well-studied questions in spectral graph theory and power engineering.

We examine the consensus model for opinion dynamics defined on both directed and undirected graphs. We will discuss proofs of convergence in both cases and show that in the case of an undirected graph the dynamics always converge to the mean of the initial condition whereas if the model is defined on a directed graph this is not the case and a consensus might not even be reached.

This talk considers time series signals in $\R^n$ as samples of an embedded space curve and proceeds to characterize such signals in terms of differential-geometric descriptors of their associated curves. In particular, a method of estimating curvature as a function of arc length is presented. Because arc length is invariant to reparameterization of a space curve, this approach provides a representation of the evolution of the time series that is invariant to local variations in the rate of the time series as well as displacement and rotation of the curve in space. The focus here is on ascertaining structural similarity of time series signals by measuring similarity of their curvatures, though extension to other applications and other geometric descriptors (e.g., torsion) is envisioned.

John Jones and David Roberts have developed a database recording invariants for p-adic extensions of low degree, see http://www.lmfdb.org/LocalNumberField/. We contribute to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of global splitting models.