Situations in which the volume of collected data overtaxes capacity to communicate and store it for centralized processing are increasingly common in applications -- in particular when the data are measurements made by high-bandwidth sensors. When the objective of data collection is to support a binary hypothesis test, one well-studied approach is to perform a set of tests each of which uses only part of the data; e.g., each node in a distributed sensor network processes its data separately and transmits only its “local” decision to a central location to a central node that synthesizes a “global” decision. The first part of this presentation will review the basics of statistical hypothesis testing and apply these to develop rules for synthesizing a global decision from the collection of local decisions. The second part will discuss some ongoing research on polling of the local decision makers by the central node.

Beginning in the 1920's, Dehn and Nielsen studied a homeomorphism invariant of manifolds called the mapping class group, which is closely related to the fundamental group. Especially in the past 50 years, the study of mapping class groups has been a very active area of research for their rich geometric and dynamic properties. In this talk, I'll give an introduction to mapping class groups and the Neilsen-Thurston classification of automorphisms of closed, oriented surfaces.

Interacting particle systems is a field of probability theory devoted to the rigorous analysis of certain types of models that arise in other fields such as statistical physics, biology, and economics. These systems are motivated by the voter model for the dynamics of opinions. A one-dimensional voter model is a stochastic process where individuals are located on the integer line who at any time can have one of two opinions denoted by 0 or 1. These individuals update their opinion at a constant rate of one based on the opinion of their two neighbors chosen uniformly at random. In my research with Drs. Lanchier and Scarlatos, we introduce an opinion graph – a finite connected graph in which the vertices represent the set of opinions. This allows for more than two opinions in the model. In addition, we also introduce a confidence threshold that dictates whether an individual interacting with a neighbor move one step towards the opinion of the other individual on the opinion graph. The main question about the general opinion model is whether the system fluctuates and clusters, leading the population to a global consensus, or fixates in a fragmented configuration. My talk will mostly focus on the background and the mathematical tools we use in this research.

Integer programming is a field of extensive study in which decision problems are interpreted as an optimization over variables constrained by linear inequalities. In my talk I will introduce the train assignment problem and discuss how to convert this problem to an integer program. Given a collection of tracks and a collection of trains with a fixed arrival-departure timetable, the train assignment problem (TAP) is to determine the maximum number of trains from that can be parked according to the timetable. Furthermore, to optimize against uncertainty in the arrival times of the trains we extend our models by stochastic and robust modelling techniques.

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In the early 1970's, Robert Riley was able to show that the Figure 8 Knot complement can be given a hyperbolic structure. He showed this fact by first showing the fundamental group of the figure 8 knot complement is isomorphic to a subgroup of $\operatorname{PSL}(2,\mathbb{C})$, and then, using the theory of Haken Manifolds, he showed that the Figure 8 Knot complement is homeomorphic to Hyperbolic 3-space, modulo the action of a discrete group of hyperbolic isometries. Riley later showed that other knot complements have hyperbolic structures and conjectured that almost all knot complements have hyperbolic structures. Later in the 70's, Riley introduced this topic to William Thurston, who proceeded to prove Riley's conjecture, that almost all knot complements can be given a hyperbolic structure. Thurston was able to come up with a more explicit construction for a hyperbolic structure on the Figure 8 knot complement, using ideal tetrahedra in the Poincaré ball, and in this talk, I will work through Thurston's construction of a hyperbolic structure on the Figure 8 knot complement.

The method of characteristics for solving first order PDEs is a tool most students encounter relatively early in their first course on PDEs. We will discuss this idea in (at least in the speaker's opinion) more detail then it is usually afforded in such a course. We will provide some observations aimed towards motivating how the characteristic method provides a useful viewpoint for first order PDEs even when it is not immediately useful in producing explicit solutions.

Both the logistic smooth transition autoregressive (LSTAR) model and the threshold autoregressive (TAR) are examples of regime-switching nonlinear time series. These models have been adopted in a wide variety of applications to model time series where the classic autoregressive moving average (ARMA) models are inappropriate. LSTAR and TAR models are formulated as a weighted combination of two or more linear autoregressive (AR) processes. The difference between these models lies in the derivation of the weights. Under a Bayesian framework, we explore alternative estimation methods to time-consuming stepwise and reversible jump Markov chain Monte Carlo (RJMCMC) schemes. These methods are simple to implement and lead to sparse estimation of more encompassing LSTAR and TAR models than often considered.

In this talk, I will introduce the basics of Ricci flow in dimension 2. In this setting, the Ricci flow equation can actually be written as a scalar PDE, which makes it much easier to deal with. In higher dimensions, existence and uniqueness of a solution to the Ricci flow equation has only been proven with the assumption that the initial manifold is complete with bounded curvature. However, in 2D, this is not the case. I will discuss existence/uniqueness of Ricci flows for this more general class of initial metrics, with proofs if time permits.