Oct

26

2020

Finiteness properties of groups can be defined using corresponding topological spaces. These properties are deeply important to group theory but can be very challenging to calculate. Bieri-Strebel Sigma Theory provides a related property of modules called "tameness" that can be found through a consistent set of steps. We will describe these steps and provide a visual intuition for the meaning of "tameness". We will show an example calculation and conclude with a conjecture about the relationship between tameness and finiteness.

Oct

26

2020

Logic is generally understood to express content-general relationships. This is problematic for the sake of teaching undergraduates logic in proof courses because their reasoning is usually content-specific. In a series of teaching experiments, I have been exploring how to guide students to apprehend logical relationships by reflecting on their mathematical reasoning. This is done by having them compare how they interpret different statements with the same logical form (intentionally varying the mathematical context) or mathematical proofs with parallel logical structure.

Oct

30

2020

In this talk, I will survey recent results obtained in joint work with Radu Dascaliuc, Tuan Pham and Ed Waymire, relating solutions of the Navier Stokes equations of fluid mechanics to a stochastic branching process naturally associated to these equations. While the equations themselves are deterministic, the scaling properties of the equations suggest a branching structure of the Fourier transform of the equations. First noted by LeJan and Sznitman in 1997, the branching structure provides a representation of the solution provided this branching process is non explosive.

Nov

02

2020

We show how coupling and stochastic dominance methods can be successfully applied to a classical problem of rigorizing Pearson's skewness. Here, we use Frechet means to define generalized notions of positive and negative skewness that we call truly positive and truly negative. Then, we apply stochastic dominance approach in establishing criteria for determining whether a continuous random variable is truly positively skewed. Intuitively, this means that scaled right tail of the probability density function exhibits strict stochastic dominance over equivalently scaled left tail.

Nov

06

2020

We will discuss recent results on the local convergence of Anderson Acceleration (also known as Pulay mixing or DIIS). This is an algorithm for accelerating the convergence of fixed point or Picard iteration and is widely used in quantum chemistry and physics codes.

In this talk we will discuss our convergence results for the method, illustrate the results with an application to radiative transport, and report on some recent results on non-differentiable nonlinearities and

global convergence.