I am an associate professor of mathematics at the University of Oregon. My research interests include random spatial processes, mathematical statistical physics and number theory.
I am currently the Secretary/Treasurer of the American Association of University Professors.
I will be teaching a summer school course on $p$-adic electrostatics at the CIMPA-CIMAT Research School on $p$-adic Numbers, Ultrametric Analysis and Applications in Guanjuato, Mexico May 23-31, 2022. We will cover the following topics.… Read More »Non-archimedean electrostatics
Some notes on foundational ideas and definitions in modern probability theory.
The set of possible outcomes of a random experiment is called the sample space. Subsets of the sample space are called events. We will eventually axiomatize probability measures, but for the moment we view probability… Read More »Sample Spaces, Events & $\sigma$-Algebras
We begin with the sample space $\Omega$ of a random experiment, and a $\sigma$-algebra $\mathcal H$ on $\Omega$ consisting of subsets of $\Omega$ to which we want to assign probabilities. Definition: A probability measure on… Read More »Measure & Probability
A measurable space consists of a set and a $\sigma$-algebra on that set. If $(E, \mathcal E)$ and $(F, \mathcal F)$ are measurable spaces, then we say $f: E \rightarrow F$ is measurable if for… Read More »Random Variables & Measurable Functions
Given a probability space $(\Omega, \mathcal H, \mathbb P)$ and positive simple random variable $X \in \mathcal H_+$, $$X = \sum_n a_n \boldsymbol 1_{A_n},$$ the expectation of $X$ is given by $$\mathbb E[X] = \sum_n… Read More »Expectation & Integral
Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space and let $(E, \mathcal E, \mu)$ be a measure space. Set $A \in \mathcal H$ and $B \in \mathcal E$ are called null sets if… Read More »Almost Surely, Almost Never & Almost Everywhere
Our expectations and integrals are examples of Lebesgue integrals. How do these relate to the Riemann integrals we learned about in calculus? The group of real numbers under addition $(\mathbb R, +)$ is an example… Read More »Haar & Lebesgue Measure
Suppose $(\Omega, \mathcal H, \mathbb P)$ is a probability space and let $X$ be a random variable. Then, we may define a measure on $(\mathbb R, \mathcal B(\mathbb R))$ by setting $$\mu_X(A) = \mathbb P\{X… Read More »Distributions & Densities