From its birth in celestial mechanics and particle physics, to its recent applications to number theory (see, for example, the celebrated Green-Tao or Einsiedler-Katok-Lindenstrauss theorems), ergodic theory has shown itself to be a powerful tool in the modern mathematical toolbox. In these first two talks, we will explore some of the basics of this field at the intersection of dynamical systems, measure theory, and functional analysis. We will introduce definitions, motivation, and some first results, including: Poincare recurrence and the Birkhoff and Von Neumann ergodic theorems. Although these talks are ``stand alone", they are envisioned as the first two in a series of at least four lectures on ergodic theory that culminate in ergodic decomposition. No background in dynamical systems will be assumed. However, a certain comfort with the terminology of measure theory, topology, and functional analysis will be assumed. Significant results and techniques in these fields will be recalled as needed.

We will discuss the Birkhoff Ergodic theorem and the definition of ergodicity. The theorem will be presented as one of the key ingredients for the Ergodic Decomposition theorem, the objective of this series of talks. Time permitting, we will begin discussing one of the other key ingredients: Riesz Representation Theorem or the theory of measurable partitions.

Groups have topology: for example GLn is a twisted product of odd spheres. How can we measure the twisting? Ordinary cohomology doesn’t see this twisting in any obvious way which is one reason we construct exotic cohomology theories. We’ll discuss the connection between complex oriented cohomology theories and formal group laws and why we care about a particular one called complex cobordism!

Last time we described how groups are twisted products of topological spaces. For this talk, we will further describe how to detect and measure twisting in the specific case of a (complex) line bundle (which is a twisted version of the cross product $X\times$ ). We will describe how to classify line bundles according to how twisted they are (construction of the first Chern class) via classifying spaces. We will end with a description of where and how generalized cohomology theories (like complex cobordism) show up in terms of line bundles. We will not assume that anyone is familiar with the material from the first talk and will not assume that the audience knows cohomology.

This will be a continuation of the talk from last week when we got as far as defining a line bundle and giving some examples. This week, we will complete last week's plan by showing how a homotopy between two spaces produces an identification of isomorphism classes of line bundles and introducing both the Chern class and the classifying space of complex line bundles. We hope to eventually discuss complex cobordism. We will be happy to review definitions for anyone who wasn't able to attend part 1, and yes, there will be donuts, real ones, from Strite's.

Paradoxes and misconceptions abound in probability and statistics. Dr. Hamdan will discuss some crucial concepts and cases, including the Monty Hall Problem, the Prisoner’s Paradox, and Simpson’s Paradox. The uses and abuses of other statistical concepts such as statistical significance and p-values, averages, the false positive paradox, and Bedford’s Law will be highlighted.

If we select an element at random from a large cyclic group, what should we expect the order to be? More generally, if we select k elements at random, what should be the order of the subgroup generated by those k elements?

Let p be a prime number and K = Fp be the finite field with p elements, and λ ∈ K such that $\lambda\neq0,\pm1$.
In this work we will study the division polynomials of the Holm elliptic curve in its original form, and a remodeled form, $H_λ : y^3 − y = λ(x^3 − x)$ and a remodeled form,$H_λ' :(u−λ)v^2=u^3−λ$. We will explicitly state their group structures together with the bi-rational correspondence to their Weierstrass models. Using these mappings, we will derive the multi-variable division polynomials and the single variable division polynomials, then investigate their properties.