# Spring 2021 Department Colloquium

Due to the pandemic, the NAU Department of Mathematics and Statistics Colloquium will be held via Zoom. The talks will typically take place on Thursdays at 4:00-5:00pm. Please contact Nandor Sieben to obtain the Zoom information or to be added to the mailing list that includes the Zoom information.

### Thursday 2/25 at 4:00-4:50

Speaker: Janos Englander (University of Colorado Boulder)

Title: Turning a coin instead of tossing it

Abstract: Given a sequence of numbers $\{p_n\}$ in $[0,1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n$, $n\geq 2$. What can we say about the distribution of the empirical frequency of heads as $n\to \infty$? We show that a number of phase transitions take place as the turning gets slower (i.e., $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n=\text{const}/n$. Among the scaling limits, we obtain some well-known special (Uniform, Gaussian, Semicircle and Arcsine) laws. The talk is intended to a general audience and no expertise in probability is assumed!

## Thursday 3/4 at 4:00-4:50

Speaker: Steve Wilson (NAU emeritus)

Title: The Diagram Quest

Abstract: A semiregular symmetry in a graph is a symmetry which moves all the vertices in cycles of the same length. The ‘diagram’ of such a symmetry has one ’node’ for each cycle and ‘links’ joining nodes, one for each orbit of edges joining vertices of those cycles. We will see what we can learn about the graph from the diagram, and how to re-construct the graph from the diagram. We will discuss what makes a diagram ‘good’ and how to find ‘good’ diagrams.

## Thursday 3/11 at 4:00-4:50

Speaker: Jim Swift (NAU) and Michael Swift (U.S. Naval Research Laboratory, Washington, D.C.)

Title: Solving the Poisson-Fermi-Dirac equation: the electrical double layer in solid-state batteries

Abstract: The Poisson-Boltzmann equation has long been used to model the changing potential in a battery at the interface between an electrode and the liquid electrolyte. A self-consistent solution of the ion concentration (given by a Boltzmann distribution) and the electrostatics (governed by the Poisson equation) gives a simple exponential decay with a characteristic “Debye” length. This basic model has been used with only slight modifications for over 100 years. However, recent advances in battery technology improve safety and reduce the size and weight of the battery by replacing the flammable liquid electrolyte with a solid. The simple Boltzmann distribution for ion concentration in liquid electrolytes no longer describes solid electrolytes, calling the applicability of the old models into question for new solid-state batteries.

In this work, we propose a model which replaces the Boltzmann distribution with a Fermi-Dirac distribution for the solid electrolyte. This leads to a new “Poisson-Fermi-Dirac” equation. We solve the Poisson-Fermi-Dirac equation by exploiting an emergent conserved quantity. The intuition is guided by an analogy to a fiendish hole of miniature golf: how hard do you hit the ball so that it comes to rest exactly at the top of a nearby hill without rolling down the other side? In addition to a numerical solution, we identify regimes in which the solution can be approximated analytically. One approximation holds for six orders of magnitude of the independent variable. The solution is then applied to various solid-state battery materials, with implications for improving the performance of future solid-state batteries.

## Thursday 3/18 at 4:00-4:50

Speaker: László Erdős (Institute of Science and Technology, Austria)

Title: Order and disorder in mathematical physics

Abstract: We review a few mathematical problems that were motivated from the quantum physics of disordered systems. They have inspired entirely new directions in mathematics such as quantum chaos and random matrices. In this talk we will present some basic results and major open conjectures in these areas. The talk will be accessible without physics background.

## Thursday 3/25

No colloquium (Department Meeting)

## Thursday 4/1 at 4:00-4:50

Speakers: Weston Loucks and Stephanie McCoy (NAU graduate students)

Title 1: Codes with The Extended Binary Hamming Code and Other Double-Circulant Minors

Abstract 1: Minors play an important role in matroid and graph theory, but have not been widely studied in the context of linear codes. In this work we establish necessary and sufficient conditions for a binary code to have another binary code as a minor, and in particular, we consider linear binary codes which have a minor isomorphic to the binary extended Hamming code, the code analogue of the identically self-dual matroid AG(3,2). We show that all optimal self-dual codes of minimum distance 4 or greater, up to length 32, have an extended binary Hamming code minor. We also study sufficient conditions with regard to substrings which indicate the presence of pure double-circulant code minors in double-circulant codes. The extended binary Golay code is given specific attention as we search for double-circulant self-dual codes which contain it as a minor 4.

Title 2: Impartial Achievement Games on Convex Geometries

Abstract 2: We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.

## Thursday 4/8 at 4:00-4:50

Speaker: Glenn Hurlbert (Virginia Commonwealth University)

Title: On Erdős–Ko–Rado Graphs and the Search for a Conjecture

Abstract: One of the cornerstones of extremal set theory is the following 1961 theorem of Paul Erdős, Chao Ko, and Richard Rado: if $\cal F$ is a family of $r$-subsets of ${1,…,n}$, with $r\le n/2$, such that every pair of sets of $\cal F$ intersect, then $|{\cal F}|\le \binom{n-1}{r-1}$. The family of all $r$-subsets containing a fixed element $x$ achieves this bound and is called a star with center $x$. There are many clever and beautiful proofs of this result; we recently discovered an injective proof.

In 2005, Fred Holroyd and John Talbot introduced a generalization to graphs, in which $\cal F$ is an intersecting family of independent $r$-sets of vertices of a graph. For a given graph, we ask if no such family is bigger than the largest star; if so, the graph is called $r$-EKR. Their conjecture is that every graph $G$ is $r$-EKR for all $r\le \mu(G)/2$, where $\mu(G)$ is the size of the smallest maximal independent set in $G$. (The Erdős-Ko-Rado theorem is the case in which $G$ is empty.)

We verified this conjecture when $G$ is a chordal graph (every cycle on at least four vertices has a chord) with an isolated vertex, and have begun to investigate graphs with few edges but no isolated vertices, such as trees and unions of paths. A central question arises as to where the center of the largest star is located. For trees, there are some natural candidates — we show that some of this intuition is right for a while but mostly wrong! In particular, we do find some answers for trees with a unique vertex of degree greater than two, among other classes. But the general situation is a grand mystery at present.

## Thursday 4/15

No colloquium (Department Meeting)

## Thursday 4/22 at 4:00-4:50

Speaker: Richard Nowakowski (Dalhousie University, Halifax)

Title: Is there a smallest positive combinatorial game? And what is it?

Abstract: Short answers to the first question: knee jerk reaction is ‘Yes; Then with more thought, ‘No’; But then on careful reflection, ‘Yes’; on even more reflection that was wrong, and the final, Final Answer is, ‘Yes’.

How do we get those answers? Combinatorial games are played between the players, Left and Right. If a game, $G$, has a positive game value this indicates how many moves advantage Left has. Games with integer-number-of-moves are easy to understand and boring to play — this is the knee-jerk answer. With more and more thought about what games are actually played, we get rest of the sequence of answers, and the games that go with them. No previous knowledge will be assumed.