By computing the volume of slices of a polytope P, we can construct its intersection body IP, a fascinating (sometimes nonconvex) object coming from convex geometry. In joint work with Katalin Berlow, Marie-Charlotte Brandenburg, and Chiara Meroni, we study the intersection body of a polytope and prove it is a semialgebraic set. We further investigate its algebraic boundary and find an upper bound on the degree of its irreducible components. This talk will include at least a few nice pictures!

Abstract: In 1909, Thue proved that when F(x,y) \in \mathbb{Z}[x,y] is irreducible, homogeneous, and has degree at least 3, the inequality |F(x,y)| \leq h has finitely many integer-pair solutions for any positive h. Because of this result, the inequality |F(x,y)| \leq h is known as Thue’s Inequality and much work has been done to find sharp bounds on the number of integer-pair solutions to Thue’s Inequality. In this talk, I will describe different techniques used by Baker, Mueller and Schmidt, Saradha and Sharma, and Akhtari and Bengoechea to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with ``the gap principle’’ and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

The absolute Galois group of the rationals has a faithful action on embeddings of 2-colored graph embeddings. As a result, we can use graph embeddings to study an otherwise mysterious, but very significant group. We can also use this relationship to construct invariants of number fields. In this talk, we'll cover some relevant background material, prove the "hard" direction of Belyi's theorem, which establishes the group action, and look at some examples with plane graphs.

We'll introduce a standard combinatorial description for cellular graph embeddings and an extension of the description to a wider class of graph embeddings. Then we'll see some applications to my own research with Peter Veerman and encounter some open permutation counting questions along the way.

Let v>k>i be non-negative integers. The generalized Johnson graph, J(v,k,i), is the graph whose vertices are the k-element subsets of a v-element set, where vertices A and B are adjacent whenever |A ∩ B| = i. In this talk, I will present joint work with students Ari Herman and Taiyo Terada in which we derive general formulas for the girth, odd girth, and diameter of J(v,k,i). Additionally, we provide a formula for the distance between any two vertices in terms of the cardinality of their intersection.

Recent years have seen the introduction of naive set theory to sudoku solving techniques. The main theorem we will cover is Phistomefel’s Ring, applicable to any sudoku puzzle and relates the digits contained in disparate regions of the sudoku grid. We will also cover two corollaries: one from PotatoHead which applies Phistomefel to sudoku puzzles with an anti-knight restriction, and one from Jay Dyer which applies to puzzles with an anti-diagonal restriction. No knowledge of advanced sudoku solving techniques is assumed for this talk.

A continuous-time quantum walk (CTQW) on a graph is a quantum version of a random walk. CTQW's have applications to quantum computing, and can be studied using the tools of algebraic graph theory. In this talk, I will define CTQW's, prove a few basic results, and show how CTQW's can be visualized with animations. The talk will assume no background in physics. It will assume some familiarity with concepts from graph theory and linear algebra (including the adjacency matrix and matrix exponentials).

We consider the results presented in the article "Some results about the connectivity of trees" by Markenzon L, Abreu N & Lee L. (2013). Algebraic connectivity is a measure of connectivity in graphs that offers more granularity than the usual measure of connectivity, but it is relatively computation intensive. The paper presents an Internal Degree Sequence (IDS) Invariant that has a simpler computation. We will explore the effectiveness of the IDS Invariant as a proxy for algebraic connectivity. Justifications and illustrations are provided. We also briefly test an Alternative Invariant and use both Invariants to motivate directions for possible future research.

Last week (in Part I) we introduced a graph to represent a knot (called the Tait graph), and a particularly nice kind of matrix, known as a spin model. This matrix helps determine if two diagrams represent the same knot. But where can we find such matrices? This week we consider the hunt for spin model matrices!

In 1996, Jaeger discovered that spin model matrices are always contained in a certain kind of matrix algebra, and this algebra is often the adjacency algebra of a graph. But which graphs have a spin model matrix in their adjacency algebra? Several families are known, and it is conjectured that the list is complete. This question remains open.

For more than a century, the theory of knots has inspired much work in algebraic topology and geometry. More recently, however, graph theory and combinatorics have begun to play an increasingly important role. In 1990, Vaughn Jones won the Fields medal for his work introducing methods from statistical mechanics to the study of knots. His work introduced not only the famous Jones polynomial, but also a more general link invariant constructed using a combination of two things: a graph representing the knot (called the Tait graph of the knot), and a particularly nice kind of matrix, known as a spin model.

But the connection to graph theory goes deeper than just the Tait graph. In 1996, Jaeger discovered that spin model matrices are always contained in a certain kind of matrix algebra, and this algebra is often the adjacency algebra of a graph. For the construction to work, however, the graph needs to be distance-regular. It is natural to ask WHICH distance-regular graphs have a spin model matrix in their adjacency algebra. Several families are known, and it is conjectured that the list is complete. This question remains open.

We focus here on establishing the persistence criteria for a single-species population, whose habitat can be represented by a Cayley tree, outside of which the species cannot survive. Persistence refers to the population’s ability to remain safely away from extinction. Here, persistence occurs if the population size, being positive at time zero, remains positive when time tends to infinity. The population is governed by a system of differential equations, that is cooperative, irreducible, and admits zero as a solution. It is shown that the population is persistent if, and only if, the stability modulus of the Jacobian of the system at zero is positive. The stability modulus is obtained analytically, and two persistence criteria are derived: i) the first states that the population per capita birth rate must be above a certain threshold, and ii) the other requires existence of the minimum patch size, which is here defined as the minimum number of levels of the Cayley tree that makes it a habitat capable of sustaining the population.

Originating in the world of 1960s chemistry, the study of chemical reaction networks describes chemical reactions as weighted, directed networks. These networks give rise to complicated polynomial systems of differential equations. This talk will discuss these systems and illustrate how their behavior may be understood through the graph Laplacian. No knowledge of chemistry is expected!

ZFC set theory conflicts with highly intuitive axioms regarding probability. This is the topic of a recent publication by John Caughman and myself. I will motivate our probability axioms and present a detailed proof of our main result.

In this talk, I will present joint research with John Caughman on the interplay between set theory and probability. Our main result: that ZFC contradicts highly intuitive axioms concerning probability. I promise napkin-sized arguments showing that both the continuum hypothesis and the axiom of choice are false...great for showing at parties!

In part II of this talk, I will first briefly recap part I of the talk from last Friday. Then I will share some of the new directions for my research project: (i) Fully characterizing the matrix, (ii) Classification (up to isomorphism) of NBLPs and corresponding family of orthogonal polynomials, (iii) DNBLPs and TNBLPs, (iv) NBLPs, and (v) Spin NBLPs.

There is a linear-algebraic object called a Leonard pair and this concept was motivated by the theory of -polynomial distance-regular graph. In this talk, I will first give a brief introduction to Leonard pairs and closely-related classes of objects: (i) totally bipartite Leonard pairs (TBLPs) and (ii) almost bipartite Leonard pairs (ABLPs). TBLPs and ABLPs as departure points, I will introduce a new class of object - nearly bipartite Leonard pairs (NBLPs) and share some of the facts I have discovered on the NBLPs.

The Vandermonde polynomial V(x₁,x₂,…,x_N) is defined as the product of (x_i - x_j) over all i < j. If the x_i are p-adic integers, then so is the Vandermonde polynomial, and hence its p-adic absolute value is either zero or |V(x₁,x₂,…,x_N)|_p = p^-n for some nonnegative integer n. If (x₁,x₂,…,x_N) is chosen uniformly randomly from (ℤ_p)^N, what is the probability that |V(x₁,x₂,…,x_N)|_p = p^-n, and how does it vary with N, p, and n? In the first half of the talk we will answer this question by developing a combinatorial formula for the probability. In the second half we will use the formula to compute the probabilities in the N = 2 and N = 3 cases (and maybe N = 4 if we have enough time), then discuss how estimates for more general N, p, and n can be made by solving coin changing problems.

A q-analog of a number N is a polynomial in q that simplifies to N when we set q = 1 and also encodes some “additional structure”. In this talk we will explore many different examples of what may be meant by “additional structure”, focusing on the classical q-analog of the factorial and moving to more recent work on a q-analog of a “Fubini number”, which gives the number of surjections from an n-element set to a k-element set.