GSS – Fall 2022

Abstract: An additive coloring of a graph is a labeling of its vertices with positive integers so that the sum of the neighbors of any two adjacent vertices differs. Given an orientation $\mathsf{D}$ of a graph $\mathsf{G}$, we introduce a new digraph $\mathsf{W(D)}$ that allows us to prove an additive coloring analog of the well-known Alon-Tarsi list coloring theorem. As a consequence of this theorem, we extend a list additive coloring result that was previously known only for bipartite graphs to a special class of tripartite graphs.

Abstract: For a finitely presented group $\mathsf{G}$, constructing the Cayley graph of $\mathsf{G}$ gives rise to interesting geometric properties that in turn can determine algorithmic properties of the group $\mathsf{G}$ itself. In this talk, we’ll talk about what the Cayley graph is, what it means for a space/group to be hyperbolic, and how hyperbolicity affects the word problem in groups.

Abstract: In a new addition to the Johnway series, we will discuss the surreal number system, first described by John Conway and later labeled surreal by Donald Knuth in 1974. In this system we describe numbers an order pair of sets, $\mathsf{x = (X_L, X_R)}$. By then defining orderings, addition, and multiplication, we can create the entire real number system as well as sets of infinitesimals. We will also discuss the somewhat surreal plot of D.E. Knuth’s novelette, Surreal Numbers, in which “two ex-students turned on to pure mathematics and found total happiness”.

Abstract: This will be a brief introduction to regular expressions, the languages they define, and their connections to finite-state machines. These are core computer science concepts that can be used to enhance mathematics in several ways. Time permitting, we will talk about one such example: automatic groups, which provide a natural way to approach computation in geometric group theory.

Abstract: Large prime numbers are essential to our security in the modern age.  Used throughout cryptography, large primes allow for secure, encrypted communication.  But how exactly do these large primes come to be?  In this talk, we investigate the history of computing large prime numbers.  Starting with computations by hand and progressing to the modern age, we will see how techniques have improved over time.

Abstract: An $\mathsf{m}$-component link is a smooth embedding of $\mathsf{m}$ disjoint circles into Euclidean 3-space, and the image of each circle is called a component of the link. The simplest invariant of links is the linking number, which counts the number of times one component winds around another. Note that the linking number only captures linking information between pairs of components, while links can have many components intertwined with one another. Thus two natural questions arise: what information does the linking number provide for 2-component links, and to what extent can the linking number measure linking information between 3 or more components? In this talk I will introduce the linking number and give a characterization of 2-component links with the same linking number. Then I will introduce Milnor’s invariants, which can be interpreted as higher-order linking numbers, in order to achieve an analogous characterization for links with 3 of more components.

Abstract: In the classification of first-order theories, many “dividing lines” have been defined in order to understand the complexity and the behavior of some classes of theories. These classes are usually defined by forbidding some specific configurations of definable subsets. In this talk, we give a brief introduction to model theory, we define the principal dividing lines and we introduce some notions of maximal complexity by requesting the presence of all the possible combinatorial patterns of definable sets.

Abstract: Associative $\mathsf{k}$-algebras are useful objects of study in mathematics, but can be unwieldy; instead, we can look at their representations to gain insights on their structure. In this talk we’ll fix $\mathbb{R}$ as a $\mathsf{k}$-algebra with a finite presentation structure over a computable field and examine decidable properties about the representations of $\mathbb{R}$, such as “does an irreducible $\mathsf{n}$-dimensional representation of $\mathbb{R}$ exist?”. To do so, we’ll introduce and borrow fundamental tools from both linear algebra and computational commutative algebra.

Abstract: This exploration will examine the presence of algebraic and topological structures in the conceptual artist Sol LeWitt’s large-scale panel, Wall Drawing 413. The algebraic structures will focus on group theory found in the permutations of the four colors used and a topological investigation will classify some of the compact 2-dimensional surfaces that can be constructed from gluing edges of matching colors of one 4×4 square in the artwork.

Abstract: Boolean functions are important objects of study in theoretical computer science, and experts have developed various measures of their complexity. One such measure, the sensitivity, proved difficult to relate to other measures. The Sensitivity Conjecture states that the sensitivity is related polynomially to other measures, but for decades no one could prove it. Finally, in 2020, Hao Huang resolved the conjecture by proving an equivalent statement, which involves analysis of induced subgraphs of hypercube graphs. In this talk, I will present Huang’s proof, and explain how his method leveraged spectral properties of hypercubes.

Abstract: Conformal dimension is a quasisymmetry invariant of metric spaces.  In this talk, we will first motivate the study of conformal dimension in the pursuit of understanding finitely generated groups.  Then, we will discuss Bourdon’s computation of the conformal dimension of a class of hyperbolic buildings.

Abstract: I’ll begin by giving an overview of Teichmüller spaces. These are spaces that can be seen as parameterizing either geometric (hyperbolic or complex structures) or algebraic objects (certain representations) associated to surfaces. I’ll then talk about Hitchin components, much more recently discovered and less well-understood analogues of Teichmüller spaces that have some remarkable properties. I will survey the progress that’s been made in understanding to what extent Hitchin components arise as spaces of geometric structures, and end by talking about how to parameterize $\mathsf{SL(3,\mathbb{R})}$ Hitchin components using higher degree complex structures, as first described by Fock and Thomas.