GSS – Spring 2022

Abstract: In this talk, we exposit Furstenberg’s $\mathsf{\times 2, \times 3}$ Conjecture – an open problem in a branch of dynamics called ergodic theory.  The conjecture states that if a Borel probability measure $\mathsf{\mu}$ on $\mathsf{[0,1)}$ is invariant and ergodic under the transformations $\mathsf{T(x) := 2x \pmod 1}$ and $\mathsf{S(x) := 3x \pmod 1}$, then it is either atomic or Lebesgue measure.  We discuss the relevant concepts necessary for understanding the problem and appreciating its significance. 

The problem has not been solved in full, but Daniel Rudolph has proved a weaker statement, imposing the additional assumption that $\mathsf{\mu}$ has positive entropy with respect to one of $\mathsf{T}$ or $\mathsf{S}$.  We review measure-theoretic entropy, and then finish with a survey of Rudolph’s methodology.

Abstract: The geodesic flow on a negatively curved manifold is a uniformly hyperbolic system: in particular it has a unique measure of maximal entropy and more generally, unique equilibrium states for Holder continuous potentials. When curvature is assumed to be non-positive, the geodesic flow becomes non-uniformly hyperbolic, and much less is known. For a rank-$\mathsf{1}$ manifold of non-positive curvature, Knieper showed the uniqueness of the measure of maximal entropy but his methods do not generalize to equilibrium states for non-zero potentials.

We will present the main result of Keith Burns, Vaughn Climenhaga, Todd Fisher, and Daniel Thompson’s paper in which they use a non-uniform version of Bowen’s specification property to establish the existence and uniqueness of equilibrium states for some non-zero potential functions. 

Abstract: Given two integral quadratic forms, it is natural to ask whether or not they are equivalent. Does there exist a procedure which, after a finite (hopefully small) number of steps, will provide an answer to this question? We will formulate this problem in terms of search bounds, and describe how to convert it to a question about the dynamics of a certain Lie group action on a homogeneous space.

Abstract: The boundary of a group or a metric space encodes all the ways you can head to infinity. For example, heading toward infinity in the Cayley graph of a group is essentially traveling along a path which gets further and further from a fixed base-point. We will present the definition of the boundary with lots of examples. Additionally, we will discuss some ways the boundary is used as a tool in proofs.

Abstract: Using the intuition of the power set of a set with the operations of union, intersection and complement, we give a brief introduction defining Boolean algebras and related concepts. We show that the above intuition is essentially everything that is going on for finite (or, more generally, complete atomic) Boolean algebras. We generalize the argument defining a topology on the set of ultrafilters of a Boolean algebra and present the Stone duality for these structures. We conclude, if time permits, with a dictionary Stone Spaces – Boolean Algebras translating some problems and properties of these two categories.

Abstract: Persistent homology is a tool which allows us to study and characterize topological features of datasets and various functions. In this talk, we’ll further develop the notion of persistence modules and outline some of their key properties, as well as introduce the persistence diagram and explain its purpose. We’ll also see some of the most commonly used abstract simplicial complexes for persistent homology, as well as key examples to understand their usefulness. Time permitting, we’ll briefly survey some interesting results achieved through the use of persistence, as well as further topics of note.

Abstract: In this talk, we will see how to use persistence modules to solve a concrete approximation problem in Morse Theory. On the way, we will introduce interleaving distance on the space of persistence modules and bottleneck distance on the space of barcodes. This will lead to the famous isometry theorem. Time permitting, we will see other invariants in persistence modules.

Abstract: In the 1970’s, John Conway defined the notion of a tangle as a tool to enumerate knots and links. In doing so he corrected and expanded the tables that were constructed by Tait, Little, and Kirkman in the late 1800’s. This talk will focus on Conway’s notion of a rational tangle and the tangle operations that allow us to realize all rational tangles as being constructed from a few simple building blocks. From there we will elaborate on the connection between the knots which arise as closures of rational tangles and the classical knot theory invariant of bridge number (a measure of a knot’s complexity). Time allowing, we will conclude with an example that highlights the remarkable utility of Conway’s combinatorial construction for the purpose of producing perfect prototypes of contemporary invariants.

Abstract: The treewidth of a graph is a number that tells us, roughly, how far a graph is from being a tree. Graphs with small treewidth exhibit certain desirable tree-like properties, so structural characterizations of graphs with small or bounded treewidth are often sought after. A famous result proved by Robertson and Seymour states that grid minors are canonical obstructions to small treewidth– graphs have large treewidth if and only if they contain a large grid minor. We present a recent result (Aboulker et al. 2021) that strengthens the grid minor theorem in the case of $\mathsf{H}$-minor-free graphs: If $\mathsf{G}$ is an $\mathsf{H}$-minor-free graph with large treewidth, then $\mathsf{G}$ must contain either a large wall or the line graph of a large wall as an induced subgraph. This result is closely related to a number of recent conjectures, which will be discussed if time permits.

Abstract: We all want privacy, especially when it comes to communication. Cryptography is the study of techniques that allow us to communicate with one another securely. In this talk, we discuss the basics of cryptography, the major drawbacks with certain cryptosystems, and how current methods overcome those drawbacks.