1/13/26 --  -- (Notes)

Abstract:  

11/14/25 -- Eric Metzger -- The Game of Hex and Connections to Topology -- (Notes)

Abstract: The game of hex involves two players each taking turns to build a path across a hexagonal grid. Unsurprisingly, somebody will always win a game of hex, but this fact — the hex theorem — turns out to be rather deep. In particular, the hex theorem, something discrete in nature can be used to give a rather elegant proof of Brouwer’s fixed point theorem, which usually requires algebraic topology to prove. In this talk we will talk about how to prove the hex theorem, its connection with Brouwer’s fixed point theorem, and the equivalence of both of them with the Jordan curve theorem, another obvious statement with nontrivial proof. 

11/21/25 -- Will Hammond -- Comparing State Space Models: Approximating WAIC Beyond MCMC

Abstract: State Space Modeling (SSM) is an exceedingly flexible modeling framework for time-series data. However, comparability is a major challenge under the SSM modeling paradigm. At present, no widely applicable and efficient framework for the comparison of SSMs (particularly those of differing dimensionality) exists due to the variety of numerical methods for estimation and the associated computational costs. Currently, many SSM practitioners forgo model selection and validation altogether due to these costs. In this talk, I will introduce SSM and current model comparison techniques and discuss my approach towards approximating the Watanabe-Akaike Information Criterion for non-MCMC SSM fitting methods.

12/5/25 -- Jonathan Pal -- Permutations & Riemann Surfaces  -- (Notes)

Abstract:  We will discuss the permutation factorization problem and its relation to the (ramified) covering space theory of Riemann surfaces.

10/18/23 -- Ziyal Jandrasi -- Linking of Knots, Linking of Manifolds, and Linking of Letters -- (Notes)

Abstract: We all likely have an intuition for what it means when two loops of string are linked.  However, it can often be difficult to tell at first glance whether this is the case.  One attempt to mathematically determine whether two loops are linked is the linking number, perhaps the most intuitive link invariant.  We will discuss how to compute the linking number and how one might expand this notion to a stronger invariant on multi-component links.  On this path, we will see how to generalize the linking number to embeddings of manifolds.  In particular, we will discuss the linking of zero-manifolds and how this gives rise to an invariant on the derived series of free groups.

2/13/24 -- Ziyal Jandrasi -- A Pointless Approach to Tychonoff’s Theorem Without AC -- (Notes)

Abstract: The Axiom of Choice (AC) is somewhat controversial among mathematicians, and is equivalent to a number of statements across many different fields.  Some to note are the Well Ordering Theorem, the existence of bases for vector spaces, the existence of spanning trees in connected graphs, and Tychonoff’s theorem.  We will discuss some consequences which arise from not assuming AC, in particular those arising in topology.  We will then outline an approach to topology in which Tychonoff holds constructively.

2/28/24 -- Thomas Carlson -- Willans' Formula and What Counts as a Formula -- (Notes)

Abstract: In 1964, Willans introduced a formula for the n-th prime number. We will take a look at this formula and explain how it works. Then, a discussion of why we might not consider this to be a formula for the n-th prime number.

4/3/24 -- Thomas Carlson -- The Wirtinger Presentations of Knot Groups -- (Notes)

Abstract: A knot invariant is an assignment of some kind (polynomial(s), yes/no, number, group…) to each knot which is invariant under ambient isotopy. In other words, a knot invariant is a function on the set of isotopy classes of knots. One such invariant is the knot group, the fundamental group of the complement of a knot embedded in S3. In this talk, we will construct this group for a general knot using the Seifert-Van Kampen theorem, then compute a few nice examples and observe a few simple consequences of this construction.

9/27/24 -- Alex Ballow -- Towards the Higher Homotopy of Spheres -- (Notes)

Despite mathematicians' love for spheres and how easy their homology groups are to compute, the homotopy groups of spheres is a very nontrivial problem which is still researched today. We will review the necessary homology concepts and discuss some of the heavy hitting theorems in the field, including the Hurewicz theorem and the Freudenthal suspension theorem. We calculate what we can, mainly when the homotopy group is zero, but will get nowhere close to them all, or even to the stable ones. This is a fascinating area of study we will only scratch the surface of, but will hopefully be a useful introduction to working with homotopy groups.

1/22/25 -- Thomas Carlson -- Nakayama’s Lemma -- (Notes)

Abstract: This talk will introduce several theorems in commutative algebra that are known (collectively or individually) as Nakayama’s Lemma. These are a sequence of results which govern the interaction of the Jacobsen radical of a ring and the finitely generated modules over that ring. We will prove these results and discuss a few corollaries which (hopefully) illustrate the utility of these results in commutative algebra and algebraic geometry.

2/5/25 -- Ziyal Jandrasi -- How Does One Quotient a Category? -- (Notes)

Abstract: A common occurrence in mathematics is that of declaring two objects “the same” when they are a priori different.  Much of the time, this comes about in the form of taking equivalence classes under an equivalence relation.  We will discuss why this is difficult in the situation of categories and see a few candidates for how to remedy the situation.