(Wednesday Geometry Seminar Talks Currently Not Included)
Spring 2022 - MSU Student-Led Seminar (WWCC) - Title: Thousand-Yard Square: Constructing nxm De Bruijn Matrices on Size-k Alphabets
Abstract: After the internet excitement surrounding an anonymous user's contribution to the study of superpermutations, inspired by the somewhat infamous Haruhi problem in 2018, I took interest in a somewhat related subject. A superpermutation being a string containing every permutation of a given length as a substring, ideally the shortest possible. I sought to construct matrices containing every matrix of a given size (with finite possible entries), ideally the smallest possible. Turns out that this is a somewhat well-researched topic, sitting comfortably between the related combinatorial ideas of De Bruijn sequences and De Bruijn torii. As such, the moniker of De Bruijn matrices has been affixed onto these oft-over-looked objects. This talk will be outlining De Bruijn sequences, an algorithm for construction of these De Bruijn matrices, discussing optimization of the size of these matrices, as well as a few other observations on the subject.
Fall 2023 - MSU Student-Led Seminar (WWCC) - Title: Fallow Fields
Abstract: In common language, the act of letting a field "fallow" refers to the practice of plowing a field but not sowing a crop in order to restore fertility to the soil. In this talk, I will be covering construction and basic concepts related to several fields (in the algebraic sense) with important and useful properties (i.e., "fertile fields" following the "fallow fields" analogy). Graduate students may have little familiarity with these fields (and therefore may be relatively "unsown fields" by local graduate students). Specifically, I plan to cover topics relating to some subset of finite fields, p-adic numbers, and/or surreal numbers approached via combinatorial game theory.
Spring 2024 - MSU Student-Led Seminar (WWCC) - Title: Polygon Spaces of \ell \in \mathbb{R}^5
Abstract: Discussion motivated by interest in Topological Robotics. We will specifically discuss the configuration space of planar linkages with 5 or fewer links which can be realized as a polygon space or as a moduli space. We will classify these spaces, note properties of their "support", and discuss degeneration of these spaces along paths in \mathbb{R}^5. Time permitting, we will have a limited discussion of some subset of the following topological invariants: dimension, homology/cohomology groups over \mathbb{Z}, cup length, Lusternik-Schnirelmann category, and topological complexity relating to these spaces.
Fall 2024 - MSU Student-Led Seminar (WWCC) - Title: Literally the BEST Theorem
Abstract: de Bruijn sequences, matrices, and torii are combinatorial objects with several properties of interest. In short, these are objects that contain all other objects of a given description as "sub"-objects exactly once. de Bruijn sequences are well understood and can be readily generated and enumerated, and, from these, de Bruijn torii may be generated. On the other hand, not all de Bruijn torii/matrices can be realized in this way, nor can they be easily enumerated. In this talk, we'll employ a graph theoretic approach to chip away at this problem and discuss an algorithm to generate all "linear" de Bruijn matrices and efficiently enumerate them literally using the BEST theorem.
Spring 2025 - MSU Math Seminar (Ph.D. proposal) - Title: Homology of Conically Smooth Manifolds via Links
Abstract: Given a "nice" smooth manifold we may affix it with additional mathematical structure to be able to apply further machinery to explicitly compute its (co)homology. Namely, we can affix a well-behaved (poset-)stratification along with the notion of conical smoothness to guarantee we can ”unzip” our space into disjoint copies of Euclidean space while remembering how these pieces interconnect using iterated blowups to construct an associated complex for the given manifold. Then, once we have successfully constructed the appropriate complex of the manifold, we may use spectral sequence methods to compute its homology. In this talk, I will provide some background to solidify these concepts as well as provide a sketch of the proof of the related proposition.
Fall 2025 - MSU Student-Led Seminar (WWCC) - Title: Metallic Ratios: Closed Forms and p-adic Behavior
Abstract: The golden ratio, (1+sqrt(5))/2, is infamous, and arises as the limit of the ratio of consecutive terms in the Fibonacci sequence, but the golden ratio is actually only one example from an entire family of ratios known as the metallic ratios. These all similarly arise from two-term recursively defined sequences of the form S_n = a*S_n-1 + S_n-2. In this talk, we will quickly compute the general form of these ratios then employ these ratios and a generating function to write a closed form equation for terms of these metallic sequences (analogous to Binet's formula). We will then investigate and prove Catalan's identity using this closed form equation. That identity then will give rise to an algorithm that helps illuminate a few unexpected convergence results of S_{p^n} in \mathbb{Z}_p, the p-adic integers.
