Table of Contents

Research

My research is in commutative algebra. My motivations are classical ideal-theoretic questions in the tradition of Gilmer, Kaplansky, Mott, and Krull, but I enjoy "cross-pollinating" different areas of mathematics to answer those questions. My most recent work (see ArXiV, jointly with Jim Coykendall) is a homological take on factorization. This paper is a bit long, because it touches on lots of areas; we establish a method of assessing the degree of pathological factorization behavior (read: non-atomic) in arbitrary integral domains. This method constructs cochain complexes and cohomology groups that are functorially associated with the group of divisibility, allowing us to obtain results about commutative rings using partially-ordered group (po-group) theory, cohomological algebra, and category theory.

Other works I am currently tinkering with include deeper categorical and topological looks at the category of po-groups. Topologically, I would like to identify the class H of integral domains (which may be empty!) so that if D is an integral domain in H then the group of divisibility G(D) admits a non-trivial Haar measure compatible with factorization in D. Categorically, we can regard the group of divisibility as a functor from the category of integral domains to the category of po-groups, leading to many questions about functoriality (identification of adjoint functors, identification of universal properties, identification of categorical equivalences, etc).

This past summer I worked with my advisor Jim Coykendall and with Sean Sather-Wagstaff looking into a new category describing ordered factorizations and their combinatorial diversity in general environments like pre-ordered monoids. This category of factorizations has a lot of interesting properties: there is a subcategory of weak equivalences and a model structure, so the category of factorizations admits a homotopy category which coincides with the positive cone of the group of divisibility. Hence, the category of factorization is a section of the group of divisibility (or rather, the group of divisibility is a retract of the category of factorization) in the sense that there exists a pair of functors, F:C → G, E:G → C, that compose to the identity in one direction (FE is the identity on G) but not the other direction (EF is not the identity on C). We show this category is a non-trivial construction by using it to prove certain classic theorems in commutative ring theory: formulate a ring-theoretic problem in the context of the category of factorizations, solve the problem in that setting, and then lift the result back into the context of ring theory. For example, we prove that an ACCP domain is necessarily atomic and we construct a proof by counter-example of an atomic domain that is not ACCP.

I also have a history of research outside of commutative algebra with mathematical ecology and computational neuroscience and I have also contributed extracurricular time toward scholarly research in cryptocurrencies. If you would like a copy of my research statement, feel free to shoot me an e-mail.

Recent Talks The above are a few of my recent talks (the one whose PDF files I could find last time I updated this website, anyway). I plan on including all my presentations here from now on.

Teaching

If you are one of my students looking for my course-specific webpages, check out my section For Students (but that will merely direct you toward blackboard for now).

I have been teaching in some capacity or another for over 8 years now. If you would like a copy of my teaching statement, feel free to contact me.

Clemson University Teaching

North Dakota State University Teaching

Roughly, my history at NDSU followed this path, although I skipped some summers and the ordering details may be slightly incorrect.

For Students

Interests

My interests and hobbies are varied, including dynamical systems in biology and neuroscience, machine learning, cryptocurrencies and practical applications of math, computational approaches to assessing cultural evolution and phenomena, and stochastic process and time series analysis, in no particular order (and with no regard to the irony built into a sentence filled with so many broad and vague taglines). I've been involved in coding projects regarding every one of these interests, some of which are available online. My master's thesis was focused on dynamical systems in neuroscience and constructing a bifurcation portrait for a computationally handy model neuron, and my research as an undergraduate involved stochastic implementations of ODEs describing ecological progression of plague in black-tailed prairie dogs and their fleas. One would call me a ``pure'' mathematician nowadays, if you accept the false dichotomy of ``pure'' vs. ``applied.''

My most recent interests have involved cryptocurrencies (the Bitcoin and Monero protocols) which I feel present novel problems for computer science and mathematics to solve. From a rather... profit-driven point of view... I have developed my own code for automatedly buying and selling bitcoins (check out my GitHub), which uses simple statistical techniques and a modified version of investment rebalancing to make profit from the high volatility of the market. It is undoubtedly broken right now, and undoubtedly inefficient, but a guy's gotta write his dissertation, naturally.

Scientifically, there are a bunch new and interesting problems in mathematics and computer science that these currencies represent, and research in these areas is currently somewhat impoverished. I have taken interest in the Cryptocurrency Difficulty Control Problem, as described here. The problem is in stochastic processes that I think is going to become more important over the coming years, and Bitcoin exemplifies one of them. How do we map a non-homogeneous Poisson process with an unknown intensity function to a stochastic process that may be regarded as a reliable clock? I am interested in approaches to compressing the blockchain of a currency to allow for greater scalability. In particular, if we regard the history of financial transactions as a partially ordered set (or an acyclic directed graph), there ought to be some order-embedding to the smallest possible acyclic directed graph, allowing for chains of undisputed transactions to, essentially, be ignored in terms of transaction validation. In other words: compose the arrows instead of keeping long strings of composable arrows just sitting around on the blockchain.

Once upon a time, I was interested in blending computational neuroscience and machine learning in Project BRAINN with a colleague, Aaron Feickert, from NDSU, but that project was terminated before it bore any intellectual fruit. My interest in using computational approaches to machine learning and cultural evolution has not waned, though. On my GitHub, (link here), you can check out my book-clustering project, wherein I download free math textbooks, use optical character recognition to convert them into text, and then attempt to blindly cluster them by topic using k-means clustering. It is a hilariously inefficient approach to a worthy problem (it takes doubly exponential time to converge upon a solution, it downloads hundreds of free math PDF textbooks to your computer, and processes them into text... well, you do the math). However, it was an interesting application of abstract data structures.

One problem that characterizes many of my interests going further back in my history is the question of how altruism evolved. This question, which seems to be directly answerable, blends my interest in stochastics, evolution, dynamical systems, game theory, and cultural phenemona. My approach to look at this question included simulating a game that hybridized the Game of Life and Prisoner's Dilemma and incorporates evolutionary theory; my simulations ended up producing fairly impressive results, which you can view here. Clustered groups of cooperators tend to form with defectors existing in the "cracks" between coooperating masses of individuals.

Contact

E-mail is bggoode at Clemson, with the usual educational suffix. You can find more detailed contact information on my CV, or you can also contact me through GitHub or through LinkedIn.