## John H. Johnson Jr.

### The Ohio State University

Welcome to my website A Little Ink!

I’m a mathematician at the Ohio State University. My research interest is on the interplay between the algebraic structure of the Stone–Čech compactification, dynamics, and combinatorics related to Ramsey theory. Connected to this work, I’ve written papers with Vitaly Bergelson, Neil Hindman (my PhD advisor), Joel Moreira, and Florian Karl Richter. See my Publications section for links to these papers.

My teaching interest is centered around using active learning and inclusive practices to promote and cultivate an environment which improves students’ abilities to construct, organize, and demonstrate their knowledge of mathematics. Recently, I’ve been working with my colleague Ranthony A.C. Edmonds developing a new service-learning course based on Margot Lee Sterley’s book Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race. I’m also currently serving as PI for the Building a Buckeye Calculus Community as part of Ohio State’s participation as a Phase II institution in SEMINAL (Engagement in Mathematics through an Institutional Network for Active Learning).

Of course, you can get a fuller sense of my professional background via my Curriculum Vitae .

### Interests

• Topological algebra
• Stone–Čech compactification
• Ramsey theory

### Education

• PhD in Mathematics, 2011

Howard University

• BS in Applied Mathematics, 2005

Texas A&M University

Current and Previous

#### The Ohio State University

Sep 2017 – Present Ohio

#### The Ohio State University

Sep 2015 – May 2017 Ohio

#### The Ohio State University

Sep 2012 – May 2015 Ohio

#### Visiting Assistant Professor

Sep 2011 – May 2012 Virginia

# Recent & Upcoming Talks

### Composing some notions of largeness and a classification problem for certain structured sets

Syndetic and thick subsets of a discrete semigroup $S$ are two well-known (and classical) "notions of largeness" that have nice algebraic characterizations in $\beta S$ related to the smallest ideal $K(\beta S)$. Moreover, the connections between these notions of largeness and van der Waerden's theorem on arithmetic progressions are among some of the earliest results in Ramsey Theory.

We'll state generalizations for both syndetic and thick subsets and show how these notions of largeness can be "composed" to produce different (and possibly new) notions of largeness. (The generalization of syndetic subset that we state is due to Shuungula, Zelenyuk, and Zelenyuk, but this notion has appeared, more or less, implicitly if unnamed in the literature connected to the algebraic structure of $\beta S$.) Some of these composite notions are well known, such as piecewise syndetic sets. Other composite notions are trivial, reducing to the collection $\{S\}$. While a few others appear to be new.

We'll sketch a suggestive visualization of these notions, outline the proof of some characterizations of these generalizations, under certain relatively mild conditions, formulate a classification problem, and outline some connections to Ramsey Theory.

### Using Data in STEM Course Redesign

A STEM course redesign project has been implemented to improve retention of students in STEM majors (particularly underrepresented students). The Center for Life Sciences Education focused on Biology 1113 (ﬁrst semester Intro Bio), with four project components: a summer institute for faculty to learn about student-centered course design; a shared database of active learning resources; peer-led team learning; and embedded undergraduate research experiences. The Department of Mathematics is conducting a redesign of ﬁrst-year calculus. This project consists of multiple interventions (sections employing active learning, ﬂipped classroom pilots, open access textbooks) viewed through many lenses (including affective surveys and conceptual pre- and post-tests) and development of a framework for the cohesive interaction of the involved faculty and staff. Data that led to redesign of speciﬁc STEM courses, the status of these course redesign projects, and recent assessment efforts regarding the success of students in the redesigned courses will be discussed.

# Recent Publications

### Revisiting the nilpotent polynomial Hales--Jewett theorem

Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of $\beta\mathbf{G}$) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

### New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups.

We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.

### A new and simpler noncommutative central sets theorem

Using dynamics, Furstenberg defined the concept of a central subset of positive integers and proved several powerful combinatorial properties of central sets. Later using the algebraic structure of the Stone–Čech compactification, Bergelson and Hindman, with the assistance of B. Weiss, generalized the notion of a central set to any semigroup and extended the most important combinatorial property of central sets to the central sets theorem. Currently the most powerful formulation of the central sets theorem is due to De, Hindman, and Strauss in [3, Corollary 3.10]. However their formulation of the central sets theorem for noncommutative semigroups is, compared to their formulation for commutative semigroups, complicated. In this paper I prove a simpler (but still equally strong) version of the noncommutative central sets theorem in Corollary 3.3.

### Images of $$C$$ sets and related large sets under nonhomogeneous spectra

Let $\alpha > 0$ and $0 < \gamma < 1$. Define $g_{\alpha, \gamma} \colon \mathbb{N} \to \mathbb{N}$ by $g_{\alpha, \gamma}(n) = \lfloor \alpha n + \gamma \rfloor$. The set $\{ g_{\alpha, \gamma}(n) {\,:\,} n \in \mathbb{N}\}$ is called the nonhomogeneous spectrum of $\alpha$ and $\gamma$. By extension, we refer to the maps $g_{\alpha, \gamma}$ as spectra. Bergelson, Hindman, and Kra showed that if $A$ is an $IP$-set, a central set, an $IP^*$-set, or a central$^*$-set, then $g_{\alpha, \gamma}[A]$ is the corresponding object. We extend this result to include several other notions of largeness: $C$-sets, $J$-sets, strongly central sets, and piecewise syndetic sets. Of these, $C$-sets are particularly interesting, because they are the sets which satisfy the conclusion of the central sets theorem (and so have many of the strong combinatorial properties of central sets) but have a much simpler elementary description than do central sets.

### A dynamical characterization of $C$ sets

Furstenberg, using tools from topological dynamics, defined the notion of a central subset of positive integers, and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-Čech compactification, this combinatorial theorem has been generalized and extended to the Central Sets Theorem. The algebraic techniques also discovered many sets, which are not central, that satisfy the conclusion of the Central Sets Theorem. We call such sets C sets. Since C sets are defined combinatorially, it is natural to ask if this notion admits a dynamical characterization similar to Furstenberg’s original definition of a central set? In this paper we give a positive answer to this question by proving a dynamical characterization of C sets.