Kellen Myers - Curriculum Vitae

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Curriculum Vitae, Kellen Myers, pdf format

Kellen Myers & Aaron Robertson, Two Color Off-Diagonal Rado-Type Numbers, Abstract:

We show that for any two linear homogeneous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z+ admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR(E0; E1) to be the smallest N such that [1,N] must admit such solutions. We determine a lower bound for RR(E0, E1) in certain cases when each Ei is of the form a1x1 + ... + anxn = z as well as find the exact value of RR(E0, E1) when each is of the form x1 + a2x2 + ... + anxn = z. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.

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Kellen Myers & Aaron Robertson, Two Color Off-Diagonal Rado-Type Numbers, Abstract:

There exists a minimum integer N such that any 2-coloring of {1, 2, ... , N} admits a monochromatic solution to x+y+kz = lw for k,l in Z+, where N depends on k and l. We determine N when l-k is in {0, 1, 2, 3, 4, 5}, and for all k, l for which (1/2)((l-k)2 -2)(l-k+1) ≤ k ≤ l - 4, as well as for arbitrary k when l = 2.

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Papers:

Kellen Myers, High Honors Thesis, abstract:

Ramsey theory is the study of the preservation of structure under finite partitions (called colorings). Ramsey theory on the integers studies particular structures (families of subsets of Z+) that are preserved under colorings of the integers (functions χ : Z+ → {1, 2,..., r} for some r in Z+). A brief survey of Ramsey theory (focusing on its application to the integers) is presented, as well as several new results.

One such result is based on a theorem of R. Rado, which states that there exists a minimum integer N such that any 2-coloring of {1, 2, ... N} admits a monochromatic solution to any linear homogenous equation E with at least 3 variables and mixed-sign coefficients. Such an N is investigated for x + y + kz = lw for k,l in Z+, where N depends on k and l. The exact such N is determined for l-k in {0, 1, 2, 3, 4, 5}, for all k,l (1/2) ((l-k)2 - 2)(l-k+1) ≤ k ≤ l - 4, as well as for arbitrary k when l = 2.

Another result is that that, given two linear homogenous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z+ admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. The 2-color off-diagonal Rado number RR(E0, E1) is defined to be the smallest N such that {1,2,...,N} must admit such solutions. A lower bound for RR(E0, E1) is determined in certain cases when each Ei is of the form a1x1 + ... + anxn = z, as is the exact value of RR(E0, E1) when each is of that form and a1=1.

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AMM Problem 11208, Summary:

The question is to compute the chromatic number of the n-th iteration of the Menger-sponge. The answer is exponential in n, and can be derived by finding a recursive formula for the genus, solving for the explicit formula, and plugging this into the Ringel-Youngs formula for the chromatic number.

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Other published works and unpublished papers:

Results on Standard and Off-Diagonal Rado Numbers, Abstract:

We will discuss an off-diagonal generalization of one of Rado’s results on the regularity of linear equations, including several relevant off-diagonal Rado numbers and bounds. Additionally, we will present new results for particular standard Rado numbers of certain linear equations. Joint work with Aaron Robertson, Colgate University.

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Off-Diagonal Rado Numbers, Abstract:

In Ramsey theory, Schur's theorem answers a fundamental question: within any coloring of the integers, there are monochromatic solutions to x+y=z (a property called regularity). Rado's theorem describes regularity conditions for any linear homogeneous equation (or system). However, Rado also showed that for 2-colorings that linear homogenous equations (and systems) are 2-regular in any nontrivial circumstance. In this talk, Rado's result will be presented and proof of an off-diagonal generalization will be given (off-diagonal indicates that a different equation governs each of the two colors). Further discussion will present the determination of some Rado numbers (off-diagonal and regular) in some specific and some general cases.

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Conferences:

Prime Time, summary:

A brief explanation of the AKS primality test.

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Research in Ramsey Theory, summary:

A presentation of my research work.

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High Honors Thesis Defense, summary:

A defense of the work found in my High Honors Thesis.

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Ramsey Theory on the Integers, Abstract:

In this talk, I will introduce Ramsey Theory - its formulation regarding both graphs and families of sets of positive integers. The most prominent results in Ramsey Theory on the integers will be stated and explained. I will define and present the ideas behind computing the Ramsey type numbers for these systems as well. Time permitting, a few "existence" proofs may be presented, as well as perhaps some computational results.

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Van der Waerden's theorem, Abstract:

In 1927, B.L. van der Waerden proved the theorem that bears his name: for given integers r and k, there exists an N so that any r-coloring of the integers 1 through N contains a monochromatic arithmetic progression of length k. Proof of this theorem will be presented, as well as an explanation of the sorts of bounds on N (w.r.t r and k) that are known. I will also discuss, if time permits, generalizations and related ideas, including the Hales-Jewett theorem and Szemerédi's theorem. If there is any extra time, I will also present an interesting new combinatorial proof of a famous conjecture of Fermat.

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Round Table Discussions:

The Round Table Discussion is an informal meeting of Rutgers Mathematics graduate students, as well as a few students from other departments, and other people affiliated with the discussion program. The goal is to informally discuss topics of interest. It is modeled on (and affiliated with) The Brooklyn School of Mathematics. No single person is speaker on any given day, but generally one person leads and guides the discussion.

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Ramey Theory on the Integers, Abstract:

Ramsey theory is the study of the structure within a set, and whether such structure is preserved under (finite) partitions (thought of as 'colorings'). A coloring with r colors is called an r-coloring. A structure preserved under colorings is called regular (or r-regular for only r-colorings). In any such problem, we can also ask what the minimal such n is (for fixed r and m), a question that is often bounded theoretically and/or computed via computer. Schur's theorem tells us that solutions to the equation x+y=z are regular. In this talk, we will consider the generalization of Schur's theorem to linear equations, done by Richard Rado, and other results (both 'existence' and bounding/computational) pertaining to linear equations - of which some solution sets are regular and some are only 2-regular.

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The Genus of Graphs and Groups: An Alliterative Survey, Abstract:

Kuratowski's theorem, a result presented in most introductory courses and texts on graph theory, characterizes precisely which graphs are planar (i.e. embeddable in the plane with no edges crossing). Extending this characterization to other surfaces, we can define the genus of a graph to be minimal genus for a surface into which the graph embeds. We can extend this definition to a group by examining the Cayley graph of th group (with respect to some set of generators), and taking the minimum over all generating sets. This talk will attempt to introduce these concepts in a broad, survey format, with examples, illustrations, and explanations for each definition, theorem, and observation. Major results will be presented to extend and motivate the elementary concepts in the talk, as well as to give some interesting level of results, but the focus of the talk will be to introduce the material at an elementary level.

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Project Euler, summary:

Project Euler is a website offering challenging computational problems of a discrete-mathematical flavor. Partcipants are encouraged to use novel, creative, and insightful methods to write programs that solve these problems.

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The Fourier Transform and Enumeration, Abstract:

This talk will provide a brief and gentle introduction into applications of the Fourier transform to enumerative problems. Some definitions and explanations will be provided, leading to comparisons between elementary enumeration techniques (that is, counting) and the techniques using the Fourier transform.

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Experimental Mathematics in Action, Abstract:

This talk will be highly audience-interactive! We will go through an overview of a few useful methods in experimental mathematics, followed by some hands-on demonstration. This will include, hopefully, audience-generated problems and ideas. See here for a link to the Mathematica content used for this talk.

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What I Did Last Summer: Pebbling at the REU, Abstract:

In combinatorics, REU problems are often very much like the best research questions -- easy to state and quite hands-on, but much deeper and more difficult to prove than they seem at first. I will introduce graph pebbling, work through some examples (audience involvement will be encouraged), and present some basic results. I will then discuss Graham's conjecture and partial results to that effect and describe some of the work the DIMACS REU students did this summer (in which I was tangentially involved).

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Developing Your Teaching Style, Abstract:

The TA Project offers a series of workshops designed to help TAs develop their teaching skills and improve their marketability. Students who attend at least four sessions of a particular series will be eligible to receive a certificate indicating their commitment to teaching. This session is designed to help teachers think about ways they can develop a style of teaching that will better engage their students.

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Other presentations, events, etc.:

Education & Service:

Tucker's Group of Genus 2:


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Tuckers Group of Genus 2, Assistantship, details:

I was fortunate enough to arrive at Colgate while Prof. Pisanski was visiting. In my first year, I was approached by Prof. Pisanski to work with him on his project, which was to create a piece of mathematical art. Specifically, we worked to help create a sculpture representation of Tucker's Group of Genus 2, named for Prof. Tucker (of Colgate). Work was directed by Prof. Pisanski and Prof. Dewitt Godfrey of the Art Department.

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Teacher's Assistant, details:

My responsibilities included grading, giving recitations, proctoring exams, and holding office hours. The goal of this position was to help in the construction of a special section of Calculus designed to meet the needs of students having trouble with the gap between high school algebra and college calculus.

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Research Assistant, details:

Taking part in a summer research program at Colgate University, I worked with Prof. Parks in her lab. Part of my time was spent attempting to work on the mathematical side of discrete signal processing. Later in the summer, my efforts shifted to fixing and updating (and as it turned out, essentially re-writing) the computer programs which operate the robotic mechanisms that control the spectroscopy aparatus.

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Employment & Teaching:

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