Lessons in Retrospect: Two PhD Co-Advisors


Back in September 2020, I identified my advisors and a project for my doctoral study in the Mathematics Department at the University of Tennessee-Knoxville. I had a very carefully thought out plan for how to identify an advisor(s) that would be a good fit for me given my interests, needs, and career goals. This plan was part of my larger idea of how to become a visible entity in the department as a graduate student relatively new to the Ph.D. program. I told this story here almost a year ago.

Now, I’m reflecting…

Like many other people, the COVID-19 pandemic has challenged me to rethink my finances considerably. It’s served as a reminder that on a given day, we might not be able to spend a dollar the way we intended. Among other things that fell apart, a scheduled Spring Break trip to Austin had to be canceled in March 2020. You’ve all heard (or even experienced) stories like this, not to mention the loss of jobs, and the loss of loved ones. …

An ordinary differential equation (ODE) is an equation involving a scalar-valued function from the real line to the real line, and some of its derivatives. A partial differential equation (PDE) is a generalization of the ordinary differential equation to the event where the argument is in multiple dimensions. In this case we have partial derivatives in each direction. For example, if the input is an ordered pair, then the input belongs to R², so we may have partial derivatives in two different directions. …

Introduction: What is Calculus of Variations?

The calculus of variations refers to the science (or the art) of solving optimization problems containing integrals. In other words, out of a collection of functions, choose the one that makes the integral of those functions as small as possible. Such a function will be called a minimizer. Sometimes the integrand will depend on the function itself alone; sometimes dependence on derivatives of the function will be allowed. Here is an abstract example of what one of these problems looks like:

Minimization problem from Rindler’s textbook
Minimization problem from Rindler’s textbook
This is an excerpt from Filip Rindler’s “Calculus of Variations” text, Springer Publications. Arguably my favorite text on the subject matter.

If you aren’t familiar with what the “W” means, don’t worry about it…

Photo courtesy of MichaelGaida via Pixabay

I’ve just finished my fourth semester in the University of Tennessee’s Mathematics Ph.D. program! That means I’m nearly at the halfway point of the program, and I am getting ever closer to the oral exam. All that being said, this seemed like a good time to reflect on how I’ve developed an answer to the question, “what do I want to do after I graduate?” Moreover, I want to evaluate if I have been spending my time and energy on suitable activities to reach this goal.

At this point I’ve finished my preliminary exams and selected a research topic and…

Image courtesy of geralt via Pixabay

As of August 2020 I had passed my preliminary examinations at the University of Tennessee-Knoxville, in analysis and partial differential equations. In the mindset of many potential advisors, this was the green light to begin working on research with them. Of course, I had to first identify an advisor and a project. While I detail this process more in a different blog post, I want to jump to what happened from a logistical standpoint after I made these important decisions.

In the Fall 2020 semester, I had extra work to do each week as my project initiated, along with one…

Introduction and outline

In this article I’m going to describe some mechanisms for carrying out an election that are mathematical and deterministic in nature (i.e. noting happens randomly). These mechanisms are called voting methods, because they are essentially ways of running an election to determine a winner. The data used in these elections come in the form of preference tables. While there are many voting methods studied in research papers and utilized as examples in recreational mathematics classes, this article will one particular voting method as a case study:

Borda Count: each person voting in the election ranks all of…


A voting method is a mechanism for determining the winner in an election, with a series of steps explicitly listed. These methods generally have some means of scoring candidates. Each voting method is used on a preference table, which is a chart that indicates people’s relative preferences between two or more options.

Voting methods are often found as a topic in the curriculum of a recreational math or math modeling course at the high school or collegiate level. …


Consider a sequence of numbers with a distinct pattern, such as

1, 4, 7, 10, 13, 16, 19, …

For this thought exercise, the sequence needs to be chosen with some purpose, rather than at complete random. When this is achieved, the sequence can be described by two types of formulas. For this specific example, those formulas take the following two forms:

S_n = 3n + 1


S_n = S_{n-1} + 3 (with S_1 = 1)

Both formulas above help to describe the sequence we started with, but there is one significant difference between them. The first formula…

Introduction and Outline

It’s just the greatest number of votes wins the election, right? Not so fast.

While a simple counting of votes can be an effective method in some situations, the theory of designing voting methods is a lot more complicated (and intriguing) than that. When one asks, “which voting method is the best,” this is both a mathematical and a philosophical question, not to mention a question that lends itself to interesting pedagogical opportunities in a math classroom. In this article I’m going to break this question into many pieces, but let’s start with defining the phrase “voting…

Joshua Siktar

Math PhD Student University of Tennessee | Academic Sales Engineer | Writer, Educator, Researcher

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