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Research Highlight: Math Isn't Just About Equations

Department of Math -- Wed, 10/26/2016 - 3:32pm

Yes, equations and math research projects go hand in hand—but for some PhD in math research opportunities, it’s all about using the equations to find meaning and applying them in a way that makes the world better.

Meet students and faculty who are using their PhD in math research opportunities to both improve the world around them and advance their fields of study.

Tracy Robin, Functional Analysis

Meet Tracy Robin, a math PhD student who plans to graduate in December 2016. His math research projects involve studying functional analysis and C*-algebras.

A C*-algebra is a certain type of space in mathematics that has both an algebraic and a topological structure. It is a complete normed algebra over the complex numbers, and the norm has very nice properties. Some basic examples of C*-algebras are: the complex numbers, the  matrices over the complex numbers, the set of bounded linear operators over an infinite dimensional Hilbert space, and the set of continuous complex valued functions over a compact interval.

“We basically study limits of sequences and series,” Tracy says. “So if you have an infinite sequence of numbers, we want to see what that sequence is converging to as the limit goes to infinity. As you go further and further down the line of the sequence, you want to know what that sequence is converging to — or if it’s diverging. As you get more abstract, it gets away from real numbers and you start looking at sequences of anything, like sequences and functions and operators.

“C*-algebras have a lot of useful properties,” he continues. “We can study the algebraic properties or the topological properties of a C*-algebra. Most of the time these properties are linked and we end up studying both. Because of the rich structure, C*-algebras are a very popular research area. C*-algebras are also used in many areas of applied math.”

Tracy chose to study functional analysis because he became more interested in analysis than the other fields of higher mathematics. Tracy likes the ideas of convergence of sequences and series, continuity of functions, and normed vector spaces.

“In functional analysis, we want to show that things are close together,” he explains. “It’s like whenever you’re looking at a set of real numbers, we think about all the numbers—we don’t think about just 1, 2, 3, 4, 5—we think about everything between there. We think about everything between the numbers, like between 0 and 1.”

In his research, he generalizes complex numbers. Each complex number has an absolute value, the complex numbers form a complete space, and every complex number has a complex conjugate. Then, they take the properties of the complex numbers and generalize them; then they move the complex numbers into infinite dimensional spaces to see which of their properties can be generalized.

That involves taking N by N matrices where N is any natural number, and then placing sets of matrices over the complex numbers to create a finite dimensional C* algebra.

"That’s your basic example of a non-communitive C* algebra where operators don’t commute by multiplication,” Tracy says. “Like if you take two matrices, AxB is not always equal to BxA. I study C* algebras in general. The properties don’t always go over from finite dimensional to infinite dimensional. The properties and terms don’t always follow along.”

Temi Gaudet, Delay Differential Equations

Temi Gaudet, a math doctoral student, is studying delay differential equations to estimate the distribution of a delay in a model.

Temi has a passion for math biology, and her dissertation is about estimating the distribution of delays, including the situations in which you can and cannot reliably estimate the distribution.

“Delays occur in biological systems when a natural time lag occurs,” Temi says. “Basically, any biological process that has a time lag as inherent property, you want to add a delay to that model to get a more accurate prediction, because in real life, it doesn’t happen instantaneously—there’s a delay.”

Right now, Temi is studying HIV infection rates and the delay in distribution in a population model.

Temi’s research also has the potential to be applied in physics, computer science, and engineering.

Read more about Temi's research >

Tingting Tang, Whale Population Modeling

Tingting, also a math PhD student, is studying the potential effects of the 2010 BP oil spill on whale populations and their habitats in the Gulf of Mexico.

Tingting is part of interdisciplinary research term that is collecting acoustic data from three areas in the Gulf of Mexico; she is helping to calculate the population level of the whales by using a model they developed to estimate the population density from the acoustic signals.

From there, she analyzes the results to understand the changes and impact on whale population models. She’s also developing mathematical models to predict the whale population dynamics in the future and to study how the population dynamic changes with regard to survivorship and birth rates.

Tingting hopes her research can help others understand what impact a large oil spill has on the whale population and to find the best strategic to help the whale population recover.

Read more about Tingting's research >

Dr. Azmy Ackleh, Population Dynamics

Dr. Azmy Ackleh, dean of the Ray P. Authement College of Sciences, is working to study the impact of the chytrid disease on the urban population of green tree frogs. To do that, they’re developing a mathematical model to determine the best time of year to inoculate the amphibians with a specific bacterium to protect them against the chytrid disease.

From 2003-2011, Dr. Ackleh co-led a study on the urban population dynamics of green tree frogs, which was, at the time, the longest study of its type based on urban capture-mark-recapture experiments on amphibians.

After collecting the data on green tree frogs’ capture-mark-recapture populations, Dr. Ackleh and his team developed a statistical model to estimate the population levels during these years and and to predict the population levels for subsequent years. What began as a research project for undergraduates grew into a research project that produced thirty trained undergraduates, six doctoral dissertations, and seventeen publications.

Now, they’re also working to develop a mathematical model to predict amphibian dynamics—but it also needs to be flexible enough so it can be adjusted to study the dynamics of other amphibian species across the world.

Read more about Dr. Ackleh's research >