Life and Math Podcast Life and Math

More than mathematics itself, Alicia Prieto enjoys learning new things. In approaching mathematics she searches first for concrete problems that interest her. Such an approach has its positives and negatives. For Dr. Prieto a main attraction is how it forces her to constantly learn new things. A potential downside is that it is slow. After all, if you start on a problem where you have to learn a bunch before you can make progress, you will never produce results as quickly as others. Using this approach Dr. Preito has worked on diverse problems ranging from modeling immune response in mathematical biology to recommender systems in data science for student course selection.

A major focus for Dr. Prieto has been agent-based modeling. This stochastic approach to modeling systems treats elements of interest as “agents” who have a some set distribution for how they move and/or interact with the surrounding systems. As an example, Dr. Prieto discusses her Ph.D. project in some detail. For this project she was modeling the immune response to a biomedical implant. The system was modeled as a grid (essentially a giant matrix) with several levels (so really several copies of the matrix stacked on each other). Each level represented some aspect of the system. For instance one level would represent the position of certain immune cells (say killer T-cells). The movement of each T-cell is stochastic, meaning at each time step there is a probability of the cell moving in each of the different directions. To make such a model work at each time step the random distribution is sampled for every cell and the cells move based on the sampling. A single run of the model means almost nothing. The point of a stochastic model is to run it many, many times (what counts as many depends on the details of the situation). The hope is that the samples run many times will reflect the range of possible outcomes for the actual biological system.

We discuss the idea of picking an appropriate model for the situation, contrasting the physics versus biology. In physics situations the degree of control and certainty over the situation often allow for deterministic models. However, in biological situations the phenomena are fundamentally uncertain and variable. We will never know exactly where all the cells are and the cells will all be unique and prone to moving randomly (though random does not mean without connection to external or internal signals). Stochastic models are often appropriate in such situations as they are more flexible and less rigid, meaning they can more readily be modified to accommodate changes in belief, something that more deterministic differential equation-type models often cannot accommodate.

We also talk about the problem of verification in doing any applied work and how Dr. Prieto was able to come full circle in verifying aspects of the model she built for her Ph.D. project. Enjoy!

As a child in Mexico, Dr. Alicia Prieto would not talk to anyone she did not already know. She also did not think she was good at math. Fearful she would fail math and never talk to anyone, her mother made her go to a math bridge program in the summer before the start of middle school. One day she and her boisterous friends annoyed the teacher so much he told them they could not leave until they solved a challenging math puzzle. To the teacher’s amazement, Dr. Prieto solved it quickly. He was so impressed he told her to join the math club. Despite her misgivings (math club did not seem like the place to make friends), she joined. Later that year she took the qualifying exam for Mexico’s math olympiad. In her own words she only stayed for the exam because there were chips and cookies after! Thankfully she did, as she became the youngest person ever to qualify for the national math olympiad training program in Mexico.

From high school she made the atypical move from home to Mexico’s elite math university CIMAT (Centro de Investigación en Matemáticas) 4 hours from home. After 3 years there she attended an REU at the University of Texas at Dallas, which she enjoyed enough that she just stayed to finish her undergraduate degree there. She shares the challenges of coming to the USA including some infuriating encounters with stupidity and prejudice in her first semester. She went from UT Dallas to UT Arlington where she earned a Ph.D. using agent-based modeling in biomedical applications.

Dr. Prieto shares these details and more (such as her regularly falling asleep on her porch at 11:30pm because her Mom would not let her come back from quinceañera parties before midnight!) in an humorous and playful reflection on her life path.

Among many lessons she highlights her struggles with an impostor syndrome where she felt like she did not belong, and the importance of learning that struggling with math (or anything in life) is normal, and not a sign of deficiency. Dr. Prieto reflects on having a bad relationship in college and a counterbalancing great friendship. She talks about Math Circles and the joy she found helping younger kids encounter the fun the interest of mathematics.

Currently a professor of mathematics at Youngstown State University, Dr. Prieto closes sharing some of the interesting surprises of coming to Youngstown State and embracing a region totally different from where she grew up and went to school.

As a mathematical biologist who specializes in modeling phenomena with differential equations, Dr. Ryan’s studies how complex biological systems organize themselves. This general topic covers things ranging from how colonies of bacteria interact in suspension to how groups of insects move in swarms to avoid predators.

Dr. Ryan highlights numerous “big picture” ideas in mathematical modeling, which he broadly splits into 3 parts: modeling, simulation, & analysis. Modeling to the act of writing down (differential) equations to capture the essential features of the physical system in question. The key here is “making the model as simple as possible, but no simpler”. Dr. Ryan considers this his favorite part, and his particular strong suite. A constant question is that of parameter estimate to ensure the terms in the model are realistic.

Once a model exists then simulation and analysis come into play. Using the tools of analysis one can work directly with the mathematical equations hoping to prove things like existence of a solution and solution uniqueness. Here model complexity matters, as a complex model may be analytically intractable, meaning it’s impossible to say much about the model using pure math.

Simulation goes the other direction from analysis. Rather than work with the differential equations, the equations are somehow discretized into a form digestible to computers, and the research can then simulate the system directly. Here there are challenges such as stability and computational efficiency. When a given model is discetized, it may be that a small change in the parameters results in a major changes in the output. The simulation is the unstable and may not be trustworthy. For computational efficiency, the actual details of how the model is programmed matter. Here Dr. Ryan highlights tricks he uses such as GPU programming that also reduces the communication cost between GPUs during a simulation.

Overall Dr. Ryan delivers a masterful overview of major aspects of mathematical modeling covering broad principles as well as specific examples from his own work.