Life and Math Podcast: Recent Episodes
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.
Dr. Shawn Ryan has a competitive, type A personality. In high school he enjoyed his science classes and did well in general. As is typical for good science students, he thought he would be an engineer, but found after his first semester in college that he enjoyed math more. Making the switch to math he was able to complete undergrad and get a master’s in 4 years at the University of Akron. While teaching students only his own age during his master’s he recognized he loved teaching along with research leading him to pursue the academic path.
Having a serious girlfriend applying to medical school he had went through a serious discernment of how to balance his professional aspirations with personal relationships. He ended up at Penn State where he found himself behind other students coming from bigger name universities. It was in that first semester that his future adviser told him point blank, “You’re starting with worse initial conditions, but it’s not about where you start, it’s about where you end. Better first derivatives can beat better initial conditions.” He put his head down and worked, succeeding in completing the PhD faster than his classmates.
While in graduate school his girlfriend became his wife, and he again had the challenge of what to do when he graduated. His hard work and some good fortune helped him first in landing a post-doc at Kent State given challenging location constraints and, later, a tenure track positions at Cleveland State, where he is currently a professor of mathematics.
Good lessons abound in this interview including the usefulness of “good” competition, the importance of being authentic, and finding balance across personal and professional life. Dr. Ryan is an example that sometimes things do work out when a person puts in the work to be in position to succeed. Enjoy!
In the math portion of the interview with Dr. Scoville, chair of the math department at Ursinus College in Pennsylvania, we hear about discrete topology. Dr. Scoville dove into discrete topology after his PhD as a way to more easily incorporate undergraduates into topology research. He talks about some of the structure involved in discrete topology including simplicial complexes. Along the way Dr. Scoville describes what a topology is as well as discussing the basics of homotopy. From there he discusses the emerging field of topological data analysis and his own specialty, discrete Morse theory. He gives an excellent description of the basic idea behind Morse theory and walks through a specific example in the discrete setting.
Dr. Scoville wanted to be a WWE-style professional wrestler ever since he was a child. His desire was such that he saw things like learning to read as a first grader or doing well in high school as impediments to his true ambition. Luckily he did not close off all option, but as he began his wrestling career he continued in community college. Over a number of years and through several important role models he discovered he could be good in school if he applied himself and unexpectedly found a love of mathematics.
Eventually he realized the seedy side of wrestling was not where he wanted to be, and he threw his whole self into math. In time he ended up at Dartmouth for a math PhD while simultaneously beginning to raise a family.
A delightful interview with a far from typical path into math. Dr. Scoville's story is a witness to the possibility of changing directions in life no matter how things have gone previously. Enjoy!
This is the "Math" part of my interview with Dr. Wanda Strychalski, an applied math professor at Case Western Reserve University specializing in computational cell biology and simulating partial differential equations. We begin discussing computational cell biology and the types of models Dr. Strychalski likes to implement. Dr. Strychalski shares different modeling methods for PDEs: the finite difference method (which she uses often), spectral methods, and finite element methods. We discuss the simple idea behind such methods being Taylor's Theorem or an analogous type of representation using different basis functions. Dr. Strychalski talks about building her own simulations, as most things have to be done custom since blackbox software is not versatile enough for her applications. Side topics include the math genealogy project and Dr. Strychalski's mathematical ancestors (Lax, Courant, and on back to Hilbert himself). There's plenty more in this entertaining and enjoyable discussion.
This is the "Life" part of my interview with Dr. Wanda Strychalski, an applied math professor at Case Western Reserve University specializing in computational cell biology and simulating partial differential equations. Dr. Strychalski encountered programming early in her high school career. Going into college she was concerned with having clear career path where she could support herself, and she focused on programming and computer science. As she progressed she found herself enjoying mathematics more and the non-simulation side of programming less. She made the jump to applied math focusing on PDE modeling. She decided to pursue a PhD, but was not sure if she would finish. She found a good culture at UNC Chapel Hill and pushed through the program finding a good niche in computational biology. Eventually she went out to California ( UC Davis) for a post-doc before landing a job as a professor at CWRU.
Dr. Strychalski's perspective of trying things out, seeing how they go, and not being overly anxious going into new situations was enjoyable and refreshing. Enjoy the interview!
This is the "Math" part of my interview with Dr. Allison Henrich, a math professor at Seattle University specializing in algebraic topology and knot theory. Knot theory is the study of closed 1-dimensional knots in 3D space. The interview starts describing origins of knot theory going back to Lord Kelvin's failed model for atoms. From there we discuss the notion of knot equivalence, isotopy, and a major theorem for knot invariants, Reidemeister's Theorem, which reduces all knot preserving transformations to 3 basic moves. The discussion moves to a generalization of knots known as virtual knots and Dr. Henrich's thesis work inventing 3 virtual knot invariants. At the end Dr. Henrich talks in depth about her recent work on knot games with college and high school students. Enjoy the interview!
This is the "Life" part of my interview with Dr. Allison Henrich, a math professor at Seattle University specializing in algebraic topology and knot theory. In high school Dr. Henrich enjoyed most subjects - theater, in particular - and had ambitions of going to the Ivy League. Her plans were derailed when she did not get into any college! She shares how she dealt with this trial and forged ahead. She discovered a love of philosophy and from there was eventually drawn into mathematics (by way of logic). Eventually she went to grad school for math at Dartmouth and became a professor at Oberlin and then Seattle University.
There are plenty of sidebars throughout the discussion highlighting lessons learned and giving unsolicited life advice to all the listeners. It was a delight spending time with Dr. Henrich. I hope listeners enjoy listening to what was an excellent discussion of Dr. Henrich's path to where she is now.
The "Math" part of our interview with Dr. Linowitz, a number theorist and differential geometer at Oberlin College. Dr. Linowitz grew up (mathematically, that is) in algebraic number theory. These days he works in inverse spectral geometry. We talk about how this field developed from a single, famous theoretical math problem, the "can you hear the shape of a drum problem". We learn that the "hearing" part of the problem refers to the spectrum (i.e. the eigenvalues) of a particular differential operator, the Laplace operator.
Because the Laplace operator exists on many Riemannian manifolds, this famous problem can be generalized to that setting. Dr. Linowitz specializes in special classes of manifolds which are negatively curved, hyperbolic manifolds.
We hear how algebraic number theory enters the picture, as many of these hyperbolic manifolds can be created using techniques from number theory. Dr. Linowitz shares an excellent example of how expertise in one field (here it's algebraic number theory) can be used to prove results in another (inverse spectral geometry).
Dr. Linowitz's gave a great interview and has a gift for explaining things lucidly. I highly recommend hearing what he has to say.
The "Life" part of our interview with Dr. Linowitz, a number theorist and differential geometer at Oberlin College. Dr. Linowitz has an atypical and interesting path for a professional mathematician. Among his twists and turns he shares how he left college shortly after enrolling and ended up joining the army, how he narrowly missed ending up in the Iraq invasion to go back to school, and how in school he dove into math because an advisor told him he shouldn't take so many math courses! Eventually he ended up graduating from Ivy League institutions for undergraduate and grad school. He describes where he was and what he was thinking at those different junctures.
Dr. Linowitz's gave a great interview and is a good example that sometimes going off the beaten path can work out quite well in the end. I hope you enjoy!
This is the second half of an interview with Nick Gurski, a mathematician specializing in algebra, and specifically in category theory. We continue talking about n-categories and the homotopy hypothesis. We learn some of Dr. Gurski's path into math, including how he realized there was actually something fascinating to study in math while taking multivariable calculus as an undergraduate. We hear some of this unsolicited life advice on learning how to engage in critical self-reflection.
A fascinating introduction to a part of math many students do not know exists. See the accompanying "What are categories, functors, and natural transformations" - part of our "What are...?" video series at lifeandmath.com - for those who feel over their head listening to this.
This is the first half of an interview with Nick Gurski, a mathematician specializing in algebra, and specifically in category theory. We dive quickly into the thick of what category theory is, beginning with its origins in algebraic topology. We run through how certain problems from other fields can be solved using category theory, using algebraic geometry as an example. We close this half talking about the homotopy hypothesis, one of Dr. Gurski's problems of interest, and higher category theory.
This is the second half of an interview with Mark Meckes, a mathematician specializing in analysis and probability. We talk about a couple of his research projects, and then delve into some of his backstory. Perspectives on things like whether to go to graduate school and a discussion of the "passion myth" come up.
This is the first half of an interview with Mark Meckes, a mathematician specializing in analysis and probability. We talk about random matrix theory, his research, and his path to becoming a mathematician. Second part coming shortly.