Georgia Institute of Technology College of Sciences

How do you picture chaos? To Alex Blumenthal, it’s a raging river, the wake behind a boat, and the infinite swirls coffee creamer makes as it’s mixed into a mug of joe. 

The chaos in these examples is in the seemingly unpredictable and unrepeatable way the fluids move– imagine predicting the exact motion of particles in a patch of whitewater, or recreating the exact way frothy water pours from a faucet into a full sink.

Now, Blumenthal, an assistant professor in the School of Mathematics, has been awarded an NSF CAREER grant to work towards just that.

The National Science Foundation Faculty Early Career Development Award is a five-year grant designed to help promising researchers establish a foundation for a lifetime of leadership in their field. Known as CAREER awards, the grants are NSF’s most prestigious funding for untenured assistant professors.

The award, for “Chaotic dynamics of systems with noise,” will help Blumenthal continue tackling some of the most difficult questions in his field– those of chaotic fluid dynamics. Because Blumenthal’s work with fluid dynamics intersects with chaos and disorder, the impacts of his work range from weather prediction to how we model economics.

The butterfly effect

Mathematicians have been interested in predicting chaotic, dynamical systems since Leonardo Da Vinci first sketched frothy, unpredictable jets of water hitting a canal, but solving this type of problem is notoriously difficult.

One reason for the challenge? The butterfly effect – where a small deviation (like a butterfly flapping its wings) can have compounding impacts on a system (like that tiny gust of wind gathering into a tornado). Because even a microscopic change in conditions can compound as the system changes, it is impossible to exactly recreate an experiment, and extremely difficult to mathematically model.

“Creating rigorous mathematical proofs is difficult in this situation, because the chaos and non-chaos coexist in these systems – and the initial conditions heavily impact the results of the mathematics,” Blumenthal explains. Imagine stirring that drop of creamer into a cup of coffee again– could you recreate exactly where the first drop is added, down to a molecular level?

It wasn’t until 2020, when Blumenthal, alongside a team of researchers, proved that it is possible to predict those folds and striations, called “swiss rolls,” that you see as the creamer is stirred into the coffee, or as two colors of paint are stirred together, providing the most rigorous mathematical proofs on turbulence to date, and proving a decades-old theory called Batchelor’s Law. Additionally, since the mathematical proof can be used to predict how fluids might mix, it can help scientists predict salinity profiles in the oceans, or atmospheric conditions.

Another key application of the research? The development of a new problem-solving framework, which Blumenthal plans to investigate with the CAREER grant as a key for unlocking new research into broader turbulence and chaos problems. 

Bringing research-level topics to students

The grant also includes funding to bring students to the forefront of the field. “One cool thing about these systems is that they lend themselves to a lot of computational projects that are accessible to undergraduates,” Blumenthal says. “I’ll be designing a short curriculum on these random dynamical systems that’s accessible to undergraduates– bringing them to research-level topics in this field in a short amount of time.” The undergraduate topics class will serve as an introduction to the kind of probabilistic perspective one takes while tackling chaotic theory, a class he plans to pilot in the Spring of 2023.

He expects the intersection with data science will be particularly interesting to undergraduates, explaining that “the principles of chaotic dynamics underlie a lot of the assumptions in data science. Data science is implicitly leveraging these ideas, so this will help students explicitly understand those implicit ideas. It’s a theoretical primer students could leverage in the data science field.”

The field is ripe for research, and Blumenthal is eager to include both graduate students and undergraduates, allocating funds for research experiences for undergraduates, alongside a graduate student workshop, where students from across the world could be invited to Georgia Tech for a weeks-long program, learning from Blumenthal and other experts about the chaotic dynamics of random systems.

“There’s a whole lot of new stuff to do, and these things lend themselves to numerical exploration – pencil and paper proofs, and computer-assisted proofs,” Blumenthal says. “There’s a growing community of people studying random dynamics, and a growing community of people doing computer proofs– it’s a great place for undergrads to have meaningful research experiences.”