For the Love of Partial Differential Equations: A Conversation with Franca Hoffmann
It was during an undergraduate exchange year in Lyon, France, that Franca Hoffmann, an assistant professor of computing and mathematical sciences, was first introduced to what has become her greatest mathematical passion: analysis of partial differential equations.
Differential equations are devised to describe how quantities change, to understand evolving processes as opposed to static realities. Ordinary differential equations (ODEs) have a single independent variable, while partial differential equations (PDEs) have multiple independent variables, making them considerably more complex but also better at describing complex realities.
Born in Germany, Hoffmann attended high school in Canada for a year, did her undergraduate and graduate work in England, attended university in France for a year, and has been involved in mathematics education in Africa since 2011. That work began when she volunteered as a tutor for summer math camps. She later became the research chair in data science for the African Institute for Mathematical Sciences (AIMS), a pan-African educational organization headquartered in Rwanda that also has branches in South Africa, Cameroon, Senegal, and Ghana. Today, Hoffmann serves as an advisor for AIMS.
Hoffmann first arrived at Caltech in 2017 as a von Kármán postdoctoral instructor. After a two-year professorship at the University of Bonn, Hoffmann returned to Caltech in 2022.
Hoffmann recently spoke about her time at Caltech, PDEs, and other things mathematical.
What do you find compelling about partial differential equations?
Within the interdisciplinary research I do, PDEs are the language with which I tend to tackle problems. They are my home base. Most processes that happen in the world are governed by some sort of change and, if you want to understand these processes, you need to understand how rates of change are related with one another. Differentials in general, and PDEs in particular, are a mathematical language for change.
For example, researchers have tried to predict how dense a crowd of people can become before a panic may cause injuries and fatalities. It turns out that the same type of equations created for human crowd control can govern the density of cells in a cluster. Obviously, these are very different applications. But the difference in mathematical modeling may simply be in the nature of one function in one component of the equation. People in a crowd may panic as a result of things they see; they are triggered by visual signals that they communicate to one another. Cells may instead rely on a chemical signal to communicate information to other cells. But once you determine that there is a function that pertains to the communication mechanism, that function can be plugged into a larger model whose structure is very similar across crowds of people or clusters of cells. It's then possible to talk about behavior in a way that applies equally to both situations.
How do you decide which processes you want to model with mathematical language?
The most common way the chain of knowledge creation works is that you have domain experts, for instance, in one of the sciences—biology, physics, chemistry—who encounter a specific question they want to answer. They try to formulate this as a model. Then, people with more experience with modeling get involved and shape things so that the domain experts have confidence that the model is reasonably useful and accurate.
But then you have to analyze or solve the model. By solving, I mean either writing down the explicit solution or being able to approximate it in some precise way. In most cases, this is either out of reach or very expensive. By analyzing, I mean understanding the properties of solutions to your question without actually being able to write them down. This is where mathematicians can help.
Usually, when a model like this makes it to the desk of a mathematician, it's just an equation. And what mathematicians love to do is to extract all the context away from it. First, you can try to nondimensionalize the model: remove or represent in a more abstract way any context-dependent variables or constants. Once you have done all the analysis you can do, the abstracted relation between quantities of interest is packaged into a theorem.
To complete the chain of knowledge creation, one final translation is necessary. Your theorem is formulated in a mathematical language that may only be understandable to other mathematicians. So, demonstrating to domain experts how the theorem is explanatory and potentially useful for them is a crucial step, however, one that mathematicians are rarely incentivized to attempt.
What first inspired you to study mathematics?
My parents ask the same question. They're both at home in the humanities, especially in art and theatre, and they can't imagine where my desire to pursue mathematics came from.
The thing I love about mathematics is that you don't really have to learn that much by heart. Once you understand the principles, you can always reproduce results or rederive the formulas you need. Along the way, you can truly internalize the "why" of how things change and evolve together.
Take the example of optimal transport. This problem originally came up as a question about how to best allocate resources during the French Revolution and the Napoleonic Wars. You had certain sites where material was extracted, and then other sites where you wanted to build. How do you transport these materials most efficiently?
You can see that this problem could relate to many different types of situations, no matter what the resources are, what the scale is, or how the materials are being moved around.
Gaspard Monge, the French mathematician who first formulated this problem, really struggled to solve it and did not succeed. It remained an open question for about 150 years. Progress was made along the way, but the really decisive work was done by a Russian mathematician, Leonid Kantorovich, who came to this problem in the 1940s. Kantorovich took a completely different approach to optimal transport. Instead of saying, "OK, I have my mass here; where do I send that mass?" he conceptualized the problem as pairs of points and asked, "How much mass do I transfer from X to Y?" This simple change in point of view enabled the development of a beautiful theory that can explicitly characterize how to optimally transport mass around in most settings.
Since Kantorovich's work, other mathematicians have pushed our understanding of optimal transport forward, but it is really only in the past 10 to 15 years that optimal transport has left the realm of theory and found its way into a broad array of applications.
For example, optimal transport is now often used in image processing. Say you have two snapshots of an action, taken just a few seconds apart, and you want to generate a video that connects these two images. Thinking in optimal transport terms, you can treat the color content in the pixels as the mass you wish to transport from the beginning image to the ending image. It turns out that actually a lot of processes in life follow an optimal transport mode of change and, in the case of image processing, you get videos that are very realistic.
What have you learned by teaching mathematics in Africa?
I've learned so much! When I first started volunteering, I worked in summer math camps, which were not designed for the students who love mathematics and want to win the next Olympiad, but rather those students who believe they can't do mathematics and never want to open a math book again.
By the time I became the research chair in data science at AIMS, we were starting a pan-African doctoral training program. It was a fantastic opportunity for me at such an early career stage to have the chance to build a program like that from scratch. AIMS is not a university, so they don't have the usual professorial faculty. We had to design completely new supervision structures where now all AIMS PhD students are being co-mentored by professors at other African universities or internationally. And data science, as we conceived it in the context of this doctoral training program, is very broad. It introduced me to a whole range of topics and interesting applications for mathematics.
You spent three years as a postdoc at Caltech before returning as an assistant professor, so you must have known what you were getting into. What do you like about Caltech?
The main thing I love about Caltech is its interdisciplinarity. People here are really interested to learn about other people's work outside of their little bubble. There are a good number of people sprinkled across Caltech's different departments who could be considered mathematicians in one way or another.
In many places, doing interdisciplinary work is not valued as much because it may mean that you're too shallow in one or more fields. That can be a danger if you have too little expertise to give meaningful input, but at the same time, it is precisely the nature of interdisciplinary research to be able to work outside your comfort zone and connect with experts in other fields. Caltech really values the kind of work I actually want to do and attracts others who want to work in a similarly interdisciplinary way.
I've been exposed to several different educational systems in the United States, Canada, the United Kingdom, France, Germany, and several African countries. These different systems divide mathematics in different ways, into different subfields. For example, my PhD work in the UK was considered pure mathematics, while in France it was seen as more applied, nearly physics rather than math.
Math in the US has closer links to industry, and Caltech is also a special case in that applied mathematics sits together with computer science.
Where do you feel that you sit on the spectrum of pure and applied mathematics?
They are so interactive, it's hard to say. Mathematics is a very concise language to express our understanding of the world. The process by which you get to a new mathematical result is very creative. Sometimes, you come to the process because you have a specific aim in mind, a specific question that you want to answer. But sometimes, even if you begin with a precise question, along the way you may get diverted because you realize that maybe the answer already exists or that the question is not the right question. And then, you realize that the overarching framework of the questions you're asking deserves to be pursued in its own right. Having the freedom to just explore and see where you can tease out these beautiful structures may have benefits down the line in both pure and applied areas in ways that we cannot predict.
