The Intuition Game
© December 2024. Tanja Wagensohn UR. All rights reserved.
They say there's nothing quite like sharing—especially if you’re sharing something with minds like Steve Jobs or Albert Einstein. Both have credited intuition as key to their success. The same could be said for Richard Höfer, Professor of Analysis and Numerics of Partial Differential Equations at the University of Regensburg. But can solving equations really begin with intuition?
Richard Höfer smiles at the question. He’s one of those who think carefully before speaking—composed, thoughtful, and gentle.
So, what does intuition mean in the context of mathematics? “I have a feeling about what might be a good approach to tackle a problem before I even fully understand why,” says the 32-year-old tenure-track professor. “At some point, you need to have an idea of what might work.” For Richard, these ideas often stem from intuition. Does he rely on it for guidance?
“Yes, quite often. But in the end, it's the proof that matters in mathematics. Still, my gut feeling is often where I start. If I don’t have intuition, it’s like groping around in the dark.” He pauses thoughtfully. “I need that feeling to gauge whether something’s close enough or if it could work.”
“We deal a lot with integrals,” he explains, “and we have to estimate them, to show they don’t get too large. An intuitive understanding of what could go wrong and how to address it really helps. I didn’t have that kind of intuition in algebra,” he adds with a grin. “But I think it depends on your character.”
A Growing Passion
Richard’s love for mathematics started early. Although he favored Latin and Greek in school, his humanistic education remains something he cherishes. However, as time passed, his passion for mathematics grew stronger. "When I began my studies, I wasn’t entirely sure what to expect. Academic math is very different from what you learn in high school. I had only a rough idea.” The turning point came when he participated in math competitions. Why? “It’s more about puzzling than calculating. It’s less about knowing why something works and more about applying it.”
As his university studies progressed, especially with calculus and linear algebra, Richard knew his heart belonged to calculus. He eventually specialized in partial differential equations.
Born in New York—“by chance,” he clarifies, as his father was a postdoctoral researcher in physics at the time—Richard returned to Germany with his family when he was just a few weeks old. They settled in Munich, later moving to Marburg. He studied in Bonn before spending three years as a postdoc in Bordeaux and Paris. “Learning French was hard work,” he laughs. “Watching French series with subtitles helped a lot.”
During his time in France, Richard fell in love with the Pyrenees, often hiking with a backpack and tent to clear his mind of numbers and equations. As for his path to becoming a professor? “It just sort of happened,” he says with a modest shrug. Many of his classmates now work in software development, management consulting, banking, or insurance. Some have become teachers. Richard, though, prefers teaching at the university level. “At university, you’re working with people who are intrinsically motivated.” He enjoys standing at the blackboard, watching his enthusiasm catch fire among students.
Fluid Mechanics and Particle Systems
Richard Höfer’s research is particularly relevant to physics. He works on fluid mechanics, focusing on multi-particle systems in liquids or gases, also known as suspensions. These involve vast numbers of particles suspended in a medium, like liquid in gas. One could model each individual particle with an equation, but when dealing with thousands, millions, or more, a macroscopic approach becomes necessary.
Rather than modeling individual particles, Richard models clouds of particles. “I use a microscopic particle model from physics, then apply mathematics to rigorously understand the macroscopic behavior of the system.”
His goal is to gain fundamental insights into interactions that are difficult to explore through experiments or numerical simulations. Experiments are often inconclusive because you can’t precisely control all parameters in the setup of such a large system. Numerically, the sheer number of particles strains even the most advanced computers. Richard’s research, in the long run, could help in fields such as medicine. For instance, drugs or fluids injected into the body containing magnetic nanoparticles could potentially be guided by an external magnetic field. He aims to improve the understanding of what factors influence these interactions. Currently, models rely largely on trial and error.
Beyond Mathematics
Richard enjoys living in Bavaria, appreciating the proximity to the Alps, the short distance from the city to the forests, and the ability to hike at a moment’s notice. “Big cities like Paris aren’t for me in the long run,” he admits. In Regensburg, he shares a flat with two housemates and two cats, who occasionally listen to him play the piano, with Chopin being a favorite. Music provides one balance to his scientific work; sports offer another. Richard runs half marathons and plays badminton competitively at both club and tournament levels.
Does he read? Absolutely. His favorite authors include John Grisham and Ken Follett. Many evenings spent reading or playing music are accompanied by a slice of quiche. In that sense, vive la France!
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