On a Sunday morning in September 2023, UC Davis mathematician Roger Casals Gutiérrez was entranced by a beautiful pattern of intersecting lines, curves and points thrown on the kitchen wall by sunlight filtered through the window.

“The moment I saw it, part of me felt ‘This is a beautiful singularity,’” recalled Casals Gutiérrez, a professor in the Department of Mathematics in the College of Letters and Science. “But then the other part of my brain was imagining the smooth surface, which actually lives in five dimensions, that projected onto that singular pattern on the wall.”

What Casals Gutiérrez witnessed that morning is called a caustic, a concept from geometric optics defined as a set of points where light rays bundle together in varying intensities. Caustics are examples of singularities, a part of Casals Gutiérrez’s research interests in the field of contact geometry.

“What I really enjoy about caustics is their dynamical nature,” Casals Gutiérrez said. “If you move the glass or the sun moves during the day, you see them evolve. They kind of come to life beyond being a static thing.”

What is a singularity?

You may not know it, but you encounter singularities regularly in daily life.

“They’re shapes that are naturally there and have sharp features to them, like a spike or a sharp razor,” Casals Gutiérrez said.

These sharp shapes are incredibly useful in engineering. Think of the design of a Formula 1 car or the hull of a large ship. The singular shapes of these things make them more aerodynamic, improving their functionality. But singularities aren't just reserved for shapes with angles. Smooth surfaces can develop a singularity. For example, when a smooth ocean wave breaks, a singularity occurs.

“Everything seems very smooth, but at some point, a transition happens,” Casals Gutiérrez said. “The theory of singularities helps us understand these transitions.”

Read more about Casals Gutiérrez’ work here.