On a Sunday morning in September 2023, UC Davis mathematician Roger Casals Gutiérrez was entranced by something he saw in his kitchen.

As sunlight filtered through the kitchen window, it cast its rays in a beautiful pattern on the wall. Comprised of lines, curves and points of varying illumination, the projected pattern appeared both circular and triangular, a hodgepodge of intersecting, nebulous shapes with various spots of brightness.

“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 at UC Davis. “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. Serendipitously, caustics, which are examples of singularities, are 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.”

View the world through Casals Gutiérrez’s eyes and you’ll realize that singularities are everywhere. They’re in rays of light, in ocean waves, in jets breaking the sound barrier and in the orbits of celestial objects. 

What is a singularity?

You may not know it, but you encounter singularities regularly in daily life. Take a table, for instance. The sharp corners of it are singularities. Or say you’re trying to parallel park and, for example’s sake, your car is leaking oil. Once you’ve successfully parked, you exit the vehicle and look at the oil trail tracing your parallel parking maneuvers. Each point in the trail where the car moved from forwards to backwards marks a cusp in that trail, a singularity.

“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. 

“There are many theorems in math that say that even when you’re studying things that are inherently smooth, like a smooth surface, or the contour of a ball, there’s going to be some point in the smooth evolution where the smooth object becomes singular,” Casals Gutiérrez said.

Think of a curved shape, like an ocean wave or airflow around a fighter jet. When that wave crashes or that jet breaks the sound barrier, 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.”

Despite their usefulness, singularity shapes can’t be modeled using run-of-the-mill functions from freshman calculus, according to Casals Gutiérrez. This led to the development of singularity theory, an area of physics and mathematics that focuses on the sudden shift-like characteristics of these shapes.

The study of singularities requires new exciting tools beyond the calculus sequence.

“Luckily, we do teach some of these methods in our more advanced courses at UC Davis.” Casals Gutiérrez said.

Making mathematics tangible

In his research, Casals Gutiérrez aims to geometrize the equations underlying singularities. The idea is to think of the set of solutions to equations as a visual space.

“The symbols and complicated structure of equations can be daunting, but one’s eyes and visual imagination can hurdle above that, using geometry to capture the true essence behind the equations,” Casals Gutiérrez said.

Once the equations are geometrized, the questions begin to concern the shape of the space. Does it have holes? Is it spherical? The specifics of the shape inform the qualitative behavior of the system.

“A lot of what my math does, and a lot of the things that I’ve put forth as a way to solve problems in algebra, in geometry and in other places, has been about making it something you can visually see, play with, simplify and manipulate until you get to the answer in a geometric way,” Casals Gutiérrez said.

Casals Gutiérrez’s hope is to develop tools that allow researchers to visualize complex systems of equations or polynomials in a 3D space.

“When you do that, you can start to manipulate things visually and get a sense of the shapes that appear. Then you start to be able to say things about the space of solutions without ever solving the equation,” he said. “You never need to compute anything. You never need to multiply. You’re solving with your hands.”

Casals Gutiérrez envisions a future class where he helps students visualize and understand calculus in a geometric way, maybe even using virtual reality headsets. Visualizing equations in this way brings mathematics to life, giving it shape and movement. The math becomes tractable, even touchable.

“It makes math more visual and to me, more personal,” he said.

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