Caltech undergradaute student Yeorgia Kafkoulis

A Mathematical Universe

Even our most reliable ideas about how the universe works break down in certain domains. They can’t account for the weirdness of quantum mechanics or the recursive chaos of fractals. Hungry for answers, many researchers—including one Caltech undergraduate and her faculty mentor—aim to come up with a better explanation.

Contributing to a grand unified theory ought to be a daunting task. But true to the Caltech spirit, mathematics scholar Yeorgia Kafkoulis thrills at the challenge.

“The fact that there are so many open questions means that there’s more to explore, more to learn about, and more to question,” says Kafkoulis, a member of the class of 2019.

Each summer, she joins up with the research team led by Caltech mathematics professor Matilde Marcolli. Kafkoulis’s task is part of an ambitious project: exploring the Swiss-cheese model of cosmology, a recalculation of the fundamental laws of nature.

Just as a Gantvoort Scholarship has helped underwrite her classroom education, her opportunities as a burgeoning investigator come courtesy of donor funding, in the form of the Summer Undergraduate Research Fellowships (SURF) program.

“I came to Caltech to learn, but I also came to do research. SURF has opened my eyes to this world of mathematical physics in particular, and also to research in general. It’s a little sneak peek into my future.”
- Yeorgia Kafkoulis

Big Cheese

As Marcolli, Kafkoulis, and colleagues seek to reconcile Einstein’s general relativity with more exotic phenomena, they double-down by questioning Newton’s assumptions.

His cosmological principle depended upon two things. One, that the rules of physics work the same way anywhere in the universe. Two, that on a large scale, the distribution of matter is about the same everywhere.

The Swiss-cheese model presents a concept of gravity that removes one of those assumptions. What if matter is not evenly distributed in the universe?

In this model, you might expect to see a cosmos made of denser stretches and pockets of emptiness—not unlike that holey Alpine cheese. You also might see the beginnings of explanations for the quantum strangeness and fractal chaos that defy the models of Einstein and Newton.

Marcolli’s team tries out new ideas in this framework and examines the effects of a modified gravity model as the universe expands over time. Elaborating on the notion of a block of cheese, this conception describes spacetime as shaped like a many-dimensioned set of bubbles.

Kafkoulis is looking for patterns in how those bubbles pack together. The arrangement seems to resemble swirling multifractals.

“Fractal-like behavior isn’t explained by the standard model,” Kafkoulis explains. “At times, the universe behaves like a fractal—in supernovae, in clusters of stars and galaxies, even in the composition of stars. To describe the universe accurately, you need to explain that fracticality.”

Awe, Excitement, and Pizza

Kafkoulis connected with her mentor early, in the first term of her freshman year. The setting was Math 20, a seminar that serves up lectures from different professors and pizza for lunch. Marcolli’s “pizza course” presentation made a big impression.

“I heard ‘Swiss-cheese model of cosmology,’ and my spider-sense started tingling,” Kafkoulis says. “As I started to get a sense for what Professor Marcolli was talking about, I was like, ‘This is awesome!’ And I mean that in the strict sense of the word. ‘This inspires awe.’”

By the next week, Kafkoulis was leaving Marcolli’s office with reading materials in hand and a newly forged SURF match that would enrich her Caltech career.

In the summers, Kafkoulis diligently proves theorems, reads countless papers, and meets with Marcolli each week to compare notes, both one-on-one and as part of her team. Her mentor’s enthusiasm for exploring Kafkoulis’s ideas has stoked her confidence.

“Professor Marcolli has helped me become a better mathematician and a better scientist,” Kafkoulis says. “She inspires me to jump forward in whatever I’m doing—just dive headfirst into the deep waters. She is, I tell people, what I want to be when I grow up.”

A Lifelong Fascination

Kafkoulis remembers her passion for understanding the universe first igniting when she was 5 years old. A public television series about string theory held her transfixed.

“I would run to my parents and explain what I had seen—even though I didn’t really understand it,” she laughs.

Academics themselves, her parents encouraged her love of science. And her father, a Caltech PhD, had one suggestion in particular for her future path, leading her far from their home in Miami.

“What he said made Caltech seem like this utopia, even when I was 5,” she says. “He described it as welcoming and intellectually stimulating. He kept saying: ‘Caltech is not the only place. But it might make a pretty good place for you.’”

Her eagerness to take on the biggest kinds of research questions suggests that the elder Kafkoulis might have been onto something.

Giving Priorities

Empower