Unlocking the Mysteries of the Universe Through Mathematics

Mathematics: The Language of the Universe

GPS navigation, cryptography, quantum computing — while some of humankind’s greatest advancements have been invented by pioneers from various cultures, they were founded upon one common grammar: mathematics.

“Mathematics is the language with which God wrote the universe,” said the famous Italian astronomer, physicist, and philosopher Galileo Galilei, who helped provide critical evidence for the heliocentric model of our solar system.

Although mostly conveyed through combinations of numbers, letters, and signs that may seem enigmatic to many, math equations hold within them countless stories — playbooks that generations of wonderers and inventors have crafted, refined, and shared in an attempt to make sense of a world full of unknown variables.

“I have faith in mathematics that, when there seems to be something special happening, when there’s some coincidence, that it’s not just a coincidence,” says mathematician Amanda Burcroff. “There’s actually some really deep, interesting, and involved reason for why that should be true.”

Burcroff’s research is focused on algebraic combinatorics, an area that provides discrete frameworks for understanding algebraic and geometric spaces that ubiquitously arise across science. This year, she joins MIT’s Department of Mathematics as a postdoc as part of the School of Science Dean’s Fellowship. Collaborating with Professor Alexander Postnikov, Burcroff is enhancing her techniques with the goal of applying them to other areas such as theoretical physics — a field that seeks to uncover the fundamental laws governing everything from subatomic particles to the cosmos itself.

“I have trust that if you keep following the path, eventually you’ll find the treasure — that is, whatever theorem or proof — that you’re looking for,” she emphasizes.

From Memorization to Creative Exploration

Like many children, Burcroff once perceived math as a subject requiring extensive memorization. Even though it came naturally to her, she found it less than exciting at first.

In high school, her exposure to calculus and geometry offered her a new perspective. Math transformed into a creative exploration of possibilities. “In most other fields, the rules are imposed on you by the world,” she explains. “But in math, you get full freedom to lay down those rules and then figure out what the implications of those rules are by using logical consequence.”

In 2015, Burcroff began her bachelor’s degree at the University of Michigan, majoring in math with a minor in computer science. There, she entered the world of combinatorics, which focuses on counting, arranging, and combining objects, forming a pivotal basis for understanding complex problems as well as the limits of computer algorithms.

“When I was starting out, I was just happy to have any mystery that anyone gave me,” she admits. To her, math was like a fun game with levels to complete. However, during a study abroad program in Budapest, Hungary — the hometown of renowned mathematician Paul Erdős — the thrill intensified when she was presented with unsolved puzzles.

Breaking Ground in Mathematical Research

Burcroff later pursued a master’s degree in pure mathematics at the University of Cambridge and a research master’s at Durham University. In 2021, she returned to the United States to begin her PhD at Harvard University, guided by Professor Lauren Williams.

Among her many contributions, Burcroff played a crucial role in unifying different mathematical approaches to understand reliability within systems. She likens this unification to discerning that two seemingly disparate sets of instructions ultimately lead to the same conclusion. By revealing these connections, her research uncovered a broader mathematical architecture — enabling her and her collaborators to delve into enduring riddles in her field.

Generalized cluster algebras provide a vital foundation for describing geometries manifesting throughout physics. For over a decade, mathematicians theorized that these fundamental building blocks were created solely by addition, yet no one could substantiate it. In 2024, Burcroff and her colleagues published groundbreaking research detailing these spaces' positive properties, using a novel method to count and organize patterns.

These revolutionary findings could influence areas extending from predicting particle collision results to elucidating string theory's geometrical spaces. As a result of her extensive work, Burcroff has received numerous esteemed accolades, including a National Science Foundation Graduate Research Fellowship, a British Marshall Scholarship, and a Jack Kent Cooke Graduate Fellowship.

The Future of Mathematics and its Implications

Despite solving a wide array of mathematical problems, new challenges continually arise. “Every time you unlock one of them, it gives you a bunch of paths to new connected mysteries,” Burcroff shares.

At MIT, she collaborates with Postnikov, who is pioneering alternative calculations in quantum field theory through combinatorial research. “Burcroff is conducting research across disciplinary boundaries,” Postnikov notes, highlighting the potential for interdisciplinary interactions.

Burcroff aims to apply combinatorial techniques in broader physical contexts and practical applications, especially in areas that bear implications for mirror symmetry — a principle in string theory that suggests mathematically equivalent spaces despite their differing appearances.

While she emphasizes that “doing math is 99 percent trying something and failing,” Burcroff finds daily motivation in the quest for discovery. To her, the journey is not just about the destination, but about embracing the continuous process of exploration and understanding.