Ever feel overwhelmed by the chaos and variability of the day to day? Maybe you’re in the wrong field. Try studying representation theory—it lets you predict things that you might otherwise never see, like how individual microscopic particles spin.

If you’re “absolutely awed” by this hack, you’re in good company. As an undergraduate, Valerio Toledano Laredo didn’t even know the field of study existed, but he was a quick convert. Now a mathematics professor at Northeastern in his third decade of research, he’s helping others, through education and outreach, begin their own journeys. 

His started with the electron.

“There’s a very complicated microscopic particle and these very complicated lab experiments you do to understand its behavior,” says Toledano Laredo, whose professor at the time put representation theory on his radar. “But you can just sit back in your chair with a paper and pen and, in half a page, actually predict some of these properties—just by pure thought.”

So what’s the secret sauce for learning about something you can’t see?

Symmetry. 

The symmetries of an object, or the transformations you can make without fundamentally changing it (think rotating a square 90 degrees), manifest in different ways. Mathematicians call these manifestations “representations,” hence “representation theory.” 

Here’s how to think about this theory in everyday terms: If I closed my eyes while you rearranged playing cards facedown, I’d have no way of knowing whether you even rearranged them. Similarly, a monochrome ball would look the same no matter how many times you spun it around. 

This quality is considered a symmetry, and it shows up—in the form of representations—in very different objects. On the subatomic level, representations of this symmetry manifest in electrons and photons, two distinct kinds of particles.

Electrons and photons are fundamentally different particles because of how they behave. You don’t have to be a physicist to recognize this: We’ve all seen lasers (“social” photons piling on top of one another) and electricity (“asocial” electrons keeping their distance from one another and stretching a signal along). 

Toledano Laredo says that knowing, through representation theory, that these particles are also somehow similar helps mathematicians and scientists understand how these tiny components of the universe behave—and prepares researchers to conduct new experiments without going in blind.

But it’s not always obvious when distinct objects share symmetries, which, Toledano Laredo says, is what makes representation theory so important to study.

Now he wants to give the next generation of researchers the tools to learn more about this complex theory, along with the related fields of algebraic geometry and mathematical physics. 

As part of a team funded by an ongoing research and training grant from the National Science Foundation, he’s working to prepare students to excel in the science, technology, engineering, and math fields through programs such as Bridge to Calculus, which prepares high schoolers in Boston to succeed in their math classes. 

Training the coming generations of mathematicians, he says, will enable our understanding of the universe to expand.

“There’s so much more to know,” says Toledano Laredo. “For me, for math, for physics—for the whole scientific community.”

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