Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations. In simple words, it lets you predict things that you might otherwise never see like how individual microscopic particles spin. Valerio Toledano Laredo didn’t even know that the field of study existed, but he was a quick convert. He is now a mathematics professor at Northeastern University, who is helping others through education and outreach.
According to him, electrons are considered to be very complicated microscopic particles upon which extensive lab experiments are conducted to understand their behaviour, but one can just sit with a paper and pen, in half a page, one can predict some of these properties, just by pure thought. Thus, the symmetries of an object, or the transformations you can make without fundamentally changing it, manifest in different ways. These are called “representations”.
One doesn’t need to be a physicist to recognise that electrons and photons are fundamentally different particles; their behaviour differs. 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. However, it’s not always obvious when distinct objects share symmetries which is what makes representation theory so important to study.
He wants to give back to world now. He wants to teach 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. He is now working to prepare students to excel in the fields of science, technology, engineering, and mathematics 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 will enable our understanding of the universe to expand”, says Toledano Laredo.
Shahjadi Jemim Rahman