A mathematician is building a dictionary that translates between abstract algebra and geometry, aiming to solve long-standing problems about the fundamental building blocks of symmetry. The project tackles modular representation theory, where the underlying number system works like a clock face rather than a number line—making the problems far more difficult than their non-modular counterparts. The core question is whether researchers can explicitly describe the irreducible representations of certain algebras, determining their structure and dimensions. This is fundamental science with no immediate practical application. But the same kind of abstract symmetry research, rooted in Emmy Noether's early 20th-century theorems, has historically underpinned discoveries in particle physics and conservation laws. If successful, this work would deepen the mathematical framework that physicists use to describe the universe's fundamental particles and forces—potentially opening new pathways for theoretical physics that could, decades from now, reshape our understanding of matter and energy.
View original technical description
At the turn of the 20th century it was difficult to draw a clear distinction between mathematicians and physicists, since many of the greatest scientists worked on every important problem in these fields. One of the most influential polymaths of the era was Emmy Noether, who developed all of the foundational theories which inspired this research project. Her greatest contribution to physics was probably Noether's first theorem, which says that if you want to understand the conservation laws of the universe then it suffices to understand the symmetries of the universe. Conservation laws are the most fundamental laws of physics, giving us clues about the nature of matter, and the shape of space, and so Noether's theorem started a wave of discovery which has been growing and growing for over a hundred years, as mathematicians and physicist seek to understand the symmetries of the universe. Today mathematicians and physicists are much easier to distinguish, however the subjects are still deeply intertwined. In modern day mathematical language, the study of symmetries is called representation theory and the goal of this project is to understand how Noether's algebraic structures can be expressed as symmetries. To rephrase this, my objective is to understand the representations of certain important families of algebras. In ancient Greece a powerful idea was born: all matter can be built up from indivisible pieces - the word "atom" literally means "indivisible" - and in the language of modern particle physics it is well-understood that all matter in the universe can be built up from the fundamental particles. In precisely the same way, the representations I seek to understand are also built from fundamental building blocks, known as irreducible representations. Can we describe these irreducible representations explicitly? Can we determine their structure and calculate their dimensions? In this research project I will answer these fundamental, elusive questions in some challenging but historically important examples. Some of the most important unanswered questions in this field pertain to algebras which we call "modular": this is because the underlying number system is not linear, like the real number line, but is circular like the numbers on the face of a clock. Questions in modular representation theory tend to be significantly harder due to the added complexity of the geometry and the arithmetic. By working with tools on the interface between abstract algebra and geometry this project will make substantial exciting progress in some of the most challenging problems in modular representation theory, showing that Noether's wave of discovery is still growing on the ocean of mathematics.
Plain English summaries and category classifications on this site are generated by AI and may not perfectly reflect the original research.
Is something wrong? Let us know