CUHK Research: Changing the world

Integrable systems, quantum symmetry and dualities Understanding the structure of quantum symmetry CUHK’s Department of Mathematics is home to a team of outstanding mathematicians who contribute expertise to the advancement of mathematics. Among them is Professor Michael McBreen who studies the intersection of rep resentat i on t heo r y, symp l ect i c geometry and mathematical physics. He was recognised by the Research Grant Council and won the Early Career Award in 2021. A young mathematician who believes in the importance of intuition, Professor McBreen’s research centres around integrable systems, quantum symmetry and dualities. Knowing more about the “chaotic world” “The grand challenge of my research is to understand how symmetries behave in a world governed by quantum mechanics. I work with tools from algebra and geometry, especially from the field of integrable systems – a class of physical systems with an extraordinary level of symmetry,” says Professor McBreen. In a world consisting of things that are chaotic and unpredictable, some special physical systems behave in an orderly and structured way. Many used to think they were an exception carrying less importance in sciences. Not until the 1960s did scientists start to realise how crucial integrable systems are in physical phenomena such as solitary waves, phase transitions in thermodynamic systems and high energy physics, and areas of pure mathematics such as number theory and representation theory. Today, they have become a central part of modern mathematics. “Almost every subject in pure mathematics has an interesting relationship with integrable systems,” notes Professor McBreen. “In the 20th century, it was realised that these systems can explain stability and self-similarity in complex phenomena. We have developed sophisticated mathematical tools for obtaining precise information about their behaviour and classification.” Professor McBreen also contributes to the study of quantum symmetry and dualities, topics that are closely associated with integrable systems. “Quantum symmetry, Michael McBreen Assistant Professor of Mathematics 74