Shedding New Light on Space and Geometry
Geometry dictates the “rules” for whatever part of the physical universe humans are anaylzing. In this sense, it is the visual study of shapes, sizes, patterns and positions in space. But, geometry is only the human brain’s way of interpreting the rules.
Dr. David Favero, Canada Research Chair in Derived Categories, believes mathematical data, referred to as “derived categories”, may offer a more accurate interpretation of the universe’s real physical properties. To get at the heart of geometry, he and his collaborators are using the latest mathematical techniques to explore the rules of physical systems.
Favero is applying “Kontsevich’s non-commutative algebraic geometry,” a field of research at the crossroads of modern math and high-energy theoretical physics. Through high-energy physics, researchers are trying to understand the nature of space and time, the forces determining the interactions between matter and energy, and the origins of elementary particles.
Favero’s research focuses on the mathematical consequences of “phase transitions” in high-energy physics, or the transformation of physical models of the universe from one stable energy state to another. For example, when water changes into ice or steam, it crosses an energy threshold, and its properties and symmetries sometimes changing drastically.
Favero is also studying “mirror symmetry” between geometric objects, and is looking for a mathematical explanation for this phenomenon. Mirror symmetry predicts that some geometric shapes will have the same physical properties, but seen in a different way.
By exploring the connection between mirror symmetry and “phase transitions” in high-energy physics, Favero’s research has the potential to solve important questions in mathematics.