Branimir Cacic (Assistant Professor)
BSc (Toronto), PhD (Caltech)
Differential geometry is a branch of mathematics that uses techniques from multivariable calculus and linear algebra to study geometric problems. In particular, it has allowed mathematicians and physicists to grapple with higher-dimensional curved spaces arising in a wide range of contexts including, most famously, Einstein’s general theory of relativity. However, by the 1930s, the development of quantum mechanics had already led mathematicians and theoretical physicists to consider generalised spaces whose coordinates need not commute (in the sense that x times y need not equal y times x). The mathematical key to constructing and studying these generalised spaces turned out to be functional analysis, an infinite-dimensional generalisation of classical linear algebra originating in the study of differential and integral equations.
The functional-analytic study of generalised spaces from the perspective of topology—properties of space preserved under continuous deformations—and of measure theory—properties of space related to integration—is well-established as the study of operator algebras, with remarkable applications in both mathematics and theoretical physics. By contrast, noncommutative differential geometry, the functional-analytic study of generalised spaces from the perspective of differential geometry, is still very much in its infancy, though intriguing connections to number theory, theoretical particle physics, and the mathematics of signal processing have all come to light in the last ten years. My own research is focussed on spectral triples and unbounded KK-theory as a framework for noncommutative differential geometry and on applications of noncommutative differential geometry to applied harmonic analysis, geometric group theory, and mathematical physics.