Graded Symmetry in Algebra and Analysis. This project will study graded symmetries in mathematics by modelling them as groupoids and inverse semigroups. Groupoids have been at the centre of mathematical interest for a long time, but have gained special prominence in recent years as a focal point for algebra, analysis and dynamics. The majority of groupoids can be naturally graded. The project introduces graded combinatorial invariants for groupoids (such as graded homology) and relates them to t ....Graded Symmetry in Algebra and Analysis. This project will study graded symmetries in mathematics by modelling them as groupoids and inverse semigroups. Groupoids have been at the centre of mathematical interest for a long time, but have gained special prominence in recent years as a focal point for algebra, analysis and dynamics. The majority of groupoids can be naturally graded. The project introduces graded combinatorial invariants for groupoids (such as graded homology) and relates them to their Steinberg and C*-algebra counterparts (such as graded K-theory). The outcome is to give sought-after unified invariants bridging algebra and analysis, and to exhaust the class of groupoids for which these much richer invariants will furnish a complete classification. Read moreRead less
A Functional Analysis of the Hypoelliptic Laplacian. Strike a bell, a sphere, or any geometrical object, and it rings. The frequencies of ringing are the mathematical spectrum, which encodes deep secrets about the shape of the object. The spectrum of the hypoelliptic laplacian is known to carry deep truths in mathematics and physics, but it remains difficult to understand. We propose a new analytic foundation, which will replace the so far non-analytical ad hoc approach, and make accessible many ....A Functional Analysis of the Hypoelliptic Laplacian. Strike a bell, a sphere, or any geometrical object, and it rings. The frequencies of ringing are the mathematical spectrum, which encodes deep secrets about the shape of the object. The spectrum of the hypoelliptic laplacian is known to carry deep truths in mathematics and physics, but it remains difficult to understand. We propose a new analytic foundation, which will replace the so far non-analytical ad hoc approach, and make accessible many new results. It is key to better understanding differential equations which lie at the boundary between quantum mechanics and the classical world. This will pave the way for Australian leadership in a new century of differential equations and geometry, and training of young mathematicians.Read moreRead less