Discovery Early Career Researcher Award - Grant ID: DE150100030
Funder
Australian Research Council
Funding Amount
$300,000.00
Summary
A new concept of independence in noncommutative probability theory. The concept of independence lies at the very core of the probability theory. Many attempts to establish the general notion of independence in noncommutative probability theory have led to only two examples so far: the classical (commutative) independence and the free one introduced by Voiculescu. Every other approach has failed to demonstrate the analogues of the key probabilistic results, such as the Law of Large Numbers and th ....A new concept of independence in noncommutative probability theory. The concept of independence lies at the very core of the probability theory. Many attempts to establish the general notion of independence in noncommutative probability theory have led to only two examples so far: the classical (commutative) independence and the free one introduced by Voiculescu. Every other approach has failed to demonstrate the analogues of the key probabilistic results, such as the Law of Large Numbers and the Central Limit Theorem. There is an urgent need for new efficient methodology. This project aims to develop an approach to the independence in terms of mixed momenta and to find new examples of independence besides the ones mentioned above.Read moreRead less
Geometry and analysis of discrete integrable systems. Whether we are looking at waves at a beach or the movement of herds of animals in a landscape, their movements and fluctuations turn out to rely on rules expressed by non-linear systems of mathematical equations. The aim of this project is to create a new mathematical theory to describe and predict the solutions of such systems.
Arithmetic hypergeometric series. Arithmetic, known nowadays as number theory, is the heart and one of the oldest parts of mathematics. The project is aimed at solving three difficult mathematical problems of contemporary mathematics by arithmetic means.
Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental th ....Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental than differential geometry in that it aims to establish an autonomous discrete analogue from which differential geometry may be derived via an appropriate continuum limit. Even though discrete differential geometry has reached a high degree of sophistication, this project seeks to deliver the first comprehensive theory in this area. Read moreRead less
Discrete differential geometry: theory and applications. Sophisticated freeform structures made of glass and metal panels are omnipresent and their architectural design has been shown to be intimately related to a new area of mathematics, namely discrete differential geometry. This project is concerned with the theoretical basis of discrete differential geometry and its real world applications.
Noncommutative analysis and geometry in interaction with quantum physics. Quantum theory has produced many advances in our understanding of the physical world for the last hundred years while mathematical breakthroughs have been made through exploiting innovative ideas from quantum physics. This project continues in this highly successful framework and will lead to advances in geometry both classical and noncommutative.