Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahl ....Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahler-Ricci flow in the closed and complete non-compact settings and the corresponding versions of Geometric Minimal Model Program; and the Kahler-Ricci flow in the Fano manifold setting and stability conditions.Read moreRead less
Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry ....Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry and algebra of a space. This is a key problem as moduli spaces describe whether a space is rigid or can be deformed. They are a central object in several fields of mathematics, including geometry and topology, gauge theory, dynamical systems, mathematical physics and invariant theory.Read moreRead less
Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the ....Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the first aim is to construct structures with good geometric properties on 3- and 4-manifolds, using triangulations. The second aim is to study combinatorial decompositions of n-manifolds, using our new technique of multisections and also searching for polyhedral metrics of non-positive curvature. The third aim is to connect quantum invariants and geometric structures, again using triangulations.Read moreRead less
Flag varieties and configuration spaces in algebra. School students learn that curves may be described by means of equations, which may therefore be solved geometrically; this is an example of the interaction of algebra and geometry. In this project geometric ideas such as simplicial geometry and cohomological representation theory will be developed, which address deep questions in modern algebra.
Generalised topological spaces. Pure mathematics creates abstractions of real-world entities; one such is the idea of a 'topological space', which abstracts from geometric forms like cubes and toruses. But topological spaces fail to capture geometric structures arising in areas like quantum physics; and this project seeks to rectify this, by developing a new more general notion.
Enriched higher category theory. At the beginning of 21st century it became clear that further progress in mathematics and physics required a development of a new powerful language, which received the name Higher Category Theory. This project is devoted to the development of this new exciting theory.
Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from ....Tractable topological computing: Escaping the hardness trap. Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project aims to defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from the field of parameterised complexity, creating powerful, practical solutions for these problems. It is expected to shed much-needed light on the vast and puzzling gap between theory and practice, and give researchers fast new software tools for large-scale experimentation and cutting-edge computer proofs.Read moreRead less
Noncommutative analysis for self-similar structure. This project in pure mathematics aims to develop novel mathematical techniques for understanding self-similar structures using operator algebras. Fractals and self-similarity have many applications both within and outside mathematics, but remain deeply mysterious, while operator algebras are the mathematical language of quantum mechanics. This project expects to provide new connections between self similarity and operator algebras advancing bot ....Noncommutative analysis for self-similar structure. This project in pure mathematics aims to develop novel mathematical techniques for understanding self-similar structures using operator algebras. Fractals and self-similarity have many applications both within and outside mathematics, but remain deeply mysterious, while operator algebras are the mathematical language of quantum mechanics. This project expects to provide new connections between self similarity and operator algebras advancing both fields. Expected outcomes include increased understanding of self-similar structures, and novel operator-algebraic phenomena and examples. Benefits include growing Australia's capacity in operator algebras and mathematics more generally, and enhanced international collaboration.Read moreRead less
Scale-Multiplicative Semigroups and Geometry. Symmetry is treated mathematically through the algebraic concept of a group. Conversely, geometric representations play a crucial role in group theory. Many classes of groups, such as the connected groups that arise in physics, have useful geometric representations, but such a representation is lacking in the case of general disconnected groups. Certain disconnected groups, closely related in algebraic terms to the connected ones, do have a geometric ....Scale-Multiplicative Semigroups and Geometry. Symmetry is treated mathematically through the algebraic concept of a group. Conversely, geometric representations play a crucial role in group theory. Many classes of groups, such as the connected groups that arise in physics, have useful geometric representations, but such a representation is lacking in the case of general disconnected groups. Certain disconnected groups, closely related in algebraic terms to the connected ones, do have a geometric representation called a 'building'. This project aims to address the lack of a representation for general disconnected groups by extending the notion of a building to create combinatorial structures on which these groups act as symmetries.Read moreRead less
Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct ....Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct computable invariants, connectivity results for triangulations, and algorithms to recognise fundamental topological properties and structures such as trisections and bundles.Read moreRead less