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
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
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
Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction ....Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction of computable invariants, solution of realisation problems and understanding degeneration of geometries.Read moreRead less
Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or n ....Triangulations: linking geometry and topology with combinatorics. Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.
Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or not at all, depending on intricate and highly involved mathematical properties.
In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications.Read moreRead less