Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere ....Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere type equation arising in applied sciences, by introducing new ideas and developing innovative cutting-edge techniques. Expected outcomes include resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.
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Additive combinatorics of infinite sets via ergodic theoretic approach. The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of ....Additive combinatorics of infinite sets via ergodic theoretic approach. The proposed project will utilise innovative ergodic theoretic approaches to enable us to address important questions in Additive Combinatorics (Number Theory) and Fractal Geometry. In particular, we will resolve long-standing inverse additive problems for infinite sets, discover sum-product phenomena in Number Theory, and find a plethora of finite configurations in fractal sets. We will also extend the structure theory of one of the most popular mathematical models of quasi-crystals to a more extensive class of groups. This project will make significant contributions to Additive Combinatorics and Ergodic Theory and will bring the Australian research in these fields to ever greater heights.Read moreRead less
New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in t ....New constructions and techniques for tensor categories. The goal of this project is to make fundamental advances in the structure theory of tensor categories. Such categories play crucial roles in numerous fields of mathematics, physics and beyond. New methods, theory and examples will be developed, inspired by algebra, representation theory and geometry. These will then be applied in the foundational study of tensor categories for (dis)proving several of the most important open conjectures in the field. This will open new perspectives for applications in other areas, most notably in representation theory. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101099
Funder
Australian Research Council
Funding Amount
$420,256.00
Summary
Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these sh ....Representation theory: studies of symmetry shadows. This project aims to solve fundamental problems in representation theory by combining cutting-edge techniques and developing novel higher level structures. Representation theory is the mathematical study of symmetry, an essential concept in science. Since the 1990s, mathematicians have been observing shadows of a more general notion of symmetry but so far have failed to explain it. Expected outcomes include a structural explanation of these shadows, new mathematical software to understand them and solutions to important conjectures. This project will make a significant contribution to the field of representation theory, with ramifications in mathematical physics and computer science.Read moreRead less
Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using pow ....Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using powerful geometric tools from conformal geometry, the project will extend this to less symmetric spaces. The knowledge generated from this project will extend to more general geometric contexts providing a concrete setting for the study of the associated natural equations in curved spaces.Read moreRead less
New frontiers in the theory of noncommutative surfaces. In the 90s, Artin launched his school of noncommutative algebraic geometry, where novel geometric methods
were used to profoundly deepen our understanding of the classical subject of noncommutative algebra. This
project aims to advance this theory by establishing several new frontiers in the theory of noncommutative
surfaces. This project expects to develop new methods involving sheaf theory, Mori's minimal model program and
moduli stacks, ....New frontiers in the theory of noncommutative surfaces. In the 90s, Artin launched his school of noncommutative algebraic geometry, where novel geometric methods
were used to profoundly deepen our understanding of the classical subject of noncommutative algebra. This
project aims to advance this theory by establishing several new frontiers in the theory of noncommutative
surfaces. This project expects to develop new methods involving sheaf theory, Mori's minimal model program and
moduli stacks, to study in particular, Artin's classification problem for noncommutative surfaces. Expected
outcomes include a much richer geometric understanding of noncommutative algebra. This project should help
ensure Australia plays a leading role in important developments in both algebra and algebraic geometry.Read moreRead less
Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the lands ....Physical realisation of enriched quantum symmetries. This project aims to investigate fundamental mathematical structures in modern category theory, providing an algebraic description of physical systems including topological order and conformal field theory. The project will study quantum symmetry, and classify and construct new classes of conformal field theories, using novel tools from enriched category theory, modular forms, and lattice gauge theory.
The main goal is to understand the landscape of topological and conformal field theories, laying the foundation for new technologies based on topological order. This timely project capitalises on the recent arrival of subfactor experts in Australia, and builds capacity in mathematical research and international links in a cutting edge field.Read moreRead less
Arithmetic and algebraic aspects of the dynamics of polynomial semigroups. This project aims to advance research in arithmetic dynamics, a rapidly developing field of mathematics that is rich in significant applications including cryptography. It will systematically study the semigroup action of several maps on an arithmetic space. The expected outcomes will extend existing concepts and theory, but also explore new mathematical phenomena related to number theory, graph theory and regular and cha ....Arithmetic and algebraic aspects of the dynamics of polynomial semigroups. This project aims to advance research in arithmetic dynamics, a rapidly developing field of mathematics that is rich in significant applications including cryptography. It will systematically study the semigroup action of several maps on an arithmetic space. The expected outcomes will extend existing concepts and theory, but also explore new mathematical phenomena related to number theory, graph theory and regular and chaotic arithmetic dynamics. There are likely applications in cryptography arising from the project and opportunities to build links between academia and national offices of information security.Read moreRead less
Working synthetically in higher categorical structures. This project aims to develop tools for synthetic reasoning in the mathematical discipline of higher category theory. Since its introduction in Euclidean geometry, synthetic reasoning has repeatedly revolutionised mathematics. The project expects to bring these benefits to important new structures such as (infinity, 2)-categories and monoidal bicategories, as well as building the expanding circle of applications of these structures. The proj ....Working synthetically in higher categorical structures. This project aims to develop tools for synthetic reasoning in the mathematical discipline of higher category theory. Since its introduction in Euclidean geometry, synthetic reasoning has repeatedly revolutionised mathematics. The project expects to bring these benefits to important new structures such as (infinity, 2)-categories and monoidal bicategories, as well as building the expanding circle of applications of these structures. The project's key outcome will be new scientific tools which will greatly benefit workers in areas of application as diverse as algebraic topology and geometry, quantum physics and computer science, while also helping to train Australia's next generation of scientific leaders.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150101415
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further ....Higher Representation Theory. Representation theory is a field of mathematics with applications across the breadth of mathematical study in fields as diverse as number theory and physics. The study of higher (or categorical) representation theory is a modern set of tools that provides new insights into representation theoretic phenomena. This project aims to study categorified quantum groups and, in particular, the categorifications provided by diagrammatic algebras. The project aims to further develop the theory of Khovanov-Lauda-Rouquier (KLR) algebras, providing important foundational results for future research to build upon.Read moreRead less