Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This proj ....Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This project will answer many fundamental questions about large, random hypergraphs. The expected outcomes of the project also include new tools for working with hypergraphs, such as efficient algorithms for sampling hypergraphs. These outcomes will benefit researchers who use hypergraphs in their work and will enhance Australia's reputation for research in this area.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE210101323
Funder
Australian Research Council
Funding Amount
$345,448.00
Summary
Random tensors and random matrices: interactions and applications. This project aims at improving knowledge on probabilistic objects having applications in, for instance, mathematical-physics, statistical physics, quantum gravity and data science. In doing so, we expect to produce new mathematical results by building upon both classical approaches and innovative ones. In particular, on one hand, the extension of classical graphical methods will be developed and, on another hand, generalized prob ....Random tensors and random matrices: interactions and applications. This project aims at improving knowledge on probabilistic objects having applications in, for instance, mathematical-physics, statistical physics, quantum gravity and data science. In doing so, we expect to produce new mathematical results by building upon both classical approaches and innovative ones. In particular, on one hand, the extension of classical graphical methods will be developed and, on another hand, generalized probability theories will be used to provide new insights. The expected outcomes include a better understanding of the generic properties of quantum states. This should significantly benefit to mathematicians and physicists whose models use those objects and may impact the broader community of engineers and technicians.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
Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recen ....Topics in triangulated categories. This project in pure mathematics, more specifically in modern homological algebra, builds on work started by the chief investigator in the last five years. What has already been done has achieved striking results, solving very different problems that have been open for two decades. And there seem to be many directions in which it could be pursued further.
The international mathematical community seems intrigued by what the chief investigator has achieved recently - judging by invitations to give prestigious talks and the feedback at these events. The expected outcome is major progress in our understanding of derived categories, as well as diverse applications. The benefit will be to enhance the international stature of Australian science.Read moreRead less
Groups, piecewise linear representations, and linear 2-representations. This project aims to address fundamental questions at the interface of two central areas of modern mathematics, geometric group theory and higher representation theory. Higher representation theory is a relatively new field, but its tools have already had a tremendous impact on mathematics. The project is expected to use these tools to address outstanding questions in geometric group theory. The expected outcomes of this pr ....Groups, piecewise linear representations, and linear 2-representations. This project aims to address fundamental questions at the interface of two central areas of modern mathematics, geometric group theory and higher representation theory. Higher representation theory is a relatively new field, but its tools have already had a tremendous impact on mathematics. The project is expected to use these tools to address outstanding questions in geometric group theory. The expected outcomes of this project include resolutions of open problems in the theory of Artin groups and the creation of a new subject, the dynamical theory of two-linear groups.Read moreRead less
Integral transforms and moduli theory. This project is in algebraic geometry, a branch of pure
mathematics. An overarching goal is a better understanding of the
algebra underlying the sophisticated geometries that arise in the
classification problems that are pervasive in mathematics and its
applications to physics. This new knowledge will then be applied to
further elucidate the geometry of these spaces.
Expected outcomes of this project include major progress in our
understanding of derived ....Integral transforms and moduli theory. This project is in algebraic geometry, a branch of pure
mathematics. An overarching goal is a better understanding of the
algebra underlying the sophisticated geometries that arise in the
classification problems that are pervasive in mathematics and its
applications to physics. This new knowledge will then be applied to
further elucidate the geometry of these spaces.
Expected outcomes of this project include major progress in our
understanding of derived categories of algebraic stacks via the
Fourier-Mukai transform.
The benefit will be to enhance the international stature of Australian
science.Read moreRead less
Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relat ....Moduli, invariants, and algebraisation. This project is in pure mathematics. It aims to address gaps in our
knowledge in the modern geometries and their associated algebraic structures that arise in classification problems that pervade mathematics and its applications.
This project expects to generate new knowledge in modern algebra and geometry.
Expected outcomes of this project include major progress in our
understanding of invariants of derived categories of algebraic stacks and the
relationship between algebraic and other geometries.
The benefit will be to enhance the international stature of Australian science.Read moreRead less
Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point ....Frobenius manifolds from a geometrical and categorical viewpoint. This project aims to provide connections between Frobenius manifolds obtained from algebraic curves in diverse ways. The different constructions, using complex geometry on the one hand and category theory on the other, provide, respectively, a quantitative and qualitative view on the same Frobenius manifold. Together, these distinct points of view allow for the calculation of previously inaccessible physical quantities, and point to deep new relations between algebraic, complex and differential geometry. These relations are expected to guide new fundamental research on the border of mathematics and physics.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