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
Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The proj ....Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The project aims are to strengthen this young field by advancing the theory and by finding applications from where its significance arises. The outcome will be new results on categorical representations and this will have benefits within mathematics, cryptography and also in physics/chemistry in the long run.Read moreRead less
Resilient Democracy for the 21st Century. This project will establish novel foundational theoretical frameworks for the design of democratic institutions that can withstand internal and external pressure towards autocratisation. It will develop state of the art dynamic models of information manipulation and political dynamics, and analyse large-scale online survey experiments, as well as contemporary and historical data. This combination will deliver new insights into the management of sensitive ....Resilient Democracy for the 21st Century. This project will establish novel foundational theoretical frameworks for the design of democratic institutions that can withstand internal and external pressure towards autocratisation. It will develop state of the art dynamic models of information manipulation and political dynamics, and analyse large-scale online survey experiments, as well as contemporary and historical data. This combination will deliver new insights into the management of sensitive information and how to protect democracy from information manipulation. Ultimately, the project will generate a body of theoretical and empirical evidence for the design of more effective and resilient democratic institutions for a more inclusive economic development. Read moreRead less
Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptua ....Universal structures in stringy extra dimensions. The project aims to study properties of extra dimensions in string theory by means of techniques from supersymmetric gauge theory. This new approach makes it possible to study areas in the landscape of stringy extra dimensions that have not been accessible before. The project expects to uncover new universal features. This will have significant impact on string theory and mathematics. Expected outcomes of this project include answers to conceptual questions in string theory, new types of extra dimensions, and new methods to compute quantum corrections in string theory. This should provide significant benefits, such as interdisciplinary collaborations at the national and international level and a strengthening of string theory in Australia.Read moreRead less
Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and sup ....Categorical geometry and perfect group schemes. The aims of this project are to construct novel geometric theories based on newly discovered tensor categories, to apply the theories to solve open problems in representation theory, algebra and category theory, and to establish profitable new connections between the influential theories of affine group schemes and classifying spaces. The geometric theories will be developed in a universal way, generalising both classical algebraic geometry and super geometry from physics, and specialising to infinitely many new theories. This universality ensures a significantly broader basis for long term applications of geometry in many areas of science. Other benefits include enhanced international collaboration and scientific capacity in Australia.Read moreRead less
Supersymmetry and supergravity: new approaches and applications. This project aims to advance our understanding of supersymmetric quantum field, gravity, and higher-spin theories. Supersymmetry and supergravity play crucial roles in modern developments in fundamental particle physics and cosmology. They also have rich connections with many branches of mathematical physics. Major conceptual questions in the description of general supergravity-matter couplings are still unsolved. By performing sta ....Supersymmetry and supergravity: new approaches and applications. This project aims to advance our understanding of supersymmetric quantum field, gravity, and higher-spin theories. Supersymmetry and supergravity play crucial roles in modern developments in fundamental particle physics and cosmology. They also have rich connections with many branches of mathematical physics. Major conceptual questions in the description of general supergravity-matter couplings are still unsolved. By performing state of the art analysis in supergravity and holographic dualities, the project will advance our understanding of quantum gravity, black holes, and cosmology placing Australia at the forefront of these important research fields.Read moreRead less
Using Abstract Networks to Study Symmetry. An operad is a mathematical tool for packaging the connection between discrete blocks of information. In other words, an operad is a type of network, particularly suited for approaching complex problems by breaking them into smaller, manageable packets. This project aims to reimagine classical objects in geometry and topology such as Teichmüller space as variations of infinity operads. This reimagining will ensure new insights into key objects across th ....Using Abstract Networks to Study Symmetry. An operad is a mathematical tool for packaging the connection between discrete blocks of information. In other words, an operad is a type of network, particularly suited for approaching complex problems by breaking them into smaller, manageable packets. This project aims to reimagine classical objects in geometry and topology such as Teichmüller space as variations of infinity operads. This reimagining will ensure new insights into key objects across three areas of mathematics: algebraic number theory (the mathematics of modern encryption), the representation theory of quantum groups and topological quantum field theories. Read moreRead less
Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic beh ....Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.Read moreRead less
Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathe ....Logarithmic conformal field theory and the 4D/2D correspondence. Conformal field theory provides powerful methods for attacking problems in theoretical physics and furnishes beautiful connections between seemingly disparate branches of pure mathematics. This proposal aims to greatly expand our knowledge of the logarithmic conformal field theories that have recently witnessed a resurgence of interest in physics. Advancing these theories is crucial to progress in high-energy physics and pure mathematics. Expected outcomes include a completely new understanding of the mathematical structure of these theories which will, in turn, facilitate applications in 4D gauge theory. This will boost research capacity and further cement Australia's reputation as an international leader in mathematical physics research.Read moreRead less