New Frontiers and Advances in Discrete Integrable Systems. Integrable systems boast a long and venerable history, and have such famous members as the Kepler system, the Korteweg-de Vries equation, and the sine-Gordon equation. More recently, interest in integrable systems has expanded to include systems with discrete time, that is, ordinary difference equations (or maps) and integrable partial difference equations. These discrete integrable systems are arguably more fundamental than the continuo ....New Frontiers and Advances in Discrete Integrable Systems. Integrable systems boast a long and venerable history, and have such famous members as the Kepler system, the Korteweg-de Vries equation, and the sine-Gordon equation. More recently, interest in integrable systems has expanded to include systems with discrete time, that is, ordinary difference equations (or maps) and integrable partial difference equations. These discrete integrable systems are arguably more fundamental than the continuous-time ones. Based upon recent breakthroughs this study will combine analysis, geometry, and computer algebra to expand and systematise this new interdisciplinary field of discrete integrable systems.Read moreRead less
Integrable Systems in Gauge and String Theories. Gauge theory describes all quantum forces except gravity. String theory aims to describe quantum gravity. Both theories are widely believed to be different limits of one unknown theory. Discoveries of integrable nonlinear partial differential equations and integrable quantum systems in gauge/string theories are among the most remarkable recent developments in mathematical physics. They have led to deep results in known gauge/string theories, as we ....Integrable Systems in Gauge and String Theories. Gauge theory describes all quantum forces except gravity. String theory aims to describe quantum gravity. Both theories are widely believed to be different limits of one unknown theory. Discoveries of integrable nonlinear partial differential equations and integrable quantum systems in gauge/string theories are among the most remarkable recent developments in mathematical physics. They have led to deep results in known gauge/string theories, as well as to viable paths towards the unknown theory that interpolates them. This project contributes to these developments by adapting and developing sophisticated technical tools and insights from integrable models to shed light on that unknown theory that transcends the gauge/string gap. Read moreRead less
Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by devel ....Multivariate polynomials: combinatorics and applications. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables. This proposal will expand our understanding of the poorly understood class of non-symmetric polynomials by studying their novel combinatorial structure. The outcomes will address the current difficulty of implementing non-symmetric polynomials in symbolic algebra packages by developing completely new computational algorithms. Secondly, this new understanding will be used to solve several challenging mathematical enumeration problems.Read moreRead less
Solvability and universality in stochastic processes. Exactly solvable stochastic processes are an important area of mathematical research, with cross-disciplinary links to quantum physics, quantum algebras and probability theory. These processes can be used to model a variety of real-world phenomena such as crystal growth and polymers in random media. This project aims to significantly expand our knowledge of exactly solvable stochastic processes by extending them to new algebraic frameworks. A ....Solvability and universality in stochastic processes. Exactly solvable stochastic processes are an important area of mathematical research, with cross-disciplinary links to quantum physics, quantum algebras and probability theory. These processes can be used to model a variety of real-world phenomena such as crystal growth and polymers in random media. This project aims to significantly expand our knowledge of exactly solvable stochastic processes by extending them to new algebraic frameworks. Among the outcomes of the project, we expect to identify new probabilistic structures which go beyond the famous Gaussian universality class. These theoretical developments allow better prediction of randomly growing interfaces, which encompass a range of phenomena from tumour growth to forest fires.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160100958
Funder
Australian Research Council
Funding Amount
$307,536.00
Summary
Quantum integrability and symmetric functions. This project aims to develop new connections between quantum integrability and a central area of pure mathematics, symmetric function theory. Quantum integrability is one of the most important areas of mathematical physics, in view of its application to modern physical theories and its mathematical richness. The project intends to use advanced symmetric function techniques to calculate quantum mechanical quantities without any approximation, and to ....Quantum integrability and symmetric functions. This project aims to develop new connections between quantum integrability and a central area of pure mathematics, symmetric function theory. Quantum integrability is one of the most important areas of mathematical physics, in view of its application to modern physical theories and its mathematical richness. The project intends to use advanced symmetric function techniques to calculate quantum mechanical quantities without any approximation, and to use the framework of quantum integrability to provide new results in symmetric function theory. The intended outcomes of the project will be new asymptotic expressions for correlation functions and more efficient computer algorithms for the calculation of a variety of symmetric functions.Read moreRead less
Discrete integrable systems. Discrete integrable systems are a fundamental generalisation of traditional integrable systems. This project, combining 5 world experts from 3 countries and 2 early career researchers, will expand and systematise this new interdisciplinary field, and will place Australia at the forefront of this intensive international activity.
From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expect ....From quantum integrable systems to algebraic geometry and combinatorics. The purpose of this project is to investigate the deep connections that have recently emerged between the study of an area of mathematical physics (quantum integrable systems) and subjects of pure mathematics (enumerative and algebraic combinatorics, and algebraic geometry). These connections have a common root, which this project plans to reveal using novel methods coming from quantum integrability. This approach is expected to illuminate these subjects leading to a new unified and interdisciplinary picture, and to resolve important open problems in the study of certain algebraic varieties and of their cohomology in the theory of symmetric functions, and related counting problems.Read moreRead less
Integrable models and topological strings. This project aims to develop advanced methods to compute n-point correlation functions in two-dimensional integrable models. The project expects to use recently discovered connections with topological strings to compute currently-inaccessible conformal blocks in conformal field theories, and their analogues in integrable massive field theories and statistical mechanical models. Expected outcomes include explicit expressions for the n-point correlation ....Integrable models and topological strings. This project aims to develop advanced methods to compute n-point correlation functions in two-dimensional integrable models. The project expects to use recently discovered connections with topological strings to compute currently-inaccessible conformal blocks in conformal field theories, and their analogues in integrable massive field theories and statistical mechanical models. Expected outcomes include explicit expressions for the n-point correlation functions, advances in the theory of topological vertices and the related representation theory, and new solutions of the Yang-Baxter equations. This should provide benefits that include a better understanding of two-dimensional integrable models and their deep connections with topological strings.Read moreRead less
Tantalizer algebras and generalized lattice models. This project exploits underlying symmetry to characterise components and flow patterns in network configurations. The project will develop tools for analysis and optimisation of systems of interacting nodes which can arise in materials, computing networks, and any social or industrial contexts with communication or product transfer between nodes.
Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and no ....Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and non-symmetric polynomials in symbolic algebra packages by developing completely new algorithms. New understanding from the project is expected to facilitate challenging computational problems of measurable quantities in quantum systems.Read moreRead less