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
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
Free parafermions: a challenge for non-Hermitian physics. This project aims to calculate and understand the physical properties of free parafermions. Parafermions have attracted interest in topological schemes for quantum computation because they are computationally more powerful than Majorana fermions. The core of this project is a fundamental model of free parafermions, which has been shown to exhibit unexplained puzzling properties. The project outcomes include an in-depth understanding of th ....Free parafermions: a challenge for non-Hermitian physics. This project aims to calculate and understand the physical properties of free parafermions. Parafermions have attracted interest in topological schemes for quantum computation because they are computationally more powerful than Majorana fermions. The core of this project is a fundamental model of free parafermions, which has been shown to exhibit unexplained puzzling properties. The project outcomes include an in-depth understanding of this model by taking the non-Hermitian features into account, establishing a connection with open quantum systems. Non-Hermitian systems are also of increasing relevance in physics, especially in quantum optics. The project also aims to contribute to training researchers in the mathematical sciences.
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