The arithmetic of supersingular elliptic curves. The proposed research will have substantial benefits both in the area of pure mathematics, and to the standing of number theory within Australia generally. If successful, the investigators envisage: - fundamental advances in the study of both elliptic curves and modular forms; - key progress in our understanding of the final Millenium Prize Problem in Mathematics; - academic software to compute special values of L-functions; - applications to com ....The arithmetic of supersingular elliptic curves. The proposed research will have substantial benefits both in the area of pure mathematics, and to the standing of number theory within Australia generally. If successful, the investigators envisage: - fundamental advances in the study of both elliptic curves and modular forms; - key progress in our understanding of the final Millenium Prize Problem in Mathematics; - academic software to compute special values of L-functions; - applications to computational mathematics, particularly elliptic curve cryptosystems; - a huge boost to the development of number theory Australia-wide.
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Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress ....Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress in a feasible time frame. In three dimensions this project will strengthen the distinguished computational topology community in Melbourne, led by pioneers such as Rubinstein, Goodman, Hodgson as well as the applicant himself.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140100088
Funder
Australian Research Council
Funding Amount
$378,628.00
Summary
Computing with matrix groups and Lie algebras: new concepts and applications. Computational algebra combines symbolic computation and pure research in algebra, and is concerned with the design of algorithms for solving mathematical problems endowed with algebraic structure. Matrix groups and Lie algebras are prominent algebraic objects describing the natural concept of symmetry. Despite being very common and important in science, there is a paucity of algorithms to study their structure. This pr ....Computing with matrix groups and Lie algebras: new concepts and applications. Computational algebra combines symbolic computation and pure research in algebra, and is concerned with the design of algorithms for solving mathematical problems endowed with algebraic structure. Matrix groups and Lie algebras are prominent algebraic objects describing the natural concept of symmetry. Despite being very common and important in science, there is a paucity of algorithms to study their structure. This project will develop deep new mathematical theories for computing with these objects, leading to ground-breaking advances in computational algebra, and providing powerful tools facilitating new research, including in other sciences. The new functionality will be used to solve two classification problems in group and Lie theory.Read moreRead less