p-Adic Methods in Arithmetic Geometry. This project concerns algorithms for determining the number of
solutions to systems of polynomial equations over finite fields
by p-adic methods. Our goal is to determine a fundamental
invariant, the zeta function, appearing in arithmetic geometry,
whose characterization was the subject of the famous Weil
conjectures.
....p-Adic Methods in Arithmetic Geometry. This project concerns algorithms for determining the number of
solutions to systems of polynomial equations over finite fields
by p-adic methods. Our goal is to determine a fundamental
invariant, the zeta function, appearing in arithmetic geometry,
whose characterization was the subject of the famous Weil
conjectures.
We seek to understand and develop p-adic methods for determining
zeta functions, taking as point of departure the methods of Satoh
and Mestre for elliptic curves. Applications of this work include
public key cryptography and coding theory, having direct impact
in e-commerce and telecommunications.
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Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture ....Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture. The project will also have a flow-on effect in other areas of mathematics and computer science where zeta functions play a central role, including cryptography, coding theory and mathematical physics.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100859
Funder
Australian Research Council
Funding Amount
$354,000.00
Summary
New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a ce ....New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a century. The expected outcomes include a deeper understanding of Weyl sums and enhanced international collaborations. Such progress will place Australia at the forefront of this important branch of number theory.Read moreRead less
Elliptic special functions. Although elliptic functions and special functions are both classical areas of mathematics, the field of elliptic special functions was only established in the last two decades. It combines ideas from analysis, modular forms and statistical mechanics to tackle problems in number theory (elliptic curves), algebra (elliptic quantum groups), mathematical physics (Seiberg duality) and more. This project aims to settle two important problems in the field of elliptic special ....Elliptic special functions. Although elliptic functions and special functions are both classical areas of mathematics, the field of elliptic special functions was only established in the last two decades. It combines ideas from analysis, modular forms and statistical mechanics to tackle problems in number theory (elliptic curves), algebra (elliptic quantum groups), mathematical physics (Seiberg duality) and more. This project aims to settle two important problems in the field of elliptic special functions: the resolution of Boyd's conjectures concerning Mahler measures and L-values of elliptic curves, and the construction of an Askey-Wilson-Koorwinder theory of elliptic biorthogonal functions for the A-type root system.Read moreRead less
Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify ....Verifying the Riemann hypothesis to large heights: theory and applications. This project aims to verify the Riemann hypothesis to a record height and apply this verification to the distribution of prime numbers. The Riemann hypothesis (an open problem for 150 years) is ubiquitous in analytic number theory and prevalent in many other areas of mathematics. This project plans to use state-of-the-art computational hardware and the mathematical and algorithmic expertise of the investigators to verify the Riemann hypothesis several orders of magnitude further than what is currently known. A secondary aim is to apply this new verification to a multitude of results in analytic number theory: this would provide future researchers with vastly superior results.Read moreRead less
Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fun ....Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fundamental science and greater understanding of classical and quantum cryptographic protocols. This project will provide the additional, and substantial, benefit of generating research output, training HDR students, and contributions towards national security.Read moreRead less
Fast algorithms for zeta functions of algebraic varieties. The project aims to develop new algorithms for counting the number of solutions to polynomial equations in several variables. This fundamental counting problem appears in many areas of mathematics and computer science, such as number theory and cryptography. The aim of the project is to develop algorithms that are more efficient and that are able to handle much larger problems than existing algorithms. The new algorithms are expected to ....Fast algorithms for zeta functions of algebraic varieties. The project aims to develop new algorithms for counting the number of solutions to polynomial equations in several variables. This fundamental counting problem appears in many areas of mathematics and computer science, such as number theory and cryptography. The aim of the project is to develop algorithms that are more efficient and that are able to handle much larger problems than existing algorithms. The new algorithms are expected to have applications to the numerical investigation of important unsolved problems in number theory, such as the Sato-Tate, Lang-Trotter and Birch-Swinnerton-Dyer conjectures.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120101293
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Counting solutions to equations over fields of large characteristic. This project will make major contributions to a fundamental problem in mathematics and computer science, namely counting the number of solutions to certain types of polynomial equations. This work has potential applications in computer security, information processing, and pure mathematics.
Algebraic methods for Markov Chain Monte Carlo and quasi-Monte Carlo. In an increasingly complex world, the requirements on computational methods for solving real world problems from areas like statistics, finance, economics, physics and others are also constantly increasing. The results from this project will significantly improve existing computational methods, thereby helping to solve existing computational challenges and further strengthening Australia's reputation as a leading scientific lo ....Algebraic methods for Markov Chain Monte Carlo and quasi-Monte Carlo. In an increasingly complex world, the requirements on computational methods for solving real world problems from areas like statistics, finance, economics, physics and others are also constantly increasing. The results from this project will significantly improve existing computational methods, thereby helping to solve existing computational challenges and further strengthening Australia's reputation as a leading scientific location. The research carried out will be in collaboration with international experts, creating and strengthening existing ties of Australian research institutions with other world class research institutes overseas.Read moreRead less
Ubiquity of Kloosterman sums in Number Theory and Beyond. This project aims to seek new methods of investigating Kloosterman sums by
combining an algebraic geometry approach with an analytic approach to develop one
powerful, unified method. Its significance lies in expected pivotal advances towards
several fundamental problems which lie at the heart of number theory such as
the Dirichlet Divisor Problem and asymptotic formulas for moments of L-functions.
The expected outcome of the project is ....Ubiquity of Kloosterman sums in Number Theory and Beyond. This project aims to seek new methods of investigating Kloosterman sums by
combining an algebraic geometry approach with an analytic approach to develop one
powerful, unified method. Its significance lies in expected pivotal advances towards
several fundamental problems which lie at the heart of number theory such as
the Dirichlet Divisor Problem and asymptotic formulas for moments of L-functions.
The expected outcome of the project is to provide a deeper understanding of the
intriguing nature of Kloosterman sums and thus open new perspectives for
applications in analytic number theory. This will provide
substantial benefits for other areas such as cryptography by deepening our understanding of pseudorandom sequences.Read moreRead less