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
Discovery Early Career Researcher Award - Grant ID: DE120100173
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
A new upper bound for the Riemann zeta-function and applications to the distribution of prime numbers. Prime numbers are known to every schoolchild and are ubiquitous in modern cryptography; some of their deepest properties relate to a function called the Riemann zeta-function. This project aims at better estimating this function, thereby improving current knowledge on the distribution of prime numbers.
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
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.
Constructive Representation Theory. A group is a mathematical structure that captures the notion of symmetry. This project will enable us to perform a deep analysis of all the ways in which the group can act on an object such as a molecule, by constructing all of its representations as a matrix group.
Discovery Early Career Researcher Award - Grant ID: DE190101222
Funder
Australian Research Council
Funding Amount
$348,070.00
Summary
Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the m ....Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the mathematical knowledge of symmetries, and show unexpected new connections between different areas of pure mathematics and mathematical physics.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