Periodic Reporting for period 3 - PRIMES (Structure in the Primes, with applications)
Berichtszeitraum: 2023-02-01 bis 2024-07-31
The main aim of the ERC Project PRIMES is to deepen our understanding of prime numbers. Prime numbers are fundamental building blocks of mathematics, and have been studied by mathematicians for thousands of years. Despite this intense interest to mathematics, prime numbers remain poorly understood and many of the oldest and most important open problems in mathematics are about the properties of prime numbers. PRIMES is an ambitious project intending to make progress on several of these key questions by developing and enhancing our mathematical techniques for studying prime numbers.
Although PRIMES is a proposal focused on improving our theoretical understanding of prime numbers within pure mathematics, the fundamental role of prime numbers is seen in many real-world situations. This is clearly seen in cryptography, where prime numbers continue to play a central role in many of the most common and widely used cryptography algorithms enabling secure communication online. Even simple questions about the effectiveness of some of these algorithms is closely related to long-standing open problems in pure mathematics. Moreover, if a flexible and general theory of prime numbers could one day be developed, this would provide a general toolkit for answering a wide variety of questions about whole numbers, which often arise in many different scientific disciplines.
The key objective of PRIMES is to develop new techniques for studying and understanding prime numbers. To do this several specific areas have been identified where new ideas appear to have the potential to make progress on long-standing problems. A common theme in each case is to identify and classify the key technical barriers to progress with our current set of techniques, and then develop new techniques that can overcome the barriers in the specific context of interest. Once these new techniques have been developed in the specific and limited situation, they can then be generalised and applied to make progress on a number of different problems.
Although PRIMES is a proposal focused on improving our theoretical understanding of prime numbers within pure mathematics, the fundamental role of prime numbers is seen in many real-world situations. This is clearly seen in cryptography, where prime numbers continue to play a central role in many of the most common and widely used cryptography algorithms enabling secure communication online. Even simple questions about the effectiveness of some of these algorithms is closely related to long-standing open problems in pure mathematics. Moreover, if a flexible and general theory of prime numbers could one day be developed, this would provide a general toolkit for answering a wide variety of questions about whole numbers, which often arise in many different scientific disciplines.
The key objective of PRIMES is to develop new techniques for studying and understanding prime numbers. To do this several specific areas have been identified where new ideas appear to have the potential to make progress on long-standing problems. A common theme in each case is to identify and classify the key technical barriers to progress with our current set of techniques, and then develop new techniques that can overcome the barriers in the specific context of interest. Once these new techniques have been developed in the specific and limited situation, they can then be generalised and applied to make progress on a number of different problems.
In the initial period a team of excellent researchers has been assembled to address the problems in questions, and significant progress has been made on a number of the key objectives of the project. Indeed, the project has so far clearly performed beyond the initial aims and has been surprisingly successful in making progress.
The most significant result of the project so far has been the proof of the famous Duffin-Schaeffer conjecture from Diophantine approximation by the PI and his collaborator Dimitris Koukoulopoulos from Universite de Montreal. This problem was known as one of the most important questions in the area of metric Diophantine approximation, but had resisted proof for 75 years. In the original proposal for PRIMES some initial work by the PI and collaborators had indicated that a more combinatorial perspective from analytic number theory might be able to offer new insights into this problem. Fortunately this succeeded in exactly the way envisaged in the proposal – we solved a ‘model problem’ which had identified a key obstruction to previous approaches by developing a new technique inspired by graph theory. Having solved the model problem, the new technique could then be generalised to solve the original conjecture in full. This completely solved the very ambitious Objective 7, and opened up new methods for attacking many other related problems, which are currently being investigated by the PI, collaborators and other researchers in the field. This result has received a significant amount of positive feedback from the field.
Objective 8 of the original proposal on fractional parts of polynomials has been successfully answered; a new iterative technique was developed by the PI to use ideas from the geometry of numbers and additive combinatorics to make progress on an old problem of Diophantine approximation. A consequence of this new way of thinking allowed the PI to also make progress (in joint work with Thomas Bloom) on a well-known problem in additive combinatorics on sets with no square differences.
Objective 3 on the vertical distribution of zeros of the Riemann Zeta function was perhaps the most speculative but also compelling question of the original proposal. The PI has made significant progress on this objective with Kyle Pratt, where we overcome the key limitations of previous zero-detection techniques whenever there is a certain amount of horizontal rigidity in the distribution of zeros of the Riemann zeta function. Moreover, we have identified a new putative arrangement of zeros which we believe is the key obstruction to extending our ideas to an unconditional improvement.
Finally, a key result has been a number of improvements on our understanding of primes in arithmetic progressions. While the PI was working on Objective 6 (Diophantine approximation with primes), he realised that some of the new techniques being developed would have significant implications for the central question of primes in arithmetic progressions. This has now become a key new strand of the project PRIMES which was completely unanticipated.
The most significant result of the project so far has been the proof of the famous Duffin-Schaeffer conjecture from Diophantine approximation by the PI and his collaborator Dimitris Koukoulopoulos from Universite de Montreal. This problem was known as one of the most important questions in the area of metric Diophantine approximation, but had resisted proof for 75 years. In the original proposal for PRIMES some initial work by the PI and collaborators had indicated that a more combinatorial perspective from analytic number theory might be able to offer new insights into this problem. Fortunately this succeeded in exactly the way envisaged in the proposal – we solved a ‘model problem’ which had identified a key obstruction to previous approaches by developing a new technique inspired by graph theory. Having solved the model problem, the new technique could then be generalised to solve the original conjecture in full. This completely solved the very ambitious Objective 7, and opened up new methods for attacking many other related problems, which are currently being investigated by the PI, collaborators and other researchers in the field. This result has received a significant amount of positive feedback from the field.
Objective 8 of the original proposal on fractional parts of polynomials has been successfully answered; a new iterative technique was developed by the PI to use ideas from the geometry of numbers and additive combinatorics to make progress on an old problem of Diophantine approximation. A consequence of this new way of thinking allowed the PI to also make progress (in joint work with Thomas Bloom) on a well-known problem in additive combinatorics on sets with no square differences.
Objective 3 on the vertical distribution of zeros of the Riemann Zeta function was perhaps the most speculative but also compelling question of the original proposal. The PI has made significant progress on this objective with Kyle Pratt, where we overcome the key limitations of previous zero-detection techniques whenever there is a certain amount of horizontal rigidity in the distribution of zeros of the Riemann zeta function. Moreover, we have identified a new putative arrangement of zeros which we believe is the key obstruction to extending our ideas to an unconditional improvement.
Finally, a key result has been a number of improvements on our understanding of primes in arithmetic progressions. While the PI was working on Objective 6 (Diophantine approximation with primes), he realised that some of the new techniques being developed would have significant implications for the central question of primes in arithmetic progressions. This has now become a key new strand of the project PRIMES which was completely unanticipated.
So far the project has developed a number of completely new techniques which go significantly beyond the state of the art. This has already led to the resolution of long-standing open problems, but the underlying techniques have the potential to go much further. It is likely that for several strands of the project much more will be achieved than was originally hoped by the end of the project, and that the full impact of these techniques will require investigation beyond the end of PRIMES. Some of the ideas developed in PRIMES are now being absorbed and carried forward by the wider research community, and some entirely new and unanticipated objectives have arisen (most notably on primes in arithmetic progressions) which means that the final scope of PRIMES is likely to be wider than originally laid out in the proposal. It was stated that the project would remain flexible and follow opportunities as they arise - it is fortunate that this has been the case
Although work is still in-progress, good progress is being made on several of the other objectives of the original proposal where full results have not yet been established. New techniques and perspectives are being developed by the PI and his team on many related fronts, particularly to do with questions on primes in thin sets and limits of sieves (objectives 1 & 2). Now that the PI has assembled a full team of exceptional personnel, he is optimistic that in the second half of the project the strong positive momentum of the first half can be maintained.
Although work is still in-progress, good progress is being made on several of the other objectives of the original proposal where full results have not yet been established. New techniques and perspectives are being developed by the PI and his team on many related fronts, particularly to do with questions on primes in thin sets and limits of sieves (objectives 1 & 2). Now that the PI has assembled a full team of exceptional personnel, he is optimistic that in the second half of the project the strong positive momentum of the first half can be maintained.