Warwick-Oxbridge-Manchester-Bristol-London (WOMBL) 1-day meeting @ Oxford 2022
Following a 2-year hiatus, the eleventh 1-day meeting in additive combinatorics and analytic number theory will take place at the University of Oxford on
Monday, 4th April 2022.
The series is funded by an LMS Scheme 3 Grant (31623).
Programme | Registration | How to get here | Participants | Bristol 2014 | Oxford 2015 | Bristol 2015 | Oxford 2016 | Imperial 2016 | Warwick 2017 | Bristol 2018 | Oxford 2018 | Cambridge 2019 | Oxford 2019
Programme
11:00-11.15: Arrival and Welcome
11:15-12:00: Ofir Gorodetsky (Oxford)
Sums of two squares are strongly biased towards quadratic residues
| Abstract: | Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This has led Knapowski and Turán to conjecture in 1962 that pi(x;4,3) > pi(x;4,1) for almost all x, in natural density sense. Here pi(x;4,a) counts primes of the form a mod 4 up to x. This conjecture was refuted eventually (under GRH) in the 90s, by Rubinstein and Sarnak and by Kaczorowski.
We investigate a similar bias for sums of two squares, and show that (under GRH) a version of the conjecture of Knapowski and Turán is true for them (with the modulus 4 replaced e.g. by 3, or any odd modulus). Specifically, we find that sums of two squares are biased towards quadratic residues in natural density sense. In the talk we shall explain the origin of the bias in both the case of primes and the case of sums of squares, and sketch the proof of the result. |
14:00-14:45: Sean Prendiville (Lancaster)
Adapting the circle method for colourings
| Abstract: | If each integer is coloured red, blue or green, how many solutions to x-y=z^2 have x,y,z all the same colour? We discuss how to adapt the Hardy-Littlewood circle method to yield a lower bound in problems of this flavour.
|
Additive problems with almost prime squares
| Abstract: | We study sums of one prime and two squares of almost-primes, that is to say, integers whose number of prime factors is less than some absolute fixed bound. We employ three main tools: an explicit formula for the number of representations of an integer by a binary quadratic form; results on additive problems with cusp forms which derive ultimately from a trace formula; and a lower-bound sieve. This is joint work with Valentin Blomer and Junxian Li.
|
16:00-16:45: Marina Iliopoulou (Birmingham)
Some remarks on the Mizohata-Takeuchi conjecture
| Abstract: | This is a conjecture on weighted estimates for the classical Fourier extension operators of harmonic analysis. In particular, let E be the extension operator associated to some surface, and f be a function on that surface. If we 'erase' part of Ef, how well can we control the 2-norm of the remaining piece? The Mizohata-Takeuchi conjecture claims some remarkable control on this quantity, involving the X-ray transform of the part of the support of Ef that we kept. In this talk we will discuss the history of the problem, and will describe a new perspective that modestly improves our knowledge (for a certain class of weights). This is joint work with A. Carbery.
|
A proof of the Erdős primitive set conjecture
| Abstract: | A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series of 1/(n*log n), ranging over n in A, is uniformly bounded over all choices of primitive sets A. In 1988 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős' conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.
|
All talks will take place in Lecture Room 3 in the Andrew Wiles Building, home of the Mathematical Institute at Oxford.
Registration
To register please fill in this google form by 30th March (you can edit your responses later, or simply email me). It is particularly important that you fill in this form ahead of time if you wish to join us for dinner. The participants list below will be updated accordingly.
How to get here
Detailed travel instructions are available from the webpage of the Oxford Mathematical Institute. For most participants, the best way to get to Oxford will be by train. If you are hoping to claim reimbursement of your expenses through WOMBL, please book your tickets as far in advance as possible to keep costs down.
Participants
Faustin Adiceam, Manchester
Daniel Altman, Oxford
Nuno Arala, Warwich
Bryan Birch, Oxford
Jonathan Bober, Bristol
Peter Bradshaw, Bristol
Jonathan Chapman, Bristol
Hou Tin Chau, Bristol
Sam Chow, Warwick
Michael Curran, Oxford
Jared Duker Lichtman, Oxford
Natalie Evans, KCL
Valeriia Gladkova, Cambridge
Ofir Gorodetsky, Oxford
*Ben Green, Oxford
Lasse Grimmelt, Oxford
*Adam Harper, Warwick
Marina Iliopoulou, Birmingham
Yifan Jing, Oxford
Oleksiy Klurman, Bristol
Valeriya Kovaleva, Oxford
Dylan King, Bristol
Samuel Mansfield, Bristol
James Maynard, Oxford
Oliver McGrath, Oxford
Yago Moreno Alonso, Bristol
Simon Myerson, Warwich
Jonathan Passant, Bristol
Kyle Pratt, Oxford
*Sean Prendiville, Lancaster
Steven Roberston, Manchester
Tom Sanders, Oxford
Andrei Seymour-Howell, Bristol
Victor Shirandami, Manchester
Ioannis Tsokanos, Manchester
Leo Versteegen, Cambridge
*Aled Walker, KCL
Adam Wells, Oxford
* Organisers