Warwick-Oxbridge-Manchester-Bristol-London (WOMBL) 1-day meeting @ Cambridge 2024
|We consider two questions. First, extending a question raised by Alon and
Krivelevich (2021): what is the minimum n such that in every labeling of the edges of the complete directed graph on n vertices by elements of Z_p^k (with p prime) there is a directed cycle whose edge labels sum to 0? Second: what is the maximum size of a multiset S with elements in Z_p^k, such that the set of subset sums of every proper multisubset S' of S is strictly smaller than that of S? I will explain the connection between these two questions, and describe our progress on both questions. This is joint work with Natasha Morrison.
|A conjecture of Marton, often known as the polynomial Freiman-Ruzsa
conjecture, states that if A is a subset of F_p^n such that |A+A| ≤ C|A|, then A is contained in a union of K cosets of a subgroup H of F_p^n of size at most |A|, where K has a polynomial dependence on C for any given p. I shall discuss a recent solution of this conjecture, focusing on the case p=2 where the argument is somewhat simpler. This is joint work with Ben Green, Freddie Manners, and Terence Tao.
| A harder version of Erdős' famous distinct-distances problem asks about
the structure of point sets in the plane that have few distances. In particular, Erdős asked if such near-optimal sets have a `lattice structure' or, given this appears very hard, if a polynomial proportion of the point set is colinear. We consider a condition related to few distances, sets with few congruent triangles, and show such sets contain either a polynomially-rich line or a positive proportion of the set on a circle. There will also be some discussion of energy results in this setting and how our methods fail for point sets with few distances. This is joint work with Sam Mansfield.
|The problem of estimating the number of common solutions to a system of
polynomial equations in an expanding box, at the interface of Diophantine Geometry and Analytic Number Theory, has a rich and fascinating history, having stimulated the development of many forms of the so-called Circle Method. We approach this problem for a new class of systems of equations - certain pairs of quadratic forms in at least 10 variables - by combining a form of the Circle Method and some tools from the theory of cusp forms.
|The discretised ring theorem, conjectured by Katz and Tao, is the fractal
version of the Erdős-Szemerédi sum-product problem. It roughly asserts that if a subset A of the reals has dimension 0 < s < 1 then at least one the sum-set A + A or the product set AA must have dimension larger than s + c, where c is a constant which depends only on s First qualitatively proved by Bourgain in 2003, more recent efforts by numerous mathematicians have given strong quantitative bounds for c. The aim of this talk is to show that one can use ideas from information theory to give a strong quantitative bound for c. The content of this talk is based on joint work with András Máthé.
|The Liouville function λ(n) is defined to be +1 if n is a product of an even
number of primes, and -1 otherwise. The statistical behaviour of λ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of +1 and -1. For example, the two-point Chowla conjecture predicts that the average of λ(n)λ(n+1) over n < x tends to zero as x goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.
All talks will take place in Meeting Room 2 at the Centre for Mathematical Sciences, Wilberforce Road, University of Cambridge.
Churchill College, which is a short walk from the CMS, have kindly agreed to host us for lunch. We will be leaving as a group promptly after the last morning talk. Please be sure to indicate your interest in lunch on the registration form below.
To register please fill in the registration form by the end of Monday, 8th January. It is particularly important that you let us know in advance, via the form, if you wish to join us for lunch. The participants list below will be updated accordingly in due course (but usually not immediately).
If you need to adjust the details of your registration after you have completed the form, please simply fill it in again - we will disregard all but your latest submission.
Detailed travel instructions are available from the webpage of the Centre for Mathematical Sciences. For most participants, the best way to get to Cambridge will be by train. From the train station you can either take a taxi to the Isaac Newton Institute on Clarkson Road (it usually costs under £10 and takes around 15 minutes), or take the UNIversal bus, which can be used by anyone (it costs £2.20 for non-University members, and takes around 20 minutes, timetable here). Entrance to the CMS is via the Faulkes Gatehouse, on the footpath between Clarkson and Madingley Road.
Please remember that 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.
Ryan Alweiss, University of Cambridge
Nuno Arala, University of Warwick
Christopher Atherfold , University of Bristol
Jonas Bayer, University of Cambridge
Benjamin Bedert, University of Oxford
Veronica Bitonti, UCL
Thomas Bloom, University of Oxford
Ned Carmichael, KCL
Davi Castro-Silva, University of Cambridge
Bruno Cavalar, University of Warwick
Jonathan Chapman, University of Bristol
Hou Tin Chau, University of Bristol
Joe Deakin, University of Cambridge
Sean Eberhard, Queen's University Belfast
David Ellis, University of Bristol
Anshula Gandhi, University of Cambridge
Yoav Gath, University of Cambridge
Val Gladkova, Cambridge University
Ofir Gorodetsky, University of Oxford
Timothy Gowers, Collège de France and University of Cambridge
*Ben Green, University of Oxford
Lasse Grimmelt, University of Oxford
Tom Gur, University of Cambridge
Merlin Haith Rowlatt, University of Bristol
Seth Hardy, University of Warwick
*Adam Harper, University of Warwick
Philip Holdridge, University of Warwick
Maria Ivan, University of Cambridge
Sidharth Jaggi, University of Bristol
Oliver Janzer, University of Cambridge
Yifan Jing, University of Oxford
Imre Leader, University of Cambridge
Shoham Letzter, UCL
Anna Luchnikov, University of Cambridge
Sophie Maclean , KCL
Samuel Mansfield, University of Bristol
Sarah Martin, University of Bristol
James Maynard, University of Oxford
Oliver McGrath, KCL
Yago Moreno Alonso, University of Bristol
Akshat Mudgal, University of Oxford
William O'Regan, University of Warwick
Jonathan Passant, University of Bristol
Cédric Pilatte, University of Oxford
*Sean Prendiville, Lancaster University
Ninad Rajgopal, University of Cambridge
Steven Robertson, University of Manchester
Misha Rudnev, University of Bristol
Besfort Shala, University of Bristol
Victor Shirandami, University of Manchester
Julia Stadlmann, University of Oxford
Barnabas Szabo, University of Warwick
Anand Tadipatri, University of Cambridge
Kate Thomas, University of Oxford
*Matthew Tointon, University of Bristol
Fred Tyrrell, University of Bristol
Peter Varju, University of Cambridge
Santiago Vazquez Saez, KCL
Leo Versteegen, University of Cambridge
*Aled Walker, KCL
*Julia Wolf, University of Cambridge
Khalid Younis, University of Warwick