Warwick-Oxbridge-Manchester-Bristol-London (WOMBL) 1-day meeting @ KCL 2022
The twelfth 1-day meeting in additive combinatorics and analytic number theory will take place at King's College London on
Tuesday, 13th September 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 | Oxford 2022
Programme
10:45-11.00: Arrival
11:00-11.05: Welcome
11:05-11:45: Akshat Mudgal (University of Oxford)
Large additive and multiplicative Sidon sets and where to find them
| Abstract: | Given a natural number s, we say that a finite set X of integers is an additive B_s[1] set if for any integer n, there is at most one distinct solution to the equation n = x_1 + ... + x_s, with x_1, ..., x_s lying in X, where we consider two solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative B_s[1] set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. For instance, even in the case s=2, wherein such sets are referred to as Sidon sets, characterising the largest additive B_{s}[1] set in {1, 2, ..., N} remains a major open problem in the area. In this talk, we consider this problem from an arithmetic combinatorial perspective, and so, we show that for every natural number s and for every finite set A of integers, writing B and C to be the largest additive and multiplicative B_s[1] sets in A, we have that max{ |B| , |C| } >> |A|^(c/s), where c >> (log log s)^(1/2 - o(1)). This is joint work with Yifan Jing. |
The n-queens problem
| Abstract: | The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^(-3))^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^(-3))^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we'll focus on the algebraic aspects of the toroidal problem and the proof. This is joint work with Peter Keevash. |
14:30-15:10: Oleksiy Klurman (University of Bristol)
Automatic semigroups
| Abstract: | The main goal of the talk is to discuss recent progress in our understanding of the following general phenomena: how does multiplicative structure correlate with "automaticity"? This is based on a joint work with J. Konieczny.
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Sumsets in Positive Density Sets
| Abstract: | In recent work with Kra, Moreira and Richter it was shown using ergodic theory that positive density sets always contain the sum of any finite number of infinite sets. In this talk I will outline our proof, focusing on the case of two sumsets and introducing the necessary ergodic theory as we go.
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16:30-17:10: Barnabás Szabó (University of Warwick)
On the existence of products of primes inside arithmetic progressions
| Abstract: | One of the most important results in 20th century number theory is Linnik's theorem, which states that if q is a large modulus, then each invertible residue class mod q contains a prime less than q^L, where L is some absolute constant. In this talk we investigate similar results concerning E_k numbers for small k, where an E_k number is a product of exactly k-many primes. In particular, we show that each invertible residue class mod q contains a product of three primes, where each prime is less than q^(6/5+epsilon).
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High-rank minors for high-rank tensors
| Abstract: | It is a standard result of linear algebra that every matrix with rank k has a k x k minor with rank k. We will first show that this property extends to the tensor rank in the optimistic way: if T is a tensor with tensor rank k, then T has a minor with size k and tensor rank k. In the case of more general tensors, the result extends asymptotically in the sense that for any fixed order d greater than or equal to 2 and for every notion of rank on order-d tensors in a class of notions which in particular contains the tensor rank, the slice rank, and the partition rank, there exist functions F and G such that if T is an order-d tensor with rank at least G(l) then T has a minor with size F(l) and rank at least l. During the talk we will prove this result in the special case of the slice rank of order-3 tensors, and then discuss some other applications of the proof techniques.
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All talks will take place in K6.29 (Old Anatomy Lecture Theatre, King's Building), on the Strand Campus of King's College London.
Registration
To register please fill in this google form by Friday, 9th September (you can edit your responses later, or simply email me). It is particularly important on this occasion that you fill in this form by the deadline, as KCL requires a list of registered participants and may deny entry to anyone not listed. It is also important that you let us know, via the form, if you wish to join us for dinner. The participants list below will be updated accordingly.
How to get here
Entrance to the meeting venue is from the Strand, into the 1970s concrete building opposite St Mary-Le-Strand church. Being a university in central London, KCL are quite security heavy; upon arrival, all participants will have to register as an 'Events visitor' at the Security desk in Reception. The local organiser is working on a way to streamline this process as much as possible, but it may take a couple of minutes before you are allowed to enter the rest of the building. Please plan your journey accordingly.
The meeting room is on the 6th floor of the King's Building. Although connected by certain passageways, the King's building is a different building to the Strand building you first entered, and it is quite easy to get lost. Avoid the 4 lifts that are just beyond the entrance gates as these are for the Strand building. Instead, continue walking to the left of the lifts, going straight on through the corridor where the message 'You are now entering the King's Building' is marked in red on the floor. Continue walking until you come to two lifts on your left (after about 100 metres). Take these lifts to the 6th floor, and follow signs on that floor to K6.29.
The nearest stations are Temple (3 mins walk), Covent Garden (9 mins walk), Charing Cross (10 mins walk), Holborn (12 mins walk), City Thameslink (13 mins walk), Blackfriars (13 mins walk), Waterloo (16 mins walk).
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.
Participants
Ardavan Afshar, KTH
Daniel Altman, Oxford
Kirsti Biggs, Uppsala
Thomas Bloom, Oxford
Candida Bowtell, Birmingham/Warwick
Jonathan Chapman, Bristol
Hou Tin Chau, Bristol
Sam Chow, Warwick
Nora Frankl, Alfred Renyi Institute
*Ben Green, Oxford
Haocong Guo, Bristol
*Adam Harper, Warwick
Philip Holdridge, Warwick
Yifan Jing, Oxford
Nikoleta Kalaydzhieva, UCL
Thomas Karam, Cambridge
Oleksiy Klurman, Bristol
Joanna Lada, LSE
Steve Lester, KCL
Albert Lopez Bruch, Oxford
Samuel Mansfield, Bristol
Paolo Marimon, Bristol
Akshat Mudgal, Oxford
Yani Pehova, LSE
*Sean Prendiville, Lancaster
Jenny Roberts, Bristol
Donald Robertson, Manchester
Steven Robertson, Manchester
Misha Rudnev, Bristol
Tom Sanders, Oxford
Victor Shirandami, Manchester
Józef Skokan, LSE
Stelios Stylianou, Bristol
Barnabás Szábo, Warwick
*Matthew Tointon, Bristol
Ioannis Tsokanos, Manchester
Leo Versteegen, Cambridge
Jackie Voros, Bristol
*Aled Walker, KCL
*Julia Wolf, Cambridge
Khalid Younis, Warwick
* Organisers