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Similar Journals
 Mathematical Proceedings of the Cambridge Philosophical SocietyJournal Prestige (SJR): 1.086 Citation Impact (citeScore): 1Number of Followers: 2      Subscription journal ISSN (Print) 0305-0041 - ISSN (Online) 1469-8064 Published by Cambridge University Press  [395 journals]
• PSP volume 170 issue 1 Cover and Front matter
• PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004121000050
Issue No: Vol. 170, No. 1 (2021)

• PSP volume 170 issue 1 Cover and Back matter
• PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004121000062
Issue No: Vol. 170, No. 1 (2021)

• Large prime factors on short intervals
• Authors: JORI MERIKOSKI
Pages: 1 - 50
Abstract: We show that for all large enough x the interval [x, x + x1/2 log1.39 x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000306
Issue No: Vol. 170, No. 1 (2021)

• Nearly free curves and arrangements: a vector bundle point of view
• Authors: SIMONE MARCHESI; JEAN VALLÈS
Pages: 51 - 74
Abstract: Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000318
Issue No: Vol. 170, No. 1 (2021)

• Vinogradov’s three primes theorem with primes having given primitive
roots
• Authors: C. FREI; P. KOYMANS, E. SOFOS
Pages: 75 - 110
Abstract: The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000331
Issue No: Vol. 170, No. 1 (2021)

• Polylogarithmic bounds in the nilpotent Freiman theorem
• Authors: MATTHEW C. H. TOINTON
Pages: 111 - 127
Abstract: We show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup $H \subset {A^{{K^{{O_s}(1)}}}}$ modulo which ${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$ contains a nilprogression of rank at most ${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$ and size at least $\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000343
Issue No: Vol. 170, No. 1 (2021)

• GL(2,+ℝ)+geometry&rft.title=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.issn=0305-0041&rft.date=2021&rft.volume=170&rft.spage=129&rft.epage=154&rft.aulast=FERAPONTOV&rft.aufirst=EVGENY&rft.au=EVGENY+FERAPONTOV&rft.au=BORIS+KRUGLIKOV&rft_id=info:doi/10.1017/S0305004119000355">Dispersionless integrable hierarchies and        class="italic">GL(2, ℝ) geometry
• Authors: EVGENY FERAPONTOV; BORIS KRUGLIKOV
Pages: 129 - 154
Abstract: Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000355
Issue No: Vol. 170, No. 1 (2021)

• Concordance to links with an unknotted component
• Authors: CHRISTOPHER W. DAVIS; JUNGHWAN PARK
Pages: 155 - 160
Abstract: We construct links of arbitrarily many components each component of which is slice and yet are not concordant to any link with even one unknotted component. The only tool we use comes from the Alexander modules.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000367
Issue No: Vol. 170, No. 1 (2021)

• Extensions of tensor categories by finite group fusion categories
• Authors: SONIA NATALE
Pages: 161 - 189
Abstract: We study exact sequences of finite tensor categories of the form Rep G → ð’ž → ð’Ÿ, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on ð’Ÿ such that ð’ž is equivalent to a certain associated crossed extension ð’Ÿ(G,Γ) of ð’Ÿ. Dually, we show that an exact sequence of finite tensor categories VecG → ð’ž → ð’Ÿ induces an Aut(G)-grading on ð’ž whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of ð’Ÿ. As an application we prove that such extensions ð’ž of ð’Ÿ are weakly group-theoretical fusion categories if and only if ð’Ÿ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S0305004119000434
Issue No: Vol. 170, No. 1 (2021)

• L-functions&rft.title=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.issn=0305-0041&rft.date=2021&rft.volume=170&rft.spage=191&rft.epage=219&rft.aulast=HEAP&rft.aufirst=WINSTON&rft.au=WINSTON+HEAP&rft_id=info:doi/10.1017/S030500411900046X">Moments of the Dedekind zeta function and other non-primitive L-functions
• Authors: WINSTON HEAP
Pages: 191 - 219
Abstract: We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.
PubDate: 2021-01-01T00:00:00.000Z
DOI: 10.1017/S030500411900046X
Issue No: Vol. 170, No. 1 (2021)

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