@article{21778, abstract = {We prove that every 𝐿-bilipschitz mapping â„€ 2 → ℝ2 canbe extended to a đ¶(𝐿)-bilipschitz mapping ℝ2 → ℝ2,and we provide a polynomial upper bound for đ¶(𝐿).Moreover, we extend the result to every separated netin ℝ2 instead of â„€ 2, with the upper bound gaininga polynomial dependence on the separation and netconstants associated to the given separated net. Thisanswers an Oberwolfach question of Navas from 2015and is also a positive solution of the two-dimensionalform of a decades old open (in all dimensions at leasttwo) problem due to Alestalo Trotsenko and VĂ€isĂ€lĂ€.}, author = {Dymond, Michael and Kaluza, Vojtech}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {Wiley}, title = {{Planar bilipschitz extension from separated nets}}, doi = {10.1112/jlms.70540}, volume = {113}, year = {2026}, } @article{21002, abstract = {The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.}, author = {Browning, Timothy D}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{The Davenport–Heilbronn method: 80 years on}}, doi = {10.1112/jlms.70371}, volume = {113}, year = {2026}, } @article{19418, abstract = {The size-Ramsey number r^(H) of a graph H is the smallest number of edges a (host) graph G can have, such that for any red/blue colouring of G, there is a monochromatic copy of H in G. Recently, Conlon, Nenadov and Trujić showed that if H is a graph on n vertices and maximum degree three, then r^(H)=O(n8/5), improving upon the upper bound of n5/3+o(1) by Kohayakawa, Rödl, Schacht and SzemerĂ©di. In this paper we show that r^(H)≀n3/2+o(1). While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.}, author = {Draganić, Nemanja and Petrova, Kalina H}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, publisher = {Wiley}, title = {{Size‐Ramsey numbers of graphs with maximum degree three}}, doi = {10.1112/jlms.70116}, volume = {111}, year = {2025}, } @article{19554, abstract = {In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse ErdƑs–RĂ©nyi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp–Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.}, author = {Glasgow, Margalit and Kwan, Matthew Alan and Sah, Ashwin and Sawhney, Mehtaab}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {Wiley}, title = {{A central limit theorem for the matching number of a sparse random graph}}, doi = {10.1112/jlms.70101}, volume = {111}, year = {2025}, } @article{18490, abstract = {For large classes of even-dimensional Riemannian manifolds (Formula presented.), we construct and analyze conformally invariant random fields. These centered Gaussian fields (Formula presented.), called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: (Formula presented.). They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field (Formula presented.), we define the Liouville Quantum Gravity measure, a random measure on (Formula presented.), heuristically given as (Formula presented.) and rigorously obtained as almost sure weak limit of the right-hand side with (Formula presented.) replaced by suitable regular approximations (Formula presented.). In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on (Formula presented.) and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's (Formula presented.) -curvature: we give a rigorous meaning to the Polyakov–Liouville measure (Formula presented.) and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel (Formula presented.) rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds. }, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {5}, publisher = {London Mathematical Society}, title = {{Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension}}, doi = {10.1112/jlms.70003}, volume = {110}, year = {2024}, } @article{18583, abstract = {There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex v, at most half of the out-neighbours of v have the same colour as . As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every d, all but finitely many d-regular graphs have a partition into two non-empty parts such that for every vertex v, at least half of the neighbours of v lie in the same part as v. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for ErdƑs–RĂ©nyi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of ‘exceptional vertices’. Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a personality-changing scheme: we ‘forget’ certain information based on the state of a Markov chain, giving us more independence to work with.}, author = {Anastos, Michael and Cooley, Oliver and Kang, Mihyun and Kwan, Matthew Alan}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {6}, publisher = {Wiley}, title = {{Partitioning problems via random processes}}, doi = {10.1112/jlms.70010}, volume = {110}, year = {2024}, } @article{17447, abstract = {Let F be a diagonal cubic form over Z in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath‐Brown, we unconditionally extract a weighted count of certain special integral zeros of F in regions of diameter X - 8 . Heath‐Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general F.}, author = {Wang, Victor}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, publisher = {Wiley}, title = {{Special cubic zeros and the dual variety}}, doi = {10.1112/jlms.12975}, volume = {110}, year = {2024}, } @article{18173, abstract = {Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided char (Fq)>3. Under the same hypotheses, we also verify weak approximation.}, author = {Glas, Jakob}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {London Mathematical Society}, title = {{Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t)}}, doi = {10.1112/jlms.12991}, volume = {110}, year = {2024}, } @article{12214, abstract = {Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. }, author = {GehĂ©r, György PĂĄl and Titkos, TamĂĄs and Virosztek, Daniel}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, keywords = {General Mathematics}, number = {4}, pages = {3865--3894}, publisher = {Wiley}, title = {{The isometry group of Wasserstein spaces: The Hilbertian case}}, doi = {10.1112/jlms.12676}, volume = {106}, year = {2022}, } @article{10772, abstract = {We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov–Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic cohomology, confirming a prediction of homological mirror symmetry.}, author = {Arguez, Nuroemuer Huelya}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, pages = {343--411}, publisher = {London Mathematical Society}, title = {{Mirror symmetry for the Tate curve via tropical and log corals}}, doi = {10.1112/jlms.12515}, volume = {105}, year = {2022}, } @article{22160, abstract = {Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed and large with respect to , what is the minimum possible degree of a polynomial with such that has zeroes of multiplicity at least at all points in ? For , a classical theorem of Alon and FĂŒredi states that the minimum possible degree of such a polynomial equals . In this paper, we solve the problem for all , proving that the answer is . As an application, we improve a result of Clifton and Huang on configurations of hyperplanes in such that each point in is covered by at least hyperplanes, but the point is uncovered. Surprisingly, the proof of our result involves Catalan numbers and arguments from enumerative combinatorics.}, author = {Sauermann, Lisa and Wigderson, Yuval}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, pages = {2379--2402}, publisher = {Wiley}, title = {{Polynomials that vanish to high order on most of the hypercube}}, doi = {10.1112/jlms.12637}, volume = {106}, year = {2022}, } @article{9586, abstract = {Consider integers 𝑘,ℓ such that 0⩜ℓ⩜(𝑘2) . Given a large graph đș , what is the fraction of 𝑘 -vertex subsets of đș which span exactly ℓ edges? When đș is empty or complete, and ℓ is zero or (𝑘2) , this fraction can be exactly 1. On the other hand, if ℓ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich, and Tyomkyn who initiated the systematic study of this question and proposed several natural conjectures. Let ℓ∗=min{ℓ,(𝑘2)−ℓ} . Our main result is that for any 𝑘 and ℓ , the fraction of 𝑘 -vertex subsets that span ℓ edges is at most log𝑂(1)(ℓ∗/𝑘)√ 𝑘/ℓ∗, which is best-possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich, and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a ‘slice’ of the Boolean hypercube.}, author = {Kwan, Matthew Alan and Sudakov, Benny and Tran, Tuan}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, pages = {757--777}, publisher = {Wiley}, title = {{Anticoncentration for subgraph statistics}}, doi = {10.1112/jlms.12192}, volume = {99}, year = {2019}, } @article{261, abstract = {Let G = SL(2, R) ⋉R2 and Γ = SL(2, Z) ⋉Z2. Building on recent work of Strömbergsson, we prove a rate of equidistribution for the orbits of a certain one-dimensional unipotent flow of Γ\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of √n mod 1.}, author = {Browning, Timothy D and Vinogradov, Ilya}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, pages = {61 -- 84}, publisher = {Wiley}, title = {{Effective ratner theorem for SL (2, R) ⋉R2 and gaps in √n modulo 1}}, doi = {10.1112/jlms/jdw025}, volume = {94}, year = {2016}, }