# Realizability and tameness of fusion systems

Proceedings of the London Mathematical Society

Published On 2023/12

A saturated fusion system over a finite p$p$‐group S$S$ is a category whose objects are the subgroups of S$S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over S$S$ is realized by a finite group G$G$ if S$S$ is a Sylow p$p$‐subgroup of G$G$ and morphisms in the category are those induced by conjugation in G$G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the …

Journal

Proceedings of the London Mathematical Society

Published On

2023/12

Volume

127

Issue

6

Page

1816-1864

## Authors

#### Jesper Møller

##### Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

#### Albert Ruiz Cirera

##### Universidad Autónoma de Barcelona

Position

Associate Professor of Mathematics

H-Index(all)

8

H-Index(since 2020)

5

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

algebraic topology

University Profile Page

### Other Articles from authors

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.09525

##### Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits of a random variable with a probability density function is considered. Under the weak condition that is almost everywhere lower semi-continuous, a coupling between and a non-negative integer-valued random variable is established so that have an interpretation as the ``sufficient digits'', since the distribution of conditioned on does not depend on . Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, becomes a Markov chain of a certain order . If then are IID with a known distribution. When and the Markov chain is uniformly geometric ergodic, a coupling is established between and a random time so that the chain after time is stationary and follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base- expansions, generalized L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

*2024/4/15*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.08387

##### The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2024/4/12*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

*2024/3*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2312.09652

##### The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2023/12/15*

Jesper Møller

Aalborg Universitet

ACM Transactions on Spatial Algorithms and Systems

##### Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions …

*2023/11/21*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel …

*2023/6/1*

Albert Ruiz Cirera

Universidad Autónoma de Barcelona

Transboundary and Emerging Diseases

##### Experimental Inoculation of Porcine Circovirus 3 (PCV-3) in Pregnant Gilts Causes PCV-3-Associated Lesions in Newborn Piglets that Persist until Weaning

Porcine circovirus 3 (PCV-3) has been detected in cases of reproductive failure but the pathogenesis of such infection is poorly understood. So far, experimental PCV-3 inoculations have been performed only in piglets. Therefore, through the experimental inoculation of pregnant gilts at two different time points (second and last third of gestation), this study aimed to evaluate the outcome of PCV-3 infection in dams and their offspring until weaning age. Two weeks postinoculation, all gilts became viremic and the infection lasted until the end of study. Farrowing occurred naturally, without evidence of reproductive disorders, and piglets showed no significant clinical signs from farrowing to weaning (21 day-old). However, majority of the delivered piglets were viremic, mostly until weaning age. Both newborn and weaned pigs showed different degrees of systemic, lymphohistiocytic arteritis and periarteritis. Lesions were more severe in the piglets infected during the second third of gestation and worsened at weaning. Additionally, PCV-3 detection in nervous and cardiac tissue and development of histopathological lesions in these tissues were gestational dependent, as only occurred in piglets infected at second third of pregnancy. Piglets with lesions raised to weaning age had less body weight than those without them. This study represents the first description of a PCV-3 experimental infection in pregnant gilts, which resulted in transplacental infection, histological lesions in piglets mimicking those of natural occurring disease, and lesser body weight in piglets with vascular lesions at weaning age. Obtained results allowed proposing a potential …

*2023/10/20*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

*2023/3*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2212.08402

##### Cox processes driven by transformed Gaussian processes on linear networks

There is a lack of point process models on linear networks. For an arbitrary linear network, we use isotropic covariance functions with respect to the geodesic metric or the resistance metric to construct new models for isotropic Gaussian processes and hence new models for various Cox processes with isotropic pair correlation functions. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

*2022/12/16*

Jesper Møller

Aalborg Universitet

Stat

##### Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

*2022/12*

Jesper Møller

Aalborg Universitet

International Statistical Review

##### Should we condition on the number of points when modelling spatial point patterns?

We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition …

*2022/12*

Jesper Møller

Aalborg Universitet

Journal of Applied Probability

##### Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

*2022/12*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step …

*2022/10/1*

Jesper Møller

Aalborg Universitet

Translational psychiatry

##### Layer III pyramidal cells in the prefrontal cortex reveal morphological changes in subjects with depression, schizophrenia, and suicide

Brodmann Area 46 (BA46) has long been regarded as a hotspot of disease pathology in individuals with schizophrenia (SCH) and major depressive disorder (MDD). Pyramidal neurons in layer III of the Brodmann Area 46 (BA46) project to other cortical regions and play a fundamental role in corticocortical and thalamocortical circuits. The AutoCUTS-LM pipeline was used to study the 3-dimensional structural morphology and spatial organization of pyramidal cells. Using quantitative light microscopy, we used stereology to calculate the entire volume of layer III in BA46 and the total number and density of pyramidal cells. Volume tensors estimated by the planar rotator quantified the volume, shape, and nucleus displacement of pyramidal cells. All of these assessments were carried out in four groups of subjects: controls (C, n = 10), SCH (n = 10), MDD (n = 8), and suicide subjects with a history of depression (SU …

*2022/9/5*

Jesper Møller

Aalborg Universitet

Graphs and Combinatorics

##### Equivariant Euler characteristics of symplectic buildings

We compute the equivariant Euler characteristics of the buildings for the symplectic groups over finite fields.

*2022/6*

Albert Ruiz Cirera

Universidad Autónoma de Barcelona

Pathogens

##### Exploratory Study of the Frequency of Detection and Tissue Distribution of Porcine Circovirus 3 (PCV-3) in Pig Fetuses at Different Gestational Ages

Porcine circovirus 3 (PCV-3) has been associated with several pig diseases. Despite the pathogenicity of this virus has not been completely clarified, reproductive disorders are consistently associated with its infection. The aim of the present work was to analyze the presence of PCV-3 DNA in tissues from pig fetuses from different gestational timepoints. The fetuses were obtained either from farms with no reproductive problems (NRP, n = 249; all of them from the last third of gestation) or from a slaughterhouse (S, n = 51; 49 of the second-third of gestation and 2 from the third one). Tissues collected included brain, heart, lung, kidney, and/or spleen. Overall, the frequency of detection of PCV-3 was significantly higher in fetuses from the last third of the gestation (69/251, 27.5%) when compared to those from the second-third (5/49, 10.2%), although the viral loads were not significantly different. Moreover, the frequency of detection in NRP fetuses (69/249, 27.7%) was significantly higher than in S ones (5/51, 9.8%). Furthermore, PCV-3 DNA was detected in all tissue types analyzed. In conclusion, the present study demonstrates a higher frequency of PCV-3 DNA detection in fetuses from late periods of the gestation and highlights wide organ distributions of the virus in pig fetuses.

*2022/1/20*

Jesper Møller

Aalborg Universitet

Journal of Computational and Graphical Statistics

##### MCMC computations for Bayesian mixture models using repulsive point processes

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well-separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the “cluster centers” is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, we use an ancillary variable method, which eliminates the problem of having intractable …

*2022/4/3*

Jesper Møller

Aalborg Universitet

Scandinavian Journal of Statistics

##### Approximate Bayesian inference for a spatial point process model exhibiting regularity and random aggregation

In this article, we propose a doubly stochastic spatial point process model with both aggregation and repulsion. This model combines the ideas behind Strauss processes and log Gaussian Cox processes. The likelihood for this model is not expressible in closed form but it is easy to simulate realizations under the model. We therefore explain how to use approximate Bayesian computation (ABC) to carry out statistical inference for this model. We suggest a method for model validation based on posterior predictions and global envelopes. We illustrate the ABC procedure and model validation approach using both simulated point patterns and a real data example.

*2022/3*

### Other articles from Proceedings of the London Mathematical Society journal

Elisa Postinghel

Loughborough University

Proceedings of the London Mathematical Society

##### Waring identifiability for powers of forms via degenerations

We discuss an approach to the secant non‐defectivity of the varieties parametrising k$k$th powers of forms of degree d$d$. It employs a Terracini‐type argument along with certain degeneration arguments, some of which are based on toric geometry. This implies a result on the identifiability of the Waring decompositions of general forms of degree kd as a sum of k$k$th powers of degree d$d$ forms, for which an upper bound on the Waring rank was proposed by Fröberg, Ottaviani and Shapiro.

*2024/1*

Alex Fink

Queen Mary University of London

Proceedings of the London Mathematical Society

##### Signed permutohedra, delta‐matroids, and beyond

We establish a connection between the algebraic geometry of the type B$B$ permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type B$B$ generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases.

*2024/3*

Anish Ghosh

Tata Institute of Fundamental Research

Proceedings of the London Mathematical Society

##### Diophantine approximation, large intersections and geodesics in negative curvature

In this paper, we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník–Besicovitch type theorem in Diophantine approximation. Our framework allows manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.

*2024/2*

Andrew Snowden

University of Michigan

Proceedings of the London Mathematical Society

##### The spectrum of a twisted commutative algebra

A twisted commutative algebra is (for us) a commutative Q$\mathbf {Q}$‐algebra equipped with an action of the infinite general linear group. In such algebras, the “GL$\mathbf {GL}$‐prime” ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct GL$\mathbf {GL}$‐primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish GL$\mathbf {GL}$‐primes. This yields an effective method for analyzing GL$\mathbf {GL}$‐primes.

*2024/1*

Yilin Wang

Massachusetts Institute of Technology

Proceedings of the London Mathematical Society

##### The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere C∖{0}$\mathbb {C} \setminus \lbrace 0\rbrace$ using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure ρ$\rho$ has finite Loewner–Kufarev energy, defined by S(ρ)=12∫∫S1×Rνt′(θ)2dθdt$$\begin{equation*} \hspace*{58pt}S(\rho) = \frac{1}{2}\iint\nolimits _{S^1 \times \mathbb {R}} \nu _t^{\prime }(\theta)^2 \, d \theta d t \end{equation*}$$whenever ρ$\rho$ is of the form νt(θ)2dθdt$\nu _t(\theta)^2 d \theta d t$, and set to ∞$\infty$ otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has …

*2024/2*

Hunter Spink

Harvard University

Proceedings of the London Mathematical Society

##### Signed permutohedra, delta‐matroids, and beyond

We establish a connection between the algebraic geometry of the type B$B$ permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type B$B$ generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases.

*2024/3*

Christopher Eur

Stanford University

Proceedings of the London Mathematical Society

##### Signed permutohedra, delta‐matroids, and beyond

We establish a connection between the algebraic geometry of the type B$B$ permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type B$B$ generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases.

*2024/3*

Ian Wanless

Monash University

Proceedings of the London Mathematical Society

##### Row‐Hamiltonian Latin squares and falconer varieties

A Latin square is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square L$L$ is row‐Hamiltonian if the permutation induced by each pair of distinct rows of L$L$ is a full cycle permutation. Row‐Hamiltonian Latin squares are equivalent to perfect 1‐factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row‐Hamiltonian and also achieve precisely one of the related properties of being column‐Hamiltonian or symbol‐Hamiltonian. This family allows us to construct non‐trivial, anti‐associative, isotopically L$L$‐closed loop varieties, solving an open problem posed by Falconer in 1970.

*2024/1*

Mattias Jonsson

University of Michigan-Dearborn

Proceedings of the London Mathematical Society

##### Birational maps with transcendental dynamical degree

We give examples of birational selfmaps of Pd,d⩾3$\mathbb {P}^d, d \geqslant 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.

*2024/1*

Fredrik Johansson Viklund

Kungliga Tekniska högskolan

Proceedings of the London Mathematical Society

##### The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere C∖{0}$\mathbb {C} \setminus \lbrace 0\rbrace$ using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure ρ$\rho$ has finite Loewner–Kufarev energy, defined by S(ρ)=12∫∫S1×Rνt′(θ)2dθdt$$\begin{equation*} \hspace*{58pt}S(\rho) = \frac{1}{2}\iint\nolimits _{S^1 \times \mathbb {R}} \nu _t^{\prime }(\theta)^2 \, d \theta d t \end{equation*}$$whenever ρ$\rho$ is of the form νt(θ)2dθdt$\nu _t(\theta)^2 d \theta d t$, and set to ∞$\infty$ otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has …

*2024/2*

Jeffrey Diller

University of Notre Dame

Proceedings of the London Mathematical Society

##### Birational maps with transcendental dynamical degree

We give examples of birational selfmaps of Pd,d⩾3$\mathbb {P}^d, d \geqslant 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.

*2024/1*

Jason P. Bell

University of Waterloo

Proceedings of the London Mathematical Society

##### Birational maps with transcendental dynamical degree

We give examples of birational selfmaps of Pd,d⩾3$\mathbb {P}^d, d \geqslant 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.

*2024/1*

Nader Masmoudi

New York University

Proceedings of the London Mathematical Society

##### Tollmien–Schlichting waves in the subsonic regime

The Tollmien–Schlichting (T‐S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T‐S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T‐S waves of temporal mode for certain boundary layer profiles with Mach number m<13$m<\frac{1}{\sqrt 3}$. In this paper, we construct the T‐S waves of both temporal mode and spatial mode to the linearized compressible Navier–Stokes system around the boundary layer flow in the whole subsonic regime m<1$m<1$, including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi‐incompressible and quasi‐compressible systems, with a key ingredient being the solution of an Orr–Sommerfeld‐type equation using a new Airy–Airy …

*2024/3*

Mark F. Demers

Fairfield University

Proceedings of the London Mathematical Society

##### Rates of mixing for the measure of maximal entropy of dispersing billiard maps

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate.

*2024/1*

Gwyn Bellamy

University of Glasgow

Proceedings of the London Mathematical Society

##### A new family of isolated symplectic singularities with trivial local fundamental group

We construct a new infinite family of four‐dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of C4$\mathbb {C}^4$ by the dihedral group of order 2d$2d$, (2) as singular points of Calogero–Moser spaces associated with dihedral groups of order 2d$2d$ at equal parameters, and (3) as singularities of a certain Slodowy slice in the d$d$‐fold cover of the nilpotent cone in sld${\mathfrak {s}}{\mathfrak {l}}_d$.

*2023/5*

Nathanael Berestycki

Universität Wien

Proceedings of the London Mathematical Society

##### Multiplicative chaos of the Brownian loop soup

We construct a measure on the thick points of a Brownian loop soup in a bounded domain D$D$ of the plane with given intensity θ>0$\theta >0$, which is formally obtained by exponentiating the square root of its occupation field. The measure is constructed via a regularisation procedure, in which loops are killed at a fix rate, allowing us to make use of the Brownian multiplicative chaos measures previously considered in Aïdékon et al. (Ann. Probab. 48 (2020), no. 4, 1785–1825), Bass et al. (Ann. Probab. 22 (1994), no. 2, 566–625) and Jego (Ann. Probab. 48 (2020), no. 4, 1597–1643), or via a discrete loop soup approximation. At the critical intensity θ=1/2$\theta = 1/2$, it is shown that this measure coincides with the hyperbolic cosine of the Gaussian free field, which is closely related to Liouville measure. This allows us to draw several conclusions which elucidate connections between Brownian multiplicative …

*2023/4*

Nir Lev

Bar-Ilan University

Proceedings of the London Mathematical Society

##### Functions tiling simultaneously with two arithmetic progressions

We consider measurable functions f$f$ on R$\mathbb {R}$ that tile simultaneously by two arithmetic progressions αZ$\alpha \mathbb {Z}$ and βZ$\beta \mathbb {Z}$ at respective tiling levels p$p$ and q$q$. We are interested in two main questions: what are the possible values of the tiling levels p,q$p,q$, and what is the least possible measure of the support of f$f$? We obtain sharp results which show that the answers depend on arithmetic properties of α,β$\alpha , \beta$ and p,q$p,q$, and in particular, on whether the numbers α,β$\alpha , \beta$ are rationally independent or not.

*2023/12*

Ratan Kr. Giri

Technion - Israel Institute of Technology

Proceedings of the London Mathematical Society

##### Positive solutions of quasilinear elliptic equations with Fuchsian potentials in Wolff class

In this paper we study the existence of non-constant positive solutions of certain non-linear elliptic equations (1.1)-div^(Vw)= 0 on Riemannian n-manifolds. Here (s4x (Vu), VM>«| Vw| p and l< p<<». The precise assumptions are given in (2.7). A typical equation of this type is the p-Laplace equation (1.2)-div (| Vu|"-2Vu)= 0.If p= 2, the equation (1.2) is the usual Laplace equation. Continuous (weak) solutions of (1.1) and (1.2) are called si-harmonic and p-harmonic, respectively. We are mainly interested in the following question. Does there exist a Riemannian n-manifold that has a positive solution of

*1992/11*

James Gabe

Syddansk Universitet

Proceedings of the London Mathematical Society

##### Dynamic asymptotic dimension and Matui's HK conjecture

We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the C∗$C^*$‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture. We also construct explicit maps from the groupoid homology groups to the K‐theory groups of their C∗$C^*$‐algebras in degrees zero and one, and investigate their properties.

*2023/4*

Alex Scott

University of Oxford

Proceedings of the London Mathematical Society

##### Erdős–Hajnal for graphs with no 5‐hole

The Erdős–Hajnal conjecture says that for every graph H$H$ there exists τ>0$\tau >0$ such that every graph G$G$ not containing H$H$ as an induced subgraph has a clique or stable set of cardinality at least |G|τ$|G|^\tau$. We prove that this is true when H$H$ is a cycle of length five. We also prove several further results: for instance, that if C$C$ is a cycle and H$H$ is the complement of a forest, there exists τ>0$\tau >0$ such that every graph G$G$ containing neither of C,H$C,H$ as an induced subgraph has a clique or stable set of cardinality at least |G|τ$|G|^\tau$.

*2023*