Lond Math Soc Research Articles

An exceptional property of the one-dimensional Bianchi–Egnell inequality

In this paper, for d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 1$$\\end{document} and s∈(0,d2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\in (0,\\frac{d}{2})$$\\end{document}, we study the Bianchi–Egnell quotient Q(f)=inff∈H˙s(Rd)\\B‖(-Δ)s/2f‖L2(Rd)2-Sd,s‖f‖L2dd-2s(Rd)2distH˙s(Rd)(f,B)2,f∈H˙s(Rd)\\B,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {Q}}(f) = \\inf _{f \\in \\dot{H}^s({\\mathbb {R}}^d) \\setminus {\\mathcal {B}}} \\frac{\\Vert (-\\Delta )^{s/2} f\\Vert _{L^2({\\mathbb {R}}^d)}^2 - S_{d,s} \\Vert f\\Vert _{L^{\\frac{2d}{d-2s}}(\\mathbb R^d)}^2}{\ ext {dist}_{\\dot{H}^s({\\mathbb {R}}^d)}(f, {\\mathcal {B}})^2}, \\qquad f \\in \\dot{H}^s({\\mathbb {R}}^d) \\setminus {\\mathcal {B}}, \\end{aligned}$$\\end{document}where Sd,s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{d,s}$$\\end{document} is the best Sobolev constant and B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document} is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d = 1$$\\end{document}, there is a neighborhood of B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document} on which the quotient Q(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {Q}}(f)$$\\end{document} is larger than the lowest value attainable by sequences converging to B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document}. This behavior is surprising because it is contrary to the situation in dimension d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document} described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d = 1$$\\end{document}, Q(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {Q}}(f)$$\\end{document} has no minimizer on H˙s(Rd)\\B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^s({\\mathbb {R}}^d) \\setminus {\\mathcal {B}}$$\\end{document}, which again would be contrary to the situation in d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 1$$\\end{document}. For d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}, this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d =1$$\\end{document} we make some numerical observations which support the conjecture stated above.

Multi-part cross-intersecting families

Let A⊆[n]a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}\\subseteq {[n]\\atopwithdelims ()a}$$\\end{document} and B⊆[n]b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}\\subseteq {[n]\\atopwithdelims ()b}$$\\end{document} be two families of subsets of [n], we say A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} and B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document} are cross-intersecting if A∩B≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\cap B\ e \\emptyset $$\\end{document} for all A∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in {\\mathcal {A}}$$\\end{document}, B∈B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\\in {\\mathcal {B}}$$\\end{document}. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.

Refined Universality for Critical KCM: Upper Bounds

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions. They are tightly linked to the monotone cellular automata called bootstrap percolation. Among the three classes of such models (Bollobás et al. in Combin Probab Comput 24(4):687–722, 2015), the critical ones are the most studied. Together with the companion paper by Marêché and the author (Hartarsky and Marêché in Combin Probab Comput 31(5):879–906, 2022), our work determines the logarithm of the infection time up to a constant factor for all critical KCM. This was previously known only up to logarithmic corrections (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). We establish that on this level of precision critical KCM have to be classified into seven categories. This refines the two classes present in bootstrap percolation (Bollobás et al. in Proc Lond Math Soc (3) 126(2):620–703, 2023) and the two in previous rougher results (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). In the present work we establish the upper bounds for the novel five categories and thus complete the universality program for equilibrium critical KCM. Our main innovations are the identification of the dominant relaxation mechanisms and a more sophisticated and robust version of techniques recently developed for the study of the Fredrickson-Andersen 2-spin facilitated model (Hartarsky et al. in Probab Theory Relat Fields 185(3):993–1037, 2023).

Equivalent birational embedding IV: reduced varieties

Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of Mella and Polastri (Bull Lond Math Soc 41(1):89–93, 2009) [20] to reducible schemes. This allows to prove a very general contractibility result for union of rational subvarieties.

Optimal projection filters with information geometry

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for the r-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019).

A geometric realization of silting theory for gentle algebras

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff and Kapranov (J Eur Math Soc 20(6):1473–1524, 2018) in the context of triangulated surfaces. We also show that the cutting of the underlying surface along a curve without self-intersections corresponds to silting reduction. This is analogous to the Calabi–Yau reduction of surface cluster categories as shown by Marsh and Palu (Proc Lond Math Soc (3) 108(2):411–440, 2014) and Qiu and Zhou (Compos Math 153(9):1779–1819, 2017).

Projections of SDEs onto submanifolds

In Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019) the authors define three projections of Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}-coordinates. We use these to show that the Itô-vector and Itô-jet projections satisfy respectively a weak and mean-square optimality criterion “for small t”: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019), which is formulated in terms of weak and strong Itô–Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.

A general family of Plotkin-optimal two-weight codes over $$\mathbb {Z}_4$$

We obtained all possible parameters of Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4,\) together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (Des Codes Cryptogr 88(12): 2493–2505, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4\) has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond Math Soc 18: 97–122, 1986).

Bounded gaps between product of two primes in imaginary quadratic number fields

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs alpha _1, alpha _2 which are product of two primes in the imaginary quadratic field K such that |sigma (alpha _1-alpha _2)|le 2 for all embeddings sigma of K if the class number of K is one and |sigma (alpha _1-alpha _2)|le 8 for all embeddings sigma of K if the class number of K is two.

Counterexample to the Laptev–Safronov Conjecture

Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009) conjectured an inequality between the magnitude of eigenvalues of a non-self-adjoint Schrödinger operator on mathbb {R}^d, dge 2, and an L^q norm of the potential, for any qin [d/2,d]. Frank (Bull Lond Math Soc 43(4):745–750, 2011) proved that the conjecture is true for qin [d/2,(d+1)/2]. We construct a counterexample that disproves the conjecture in the remaining range qin ((d+1)/2,d]. As a corollary of our main result we show that, for any q>(d+1)/2, there is a complex potential in L^qcap L^{infty } such that the discrete eigenvalues of the corresponding Schrödinger operator accumulate at every point in [0,infty ). In some sense, our counterexample is the Schrödinger operator analogue of the ubiquitous Knapp example in Harmonic Analysis. We also show that it is adaptable to a larger class of Schrödinger type (pseudodifferential) operators, and we prove corresponding sharp upper bounds.

The Dynamics of Complex Box Mappings

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);the QC-Criterion and the Spreading Principle from KSS. The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:puzzle pieces shrink to points,(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting.We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

A Positive Formula for Type A Peterson Schubert Calculus

The Peterson variety is a special case of a nilpotent Hessenberg variety, a class of subvarieties of G/B that have appeared in the study of quantum cohomology, representation theory and combinatorics. In type A, the Peterson variety Y is a subvariety of $$Fl(n;{\mathbb {C}})$$ , the set of complete flags in $${\mathbb {C}}^n$$ , and comes equipped with an action by a one-dimensional torus subgroup S of a standard torus T that acts on $$Fl(n;{\mathbb {C}})$$ . Using the Peterson Schubert basis introduced in Harada and Tymoczko (Proc Lond Math Soc 103(1):40–72, 2011) and obtained by restricting a specific set of Schubert classes from $$H_T^*(Fl(n; {\mathbb {C}}))$$ to $$H_S^*(Y)$$ , we describe the product structure of the equivariant cohomology $$H_{S}^*(Y)$$ . In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit, positive, combinatorial formula for its structure constants. A key step in our proof requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde’s identity, and merits independent interest.

Schrödinger Operators with Complex Sparse Potentials

We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S. Bögli (Commun Math Phys 352:629–639, 2017), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull Lond Math Soc 43:745–750, 2011 and Trans Am Math Soc 370:219–240, 2018) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann Inst H Poincaré Sect A (N.S.) 38:7–13, 1983) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.

A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.

A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis

The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559–584, 2009) proves a quantification of the original Besicovitch projection theorem.

The Spectra of Banach Algebras of Holomorphic Functions on Polydisk-Type Domains

Aron et al. (Math Ann 353:293–303, 2012) proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $$\mathcal {H}^\infty (B_{c_0})$$ . On the other hand, Cole and Gamelin (Proc Lond Math Soc 53:112–142, 1986) showed that $$\mathcal {H}^\infty (\ell _2 \cap B_{c_0})$$ is isometrically isomorphic to $$\mathcal {H}^\infty (B_{c_0})$$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets U of a Banach space X for which $$\mathcal {H}^\infty (U)$$ is isometrically isomorphic to $$\mathcal {H}^\infty (B_{c_0})$$ . We prove that there exist polydisk-type domains U of any infinite dimensional Banach space X with a Schauder basis such that $$\mathcal {H}^\infty (U)$$ is isometrically isomorphic to $$\mathcal {H}^\infty (B_{c_0})$$ , which generalizes the result by Cole and Gamelin (Proc Lond Math Soc 53:112–142, 1986). Furthermore, we study the analytic and algebraic structure of the spectrum of $$\mathcal {H}^\infty (U)$$ and show that the Cluster Value Theorem is true for $$\mathcal {H}^\infty (U)$$ .

Primitive normalisers in quasipolynomial time

The normaliser problem has as input two subgroups H and K of the symmetric group mathrm {S}_n, and asks for a generating set for N_K(H): it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 2020) that if H is primitive, then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups H and K of mathrm {S}_n, in quasipolynomial time, we can decide whether N_{mathrm {S}_n}(H) is primitive, and if so, compute N_K(H). Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser in mathrm {S}_n is known not to be primitive.

Non-negatively curved GKM orbifolds

In this paper we study non-negatively curved and rationally elliptic GKM_4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in Wiemeler (J Lond Math Soc 91(3): 667–692, 2015) and was used there to obtain a classification of non-negatively curved torus manifolds.

Hyperrigidity of C*-Correspondences

We show that hyperrigidity for a C*-correspondence (A, X) is equivalent to non-degeneracy of the left action of the Katsura ideal mathcal {J}_X on X. This extends the work of Kakariadis (Bull Lond Math Soc 45(6):1119–1130, 2013, Theorem 3.3) and Dor-On and Salomon (J Lond Math Soc 98(2):416–438, 2018, Theorem 3.5) who establish this equivalence for discrete graphs as well as the work of Katsoulis and Ramsey (The non-selfadjoint approach to the Hao–Ng isomorphism, arXiv preprint arXiv:1807.11425, 2018, Theorem 3.1), who establish one direction of this equivalence.