TY - JOUR AB - We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k). Let k be an algebraically closed field of characteristic p>n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bunn,P the moduli stack of (quasi-)parabolic vector bundles on X, and by Locn,P the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category Db(Qcoh(Loc0n,P)) of quasi-coherent sheaves on an open substack Loc0n,P⊂Locn,P, and the bounded derived category Db(D0Bunn,P-mod) of D0Bunn,P-modules, where D0Bunn,P is a localization of DBunn,P the sheaf of crystalline differential operators on Bunn,P. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles . AU - Shen, Shiyu ID - 14986 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Tamely ramified geometric Langlands correspondence in positive characteristic ER - TY - JOUR AB - he approximate graph coloring problem, whose complexity is unresolved in most cases, concerns finding a c-coloring of a graph that is promised to be k-colorable, where c≥k. This problem naturally generalizes to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyze the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph coloring and promise graph homomorphism problems. AU - Krokhin, Andrei AU - Opršal, Jakub AU - Wrochna, Marcin AU - Živný, Stanislav ID - 12563 IS - 1 JF - SIAM Journal on Computing KW - General Mathematics KW - General Computer Science SN - 0097-5397 TI - Topology and adjunction in promise constraint satisfaction VL - 52 ER - TY - JOUR AB - John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^{d}$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called Löwner’s ellipsoid, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the Löwner problems is more intricate than it is in the setting of convex bodies. AU - Ivanov, Grigory AU - Naszódi, Márton ID - 14737 IS - 23 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Functional John and Löwner conditions for pairs of log-concave functions VL - 2023 ER - TY - JOUR AB - The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor–Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role. AU - Wang, B. AU - Mellibovsky, F. AU - Ayats López, Roger AU - Deguchi, K. AU - Meseguer, A. ID - 14754 IS - 2246 JF - Philosophical Transactions of the Royal Society A KW - General Physics and Astronomy KW - General Engineering KW - General Mathematics SN - 1364-503X TI - Mean structure of the supercritical turbulent spiral in Taylor–Couette flow VL - 381 ER - TY - JOUR AB - We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂ Ω. The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [ 0 , T ] for some time T > 0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)). AU - Moser, Maximilian ID - 14755 IS - 3-4 JF - Asymptotic Analysis KW - General Mathematics SN - 0921-7134 TI - Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result VL - 131 ER - TY - JOUR AB - Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences. AU - Shipman, Barbara A. AU - Stephenson, Elizabeth R ID - 12307 IS - 5 JF - PRIMUS KW - Education KW - General Mathematics SN - 1051-1970 TI - Tangible topology through the lens of limits VL - 32 ER - TY - JOUR AB - Empirical essays of fitness landscapes suggest that they may be rugged, that is having multiple fitness peaks. Such fitness landscapes, those that have multiple peaks, necessarily have special local structures, called reciprocal sign epistasis (Poelwijk et al. in J Theor Biol 272:141–144, 2011). Here, we investigate the quantitative relationship between the number of fitness peaks and the number of reciprocal sign epistatic interactions. Previously, it has been shown (Poelwijk et al. in J Theor Biol 272:141–144, 2011) that pairwise reciprocal sign epistasis is a necessary but not sufficient condition for the existence of multiple peaks. Applying discrete Morse theory, which to our knowledge has never been used in this context, we extend this result by giving the minimal number of reciprocal sign epistatic interactions required to create a given number of peaks. AU - Saona Urmeneta, Raimundo J AU - Kondrashov, Fyodor AU - Khudiakova, Kseniia ID - 11447 IS - 8 JF - Bulletin of Mathematical Biology KW - Computational Theory and Mathematics KW - General Agricultural and Biological Sciences KW - Pharmacology KW - General Environmental Science KW - General Biochemistry KW - Genetics and Molecular Biology KW - General Mathematics KW - Immunology KW - General Neuroscience SN - 0092-8240 TI - Relation between the number of peaks and the number of reciprocal sign epistatic interactions VL - 84 ER - TY - JOUR AB - We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set. In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”. Our main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations. AU - Drach, Kostiantyn AU - Schleicher, Dierk ID - 11717 IS - Part A JF - Advances in Mathematics KW - General Mathematics SN - 0001-8708 TI - Rigidity of Newton dynamics VL - 408 ER - TY - JOUR AB - The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies ∥kt∥1≍t,∥kt∥∞≍1, so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s. AU - Akylzhanov, Rauan AU - Kuznetsova, Yulia AU - Ruzhansky, Michael AU - Zhang, Haonan ID - 12210 IS - 4 JF - Mathematische Zeitschrift KW - General Mathematics SN - 0025-5874 TI - Norms of certain functions of a distinguished Laplacian on the ax + b groups VL - 302 ER - TY - JOUR AB - 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. AU - Gehér, György Pál AU - Titkos, Tamás AU - Virosztek, Daniel ID - 12214 IS - 4 JF - Journal of the London Mathematical Society KW - General Mathematics SN - 0024-6107 TI - The isometry group of Wasserstein spaces: The Hilbertian case VL - 106 ER - TY - JOUR AB - We review our theoretical results of the sound propagation in two-dimensional (2D) systems of ultracold fermionic and bosonic atoms. In the superfluid phase, characterized by the spontaneous symmetry breaking of the U(1) symmetry, there is the coexistence of first and second sound. In the case of weakly-interacting repulsive bosons, we model the recent measurements of the sound velocities of 39K atoms in 2D obtained in the weakly-interacting regime and around the Berezinskii–Kosterlitz–Thouless (BKT) superfluid-to-normal transition temperature. In particular, we perform a quite accurate computation of the superfluid density and show that it is reasonably consistent with the experimental results. For superfluid attractive fermions, we calculate the first and second sound velocities across the whole BCS-BEC crossover. In the low-temperature regime, we reproduce the recent measurements of first-sound speed with 6Li atoms. We also predict that there is mixing between sound modes only in the finite-temperature BEC regime. AU - Salasnich, Luca AU - Cappellaro, Alberto AU - Furutani, Koichiro AU - Tononi, Andrea AU - Bighin, Giacomo ID - 12154 IS - 10 JF - Symmetry KW - Physics and Astronomy (miscellaneous) KW - General Mathematics KW - Chemistry (miscellaneous) KW - Computer Science (miscellaneous) SN - 2073-8994 TI - First and second sound in two-dimensional bosonic and fermionic superfluids VL - 14 ER - TY - JOUR AB - Partially observable Markov decision processes (POMDPs) are standard models for dynamic systems with probabilistic and nondeterministic behaviour in uncertain environments. We prove that in POMDPs with long-run average objective, the decision maker has approximately optimal strategies with finite memory. This implies notably that approximating the long-run value is recursively enumerable, as well as a weak continuity property of the value with respect to the transition function. AU - Chatterjee, Krishnendu AU - Saona Urmeneta, Raimundo J AU - Ziliotto, Bruno ID - 9311 IS - 1 JF - Mathematics of Operations Research KW - Management Science and Operations Research KW - General Mathematics KW - Computer Science Applications SN - 0364-765X TI - Finite-memory strategies in POMDPs with long-run average objectives VL - 47 ER - TY - JOUR AB - Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell. AU - Brown, Adam AU - Romanov, Anna ID - 8773 IS - 1 JF - Proceedings of the American Mathematical Society KW - Applied Mathematics KW - General Mathematics SN - 0002-9939 TI - Contravariant forms on Whittaker modules VL - 149 ER - TY - JOUR AB - In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space. AU - Virosztek, Daniel ID - 9036 IS - 3 JF - Advances in Mathematics KW - General Mathematics SN - 0001-8708 TI - The metric property of the quantum Jensen-Shannon divergence VL - 380 ER - TY - JOUR AB - A tight frame is the orthogonal projection of some orthonormal basis of Rn onto Rk. We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes. AU - Ivanov, Grigory ID - 10860 IS - 4 JF - Canadian Mathematical Bulletin KW - General Mathematics KW - Tight frame KW - Grassmannian KW - zonotope SN - 0008-4395 TI - Tight frames and related geometric problems VL - 64 ER - TY - JOUR AB - In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 10867 IS - 3 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Waist of balls in hyperbolic and spherical spaces VL - 2020 ER - TY - JOUR AB - In order to provide a local description of a regular function in a small neighbourhood of a point x, it is sufficient by Taylor’s theorem to know the value of the function as well as all of its derivatives up to the required order at the point x itself. In other words, one could say that a regular function is locally modelled by the set of polynomials. The theory of regularity structures due to Hairer generalizes this observation and provides an abstract setup, which in the application to singular SPDE extends the set of polynomials by functionals constructed from, e.g., white noise. In this context, the notion of Taylor polynomials is lifted to the notion of so-called modelled distributions. The celebrated reconstruction theorem, which in turn was inspired by Gubinelli’s \textit {sewing lemma}, is of paramount importance for the theory. It enables one to reconstruct a modelled distribution as a true distribution on Rd which is locally approximated by this extended set of models or “monomials”. In the original work of Hairer, the error is measured by means of Hölder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labbé. It is the aim of this work to adapt the analytic part of the theory of regularity structures to the scale of Triebel–Lizorkin spaces. AU - Hensel, Sebastian AU - Rosati, Tommaso ID - 9196 IS - 3 JF - Studia Mathematica KW - General Mathematics SN - 0039-3223 TI - Modelled distributions of Triebel–Lizorkin type VL - 252 ER - TY - JOUR AB - We study the unique solution m of the Dyson equation \( -m(z)^{-1} = z\1 - a + S[m(z)] \) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving linear operator on A. We show that m is the Stieltjes transform of a compactly supported A-valued measure on R. Under suitable assumptions, we establish that this measure has a uniformly 1/3-Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of m near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases. AU - Alt, Johannes AU - Erdös, László AU - Krüger, Torben H ID - 14694 JF - Documenta Mathematica KW - General Mathematics SN - 1431-0635 TI - The Dyson equation with linear self-energy: Spectral bands, edges and cusps VL - 25 ER - TY - JOUR AB - In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’. AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8419 IS - 2131 JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences KW - General Engineering KW - General Physics and Astronomy KW - General Mathematics SN - 1364-503X TI - On the integrability of Birkhoff billiards VL - 376 ER - TY - JOUR AB - In this paper, we study small perturbations of a class of non-convex integrable Hamiltonians with two degrees of freedom, and we prove a result of diffusion for an open and dense set of perturbations, with an optimal time of diffusion which grows linearly with respect to the inverse of the size of the perturbation. AU - Bounemoura, Abed AU - Kaloshin, Vadim ID - 8501 IS - 2 JF - Moscow Mathematical Journal KW - General Mathematics SN - 1609-3321 TI - Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom VL - 14 ER -