We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and definable subcategories, and as such appear in a wide range of algebraic and topological settings. Firstly we investigate and characterise purity preserving functors from a triangulated category into a finitely accessible category with products, which we term coherent functors. This yields a new property for the restricted Yoneda embedding as the universal coherent functor. We build upon the utility of coherent functors to provide several equivalent conditions for an additive, not necessarily triangulated, functor between triangulated categories to be definable: a functor is definable if and only if it preserves filtered homology colimits and products, if and only if it uniquely extends along the restricted Yoneda embedding to a definable functor between the corresponding module categories. We apply these results to the functoriality of the Ziegler spectrum, an object of study in pure homological algebra and representation theory.
Let k be a field of characteristic not 2 and X a smooth intersection of two quadrics in ℙ_k^5 . In this note we show that if the 2-cohomological dimension of k is 1, then X has a rational point.
We consider a homogeneous Bose gas composed of N identical bosons occupying an infinite two-dimensional space. This gas exhibits both an attractive two-body interaction and a repulsive three-body interaction. Via the intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the model. Our investigation focuses on the system’s behavior in relation to the two-body interaction.
Let (a(j)) be an unbounded convex sequence of natural numbers. In 1983, Carleson [7] proved that a necessary and sufficient condition for the (C, 1) means of (S_a(j)f) (partial Fourier sums) to converge uniformly to f in the supremum norm is sup _j j^-1/2log a(j) <+∞ . In this paper, we prove that if we consider Riesz logarithmic means instead of (C, 1) means, then almost everywhere convergence holds for any integrable function and for any convex (a(j)). That is (see Corollary 1.3): Let (a(j)) be any unbounded convex sequence of natural numbers and f∈ L^1 . Then the following convergence holds almost everywhere: 1/log n∑ _j=1^nS_a(j)f/j→ f. In fact, we verify a more general statement (see Theorem 1.1).
Abstract Using algorithms implicit in the classification of $$\text {SL}(2,\mathbb {Z})$$ SL ( 2 , Z ) -orbits of primitive origamis in the stratum $$\mathcal {H}(2)$$ H ( 2 ) due to Hubert–Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $$\mathcal {H}(2)$$ H ( 2 ) is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in $$\mathcal {H}(4)$$ H ( 4 ) and $$\mathcal {H}(6)$$ H ( 6 ) , we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form $$O(N^{2/3}\log N)$$ O ( N 2 / 3 log N ) , where N is the size of the orbit graph.
We introduce and study Zoll manifolds with boundary: compact Riemannian manifolds with smooth boundary such that every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. We first show that all such free boundary geodesics are embedded and have a common length, and that the boundary has at most two connected components. If there are two components, we prove that the manifold is a product of an interval with a closed manifold. When the boundary is connected, we show that the manifold is a tubular neighborhood of a closed embedded submanifold, the “soul”, and that the complement of the soul is diffeomorphic to a half-open cylinder over the boundary. We further prove that all free boundary geodesics are maximally degenerate critical points of the energy functional and have the same Morse index, which equals the multiplicity of the unique focal point occurring at the midpoint of each geodesic. The projection from the boundary to the soul is then either a nontrivial two-fold covering or a smooth sphere bundle, according to the value of this index. As applications, we obtain a complete classification of Zoll surfaces with boundary and of three-dimensional Zoll manifolds with boundary.
Generalizing work of Marin (Artin groups and Yokonuma–Hecke algebras. Int Math Res Not IMRN 13:4022–4062, 2018), we construct in a unified way all the “braids and ties” algebras available in literature and new ones.
We establish the boundedness of the Riesz transform from the Hardy space associated with the operator to the Lebesgue space L^1 of integrable functions. For the standard Euclidean Laplace operator, this is a classical result that plays a significant role in harmonic analysis and theory of singular integral operators. Here, we consider a one-dimensional model of manifolds with ends and exterior Dirichlet boundary conditions. This setting extends the work of Hassell and the third author. Specifically, we examine the real line with the measure |x|^d-1dx leading to various versions of Bessel operators. For integer d, this mimics the measure on Euclidean d-dimensional space and the obtained results are expected to provide good predictions for a class of Riemannian manifolds with Euclidean ends.
Let C be a Krull-Schmidt triangulated category with shift functor [1] and R be a rigid subcategory of C. We are concerned with the mutation of two-term weak R[1]-cluster tilting subcategories. We show that any almost complete two-term weak R[1]-cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of tau-tilting theory in functor categories and abelian categories.
The purpose of this paper is to study noncommutative Bochner–Riesz means associated with Fourier–Bessel expansions. More precisely, we establish the noncommutative maximal inequalities for this operator and then prove the corresponding pointwise convergence theorems. Moreover, by using the noncommutative Hilber-valued Calderón–Zygmund theory, we also investigate the mapping properties for noncommutative Littlewood–Paley–Stein g_k -functions related to the Poisson semigroup of Fourier–Bessel expansions for each k≥ 1 .
Let ℬ(ℋ) be the algebra of all bounded linear operators on an infinite dimensional complex separable Hilbert space ℋ . We prove that all operators T in ℬ(ℋ) satisfying the following two conditions constitute a co-meager subset of ℬ(ℋ) : (i) For every n∈{± 1,± 2,± 3,⋯ ,±∞} , there exists a complex number λ such that T-λ is semi-Fredholm with index n. (ii) The Wolf spectrum of T has an empty interior. As an application, we show that a typical operator on ℋ has a thick point spectrum, a thick residual spectrum and a rare continuous spectrum. This solves a problem raised by T. Eisner and T. Mátrai in the affirmative, and complements a recent result of M. Scherer. Also the spectra of typical operators in certain special classes of operators are studied.
Ghostly ideals are among the most mysterious objects in coarse index theory. In this paper we show that if a metric space X with bounded geometry admits a coarse embedding into an ℓ ^p -space ( 1 ≤ p < ∞ ), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in K-theory. As consequences, we deduce that such spaces satisfy the relative coarse Baum-Connes conjectures introduced in [12], as well as the operator norm localization property for finite rank projections ( ONL_𝒫_Fin ) as introduced in [1].
We prove a stability result for generalized Kähler–Ricci flow on a toric Fano manifold.
We review Hom -infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen–King–Su’s Frobenius categorification of the Grassmannian: (1) we determine the g-vectors of the Plücker coordinates with respect to the triangular initial seed and (2) we express the F-polynomials associated with the Donaldson–Thomas transformation in terms of 3-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.
We introduce the new concept of weighted K-k-Schur functions—a novel family within the broader class of Katalan functions—that unifies and extends both K-k-Schur functions and closed k-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed k-bounded partitions. As a central application, we resolve the K-k-Schur alternating conjecture—posed by Blasiak, Morse, and Seelinger in 2022—for a wide class of k-bounded partitions, including all strictly decreasing k-bounded partitions. Our results shed new light on the combinatorial structure of K-theoretic symmetric functions.
We study the discrete nonlinear random wave equation u_tt -(εΔ -V) u+δ |u|^2 p u =0 (p∈ℕ^+) on ℤ^d× [0, ∞ ) , where 0<ε , δ≪ 1, Δ is the discrete Laplacian and V is the random potential. We fix the random potential V in a good set, then we use the small amplitudes as parameters to construct quasi-periodic solutions of the nonlinear random wave equation.
Using algorithms implicit in the classification of SL(2, Z)-orbits of primitive origamis in the stratum H(2) due to Hubert-Lelievre and McMullen, we give diameter bounds on the resulting orbit graphs. Since themachinery of McMullen from H(2) is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in H(4) and H(6), we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form O(N-2/3 log N), where N is the size of the orbit graph.
