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In the second paper [16] of this series, we obtained an analog of the prime number theorem for a class of branched covering maps on the 2-sphere S 2 called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function on S 2 satisfying the α -strong non-integrability condition, is asymptotically the same as the well-known logarithmic integral, with an exponential error bound. In this third and last paper of the series, we show that the α -strong non-integrability condition is generic in the class of α -Hölder continuous functions. [ABSTRACT FROM AUTHOR]

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure. [ABSTRACT FROM AUTHOR]

In this paper, we prove the categorical wall-crossing formula for certain quivers containing the three loop quiver, which we call DT/PT quivers. These quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories studied in our previous papers, and we regard it as a categorical analogue of the numerical DT/PT correspondence. As an application, we prove a categorical DT/PT correspondence for sheaves supported on reduced plane curves in the affine three dimensional space. [ABSTRACT FROM AUTHOR]

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has several distinct choices of forward orbit. As is demonstrated in this paper, this non-uniqueness leads to different behaviour to that observed in the traditional setting where every point has a unique forward orbit. We prove several almost sure results on the Lebesgue measure of the set of points satisfying a given recurrence rate, and on the Lebesgue measure of the set of points returning to a shrinking target infinitely often. In certain cases, when the Lebesgue measure is zero, we also obtain Hausdorff dimension bounds. One interesting aspect of our approach is that it allows us to handle targets that are not simply balls, but may have a more exotic geometry. [ABSTRACT FROM AUTHOR]

In this paper, we study the prescribed Q -curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M , g) , which was introduced by S. Brendle in [3] , where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫ M Q d μ < (n − 1) ! Vol (S n). In this paper we study the critical case that ∫ M Q d μ = (n − 1) ! Vol (S n) , we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in [21] in dimension 4 and extend the work of Li-Zhu [22] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs. [ABSTRACT FROM AUTHOR]

For each cardinal κ , let B (κ) be the ideal of bounded subsets of κ and P κ (κ) be the ideal of subsets of κ of cardinality less than κ. Under determinacy hypothesis, this paper will completely characterize for which cardinals κ there is a nontrivial maximal B (κ) almost disjoint family. Also, the paper will completely characterize for which cardinals κ there is a nontrivial maximal P κ (κ) almost disjoint family when κ is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown. Assuming AD + , for all κ < Θ , there are no maximal B (κ) almost disjoint families A such that ¬ (| A | < cof (κ)). For all κ < Θ , if cof (κ) > ω , then there are no maximal P κ (κ) almost disjoint families A so that ¬ (| A | < cof (κ)). Assume AD and V = L (R) (or more generally, AD + and V = L (P (R))). For any cardinal κ , there is a maximal B (κ) almost disjoint family A so that ¬ (| A | < cof (κ)) if and only if cof (κ) ≥ Θ. For any cardinal κ with cof (κ) > ω , there is a maximal P κ (κ) almost disjoint family if and only if cof (κ) ≥ Θ. [ABSTRACT FROM AUTHOR]

This paper constitutes the first part of a study that addresses three rich historical themes in harmonic analysis that, in the non-zero curvature setting, investigate the boundedness properties along variable curves for (I) the linear Hilbert transform and analogous maximal operator, (II) the Carleson-type operators, and (III) the bilinear Hilbert transform and analogous maximal operator. Our Main Theorem in this paper states that, given a general variable curve γ (x , t) in the plane that is assumed only to be measurable in x and to satisfy suitable non-zero curvature in t and extra (mild) non-degeneracy conditions, the operators at (I) and (II) defined along the curve γ are L p -bounded for 1 < p < ∞. Our result provides a new and unified treatment of the first two themes as well as a unitary approach within the x -variable setting for both the singular integral and the maximal operator versions of (I). At the heart of our approach is the LGC–methodology introduced here, which encompasses three key ingredients: (1) phase/space discretization that confines the phase of the multiplier to oscillate at the linear level, (2) adapted Gabor frame discretization of the input function(s), and (3) cancelation extraction via the analysis of the time-frequency correlation(s) capturing the non-zero curvature of γ. [ABSTRACT FROM AUTHOR]

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the D ˜ 4 -type GDAHA to representations of the E ˜ 6 -type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D ˜ 4 and E ˜ 6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces. For E ˜ 6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations. [ABSTRACT FROM AUTHOR]

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [18] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability of Beckner's [4] restrictive Sobolev inequalities of fractional order s with 0 < s < n 2 on the flat sub-manifold R n − 1 and the sphere S n − 1 with the explicit lower bound. When s = 1 , this implies the explicit lower bound for the stability of Escobar's first order Sobolev trace inequality [19] which has remained unknown in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a conformally flat and complete manifold (M , g) = (R n , e 2 u | d x | 2) with finite total Q-curvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that the metric g is normal if and only if the volume entropy is finite. Furthermore, we establish an identity for the volume entropy utilizing the integrated Q-curvature. Additionally, under normal metric assumption, we get a result concerning the behavior of the geometric distance at infinity compared with Euclidean distance. With help of this result, we prove that each polynomial growth polyharmonic function on such manifolds is of finite dimension. Meanwhile, we prove several rigidity results by imposing restrictions on the sign of the Q-curvature. Specifically, we establish that on such manifolds, the Cohn-Vossen inequality achieves equality if and only if each polynomial growth polyharmonic function is a constant. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sharp and quantitative criterion, which focuses solely on Q curvature, to demonstrate the Chern-Gauss-Bonnet integral. In contrast to the previous results [6,7,11] , we use a new approach that involves estimating the singular integral. Furthermore, we derive the asymptotic formula for the solution to the general Q curvature equation. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory; this operad is equivalent to the framed E 2 operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus 0 curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic 0, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic 0, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy BV algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first Szegő limit theorems on d -torus T d for 1 ≤ d ≤ + ∞. Given φ ∈ L 1 (T d) and a finite subset σ = { ξ 1 ⋯ , ξ n } of the dual group Z d of T d , the truncated Toeplitz matrix with respect to σ is T σ φ = { φ ˆ (ξ j − ξ i) } 1 ≤ i , j ≤ n. For any Følner sequence { σ N } of Z d and φ ∈ L + 1 (T d) , it is shown that lim N → ∞ ⁡ (det ⁡ T σ N φ) 1 | σ N | = exp ⁡ (∫ T d log ⁡ φ d m d). In the case d = + ∞ , we are associated with multiplicative Toeplitz matrix T φ = { φ ˆ (j / i) } i , j ∈ N and the most relevant non-Følner truncation, that is, T N φ = { φ ˆ (j / i) } 1 ≤ i , j ≤ N , where σ N = { 1 , ... , N }. It is shown that for each φ ∈ L R ∞ (T ∞) and f ∈ C [ ess-inf φ , ess-sup φ ] , the limit lim N → ∞ ⁡ 1 N Tr f ( T N φ) exists. Moreover, it is proven that the limit lim N → ∞ ⁡ (det ⁡ T N φ) 1 N exists for any φ ∈ L + 1 (T ∞) with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski in [30]. [ABSTRACT FROM AUTHOR]

Intermediate dimensions were introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford–McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions dim θ ⁡ Λ of any Bedford–McMullen carpet Λ for the whole spectrum of θ ∈ [ 0 , 1 ] , in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in θ , and the function θ ↦ dim θ ⁡ Λ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent. [ABSTRACT FROM AUTHOR]

We reduce the chromatic Iwasawa main conjecture for elliptic curves to the conjectured existence of certain Beilinson–Flach classes in the supersingular case. This generalizes the method of Wan in which a p = 0 was assumed; the main innovation of this paper consists in the new 2-dimensional technique to overcome this condition. We also derive the 3-part of the leading term formula in the Birch and Swinnerton-Dyer conjecture for analytic rank 0 or 1 from the main conjecture. Another consequence is that among those elliptic curves with a prescribed number of solutions modulo any fixed prime, infinitely many would satisfy the full Birch and Swinnerton-Dyer conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we shall provide a purely ∞-categorical construction of the mixed graded structure (in the sense of Calaque, Pantev, Toën, Vaquié and Vezzosi) of Chevalley-Eilenberg complexes computing homology and cohomology of Lie algebras defined over a field k of characteristic 0. While this additional piece of structure on Chevalley-Eilenberg complexes is expected and described in work by Calaque and Grivaux in terms of explicit models, there is not a formal and model independent description of the mixed graded Chevalley-Eilenberg ∞-functors in available literature. After constructing in all details the Chevalley-Eilenberg ∞-functors and studying their main formal properties, we present some further conjectures on their behaviour. [ABSTRACT FROM AUTHOR]

Let M (S n + 1) denote the Möbius transformation group of the (n + 1) -dimensional sphere S n + 1. A hypersurface x : M n → S n + 1 is called a Möbius homogeneous hypersurface if there exists a subgroup G of M (S n + 1) such that the orbit G ⋅ p = x (M n) , p ∈ x (M n). In this paper, the Möbius homogeneous hypersurfaces are classified completely up to a Möbius transformation of S n + 1. [ABSTRACT FROM AUTHOR]

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument. [ABSTRACT FROM AUTHOR]

Given a vertex operator algebra V , one can attach a graded Poisson algebra called the C 2 -algebra R (V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C 2 -cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R (V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras. [ABSTRACT FROM AUTHOR]

Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The aim of this paper is to study the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of generalized m -Jacobi-Rogers triangles and their applications. Using the theory of production matrices, we present the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of the output matrices of certain production matrices. In particular, we gain a criterion for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the generalized m -Jacobi-Rogers triangle. This immediately implies that the corresponding generalized m -Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex of higher order in all the indeterminates. In consequence, for m = 1 , we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m , combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of exponential Rirodan arrays. In addition, we also derive some results for coefficientwise Hankel-total positivity in terms of compositional functions and m -branched Stieltjes-type continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) r th-order Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total positivity of r th-order Eulerian polynomials, which in particular implies the conjecture of Sokal on the coefficientwise Hankel-total positivity of reversed 2th-order Eulerian polynomials), (4) multivariate Ward polynomials, labeled series-parallel networks and nondegenerate fanout-free functions, (5) an array from the Lambert function and a generalization of Lah numbers and associated triangles, and so on. [ABSTRACT FROM AUTHOR]

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation. [ABSTRACT FROM AUTHOR]

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight λ + 1 / 2 and level N has weight 2 λ and level 6 N , and is new at the primes 2 and 3 with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes ℓ which give rise to quadratic congruences modulo arbitrary powers of ℓ. [ABSTRACT FROM AUTHOR]

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques. [ABSTRACT FROM AUTHOR]

In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi–Trudi-like formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials. [ABSTRACT FROM AUTHOR]

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring R and a module-finite R -algebra Λ, we study the set tors Λ (respectively, torf Λ) of torsion (respectively, torsionfree) classes of the category of finitely generated Λ-modules. We construct a bijection from torf Λ to ∏ p torf (κ (p) ⊗ R Λ) , and an embedding Φ t from tors Λ to T R (Λ) : = ∏ p tors (κ (p) ⊗ R Λ) , where p runs over all prime ideals of R. When Λ = R , these give classifications of torsionfree classes, torsion classes and Serre subcategories of mod R due to Takahashi, Stanley-Wang and Gabriel. To give a description of Im Φ t , we introduce the notion of compatible elements in T R (Λ) , and prove that all elements in Im Φ t are compatible. We give a sufficient condition on (R , Λ) such that all compatible elements belong to Im Φ t (we call (R , Λ) compatible in this case). For example, if R is semi-local and dim ⁡ R ≤ 1 , then (R , Λ) is compatible. We also give a sufficient condition in terms of silting Λ-modules. As an application, for a Dynkin quiver Q , (R , R Q) is compatible and we have a poset isomorphism tors R Q ≃ Hom poset (Spec R , C Q) for the Cambrian lattice C Q of Q. [ABSTRACT FROM AUTHOR]

The classification of Fano varieties is an important open question, motivated in part by the MMP. Smooth Fano varieties have been classified up to dimension three: one interesting feature of this classification is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). There is a program to use mirror symmetry to classify Fano varieties in higher dimensions. Fano varieties are expected to correspond to certain Laurent polynomials under mirror symmetry; given such a Fano toric complete intersections, one can produce a Laurent polynomial via the Hori–Vafa mirror. In this paper, we give a method to find Laurent polynomial mirrors to Fano quiver flag zero loci in Y -shaped quiver flag varieties. To do this, we generalize the Gelfand–Cetlin degeneration of type A flag varieties to Fano Y -shaped quiver flag varieties, and describe these degenerations as toric quiver moduli spaces. We find conjectural mirrors to 99 four dimensional Fano quiver flag zero loci, and check them up to 20 terms of the period sequence. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself. [ABSTRACT FROM AUTHOR]

In this paper, we determine the finite groups with a Sylow r -subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case r = 2. Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem. [ABSTRACT FROM AUTHOR]

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of p -Laplace type. The main results include characterizations for the limiting parabolic A ∞ and A 1 classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón–Zygmund decompositions, covering and chaining arguments appear in the proofs. [ABSTRACT FROM AUTHOR]

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [27] , predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair (X , D) , where X is a smooth Fano variety and D is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of D. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev [32]. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it. [ABSTRACT FROM AUTHOR]

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t → ∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points. [ABSTRACT FROM AUTHOR]

In this paper we establish affinizations and R-matrices in the language of pro-objects, and as an application, we construct reflection functors over the localizations of quiver Hecke algebras of arbitrary finite types. This reflection functor categorifies the braid group action on the half of a quantum group and the Saito reflection. [ABSTRACT FROM AUTHOR]

Let Σ be a compact oriented 2-manifold (possibly with boundary), and let G Σ be the linear span of free homotopy classes of closed oriented curves on Σ equipped with the Goldman Lie bracket [ ⋅ , ⋅ ] Goldman defined in terms of intersections of curves. A theorem of Goldman gives rise to a Lie homomorphism Φ even from (G Σ , [ ⋅ , ⋅ ] Goldman) to functions on the moduli space of flat connections M Σ (G) for G = U (N) , G L (N) , equipped with the Atiyah-Bott Poisson bracket. The space G Σ also carries the Turaev Lie cobracket δ Turaev defined in terms of self-intersections of curves. In this paper, we address the following natural question: which geometric structure on moduli spaces of flat connections corresponds to the Turaev cobracket? We give a constructive answer to this question in the following context: for G a Lie supergroup with an odd invariant scalar product on its Lie superalgebra, and for nonempty ∂Σ, we show that the moduli space of flat connections M Σ (G) carries a natural Batalin-Vilkovisky (BV) structure, given by an explicit combinatorial Fock-Rosly formula. Furthermore, for the queer Lie supergroup G = Q (N) , we define a BV-morphism Φ odd : ∧ G Σ → Fun (M Σ (Q (N))) which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map Φ odd is constructed using the "odd trace" function on Q (N). [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The "almost" is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite. For a graph G , let I G denote its edge ideal, the monomial ideal generated by x i x j for every edge ij of G. In this paper we study the graded Betti numbers of I G , which are combinatorial invariants that measure the complexity of a minimal free resolution of I G. The Betti numbers of the form β i , 2 i + 2 constitute the "main diagonal" of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let β i , j be a parabolic Betti number on the r -th row of the Betti table, for r ≥ 3. We prove that almost all graphs G with β i , j (I G) = 0 can be partitioned into r − 2 cliques and one independent set. In particular, for almost all graphs G with β i , j (I G) = 0 , the regularity of I G is r − 1. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of investigations of the pluripotential theory on the Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on the Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type theorem on the boundary behavior of the Poisson integral. We will see a relationship between the pluriharmonic measures and the Patterson-Sullivan measures discussed by Athreya, Bufetov, Eskin and Mirzakhani. [ABSTRACT FROM AUTHOR]

Pyramids introduced by Gromov are generalized objects of metric spaces with Borel probability measures. We study non-trivial pyramids, where non-trivial means that they are not represented as metric measure spaces. In this paper, we establish general theory of invariants of pyramids and construct several invariants. Using them, we distinguish concrete pyramids. Furthermore, we study a space consisting of non-trivial pyramids and prove that the space have infinite dimension. [ABSTRACT FROM AUTHOR]

In this paper, we consider coalgebra measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between stable anti-Yetter Drinfeld (SAYD) modules over Hopf algebroids. These give an enrichment of the global category of SAYD modules over comodules. These measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we consider non-symmetric operads with multiplication and modules over them which have both cyclic and Gerstenhaber type structures, known as cyclic unital comp modules. We obtain an enrichment of cyclic unital comp modules over comodules, as well as morphisms on cyclic homology induced by comodule measurings of comp modules over operads with multiplication. [ABSTRACT FROM AUTHOR]

In this paper, we prove that convex hypersurfaces under the flow by powers α > 0 of the Gauss curvature in space forms N n + 1 (κ) of constant sectional curvature κ (κ = ± 1) contract to a point in finite time T ⁎. Moreover, convex hypersurfaces under the flow by power α > 1 n + 2 of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in N n + 1 (κ). This extends the known results in Euclidean space to space forms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce quantum Demazure–Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We provide explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of smooth, origin-symmetric, strictly convex solutions to the prescribed even L p curvature problem. [ABSTRACT FROM AUTHOR]

Let X be a smooth Artin stack with properly stable good moduli space X → π X. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space X is smooth and the morphism π is flat. [ABSTRACT FROM AUTHOR]

This is the second of a series of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo-holomorphic disks. Using the notion of obstruction bundle data introduced in [10] , we give a systematic way of constructing a system of Kuranishi structures on the moduli spaces of pseudo-holomorphic disks which are compatible at the boundary and corners. More specifically, it defines a tree-like K-system in the sense of [8, Definition 21.9] , [11, Definition 21.9]. The method given in this paper does not only simplify the description of the constructions in the earlier literature, but also is designed to provide a systematic utility tool for the construction of a system of Kuranishi structures in the future research. We also establish its uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a coherent subsheaf of Saito's S -sheaf, which combines the S -sheaf and the multiplier ideal sheaf. We construct its L 2 -Dolbeault resolution, which generalizes MacPherson's conjecture on the L 2 resolution of the Grauert-Riemenschneider sheaf. We also prove various transcendentally-based vanishing theorems for the S -sheaf, including Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem, and new ones such as the Nadel vanishing theorem. Finally, we discuss the applications of our results on the relative version of Fujita's conjecture, such as Kawamata's conjecture. [ABSTRACT FROM AUTHOR]

As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the Z -invariants of the R-matrices between cuspidal modules is quite significant. In this paper, we prove that the (q , t) -Cartan matrix specialized at q = 1 of any finite type, called the t-quantized Cartan matrix , inform us of the invariants of R-matrices. To prove this, we use combinatorial AR-quivers associated with Dynkin quivers and their properties as crucial ingredients. [ABSTRACT FROM AUTHOR]

There are two combinatorial ways of parameterizing the J b (F) -orbits of the irreducible components of affine Deligne-Lusztig varieties for GL n and superbasic b. One way is to use the extended semi-modules introduced by Viehmann. The other way is to use the crystal bases introduced by Kashiwara and Lusztig. In this paper, we give an explicit correspondence between them using the crystal structure. [ABSTRACT FROM AUTHOR]

This paper defines quantum toric varieties associated with an arbitrary fan in a finitely generated subgroup of some R d. This is a generalization of the results of the article [14] of Katzarkov, Lupercio, Meersseman and Verjovsky. [ABSTRACT FROM AUTHOR]

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem , which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called N -generic condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section. [ABSTRACT FROM AUTHOR]

This paper proposes a framework to show that the Fukaya category of a symplectic manifold X determines the open Gromov-Witten invariants of Lagrangians L ⊂ X. We associate to an object in an A ∞ -category an extension of the negative cyclic homology, called relative cyclic homology. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian L in the Fukaya category of a symplectic manifold X to the S 1 -equivariant relative quantum homology of (X , L). Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane. This in particular includes the open Gromov-Witten invariants of the real locus of the quintic threefold considered in [23]. [ABSTRACT FROM AUTHOR]