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Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper

Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

This paper considers the problem of establishing L p -improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29] , and on recent results for geometric nonconcentration inequalities [11] . As an initial application of this new approach, this paper establishes sharp restricted strong type L p -improving inequalities for certain model quadratic submanifolds in the range k n ≤ 2 k .

In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper [36] , based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper [35] we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of the conjectures in [36] for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for ( 2 , 4 ) complete intersection curves.

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

This paper establishes a version of Nevanlinna theory based on Askey–Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C . A second main theorem that we have derived allows us to define an Askey–Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey–Wilson type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey–Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey–Wilson. This paper concludes with an application to difference equations generalising the Askey–Wilson second-order divided difference equation.

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ).

Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)

In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2] . In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1] .

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.

In this paper, we prove that the algebra of an \'etale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal \'etale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an \'etale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of \'etale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and \'etale groupoids with simple algebras outside of a prescribed set of prime characteristics., Comment: Revisions after a referee report. We define the essential algebra of an ample groupoid as the quotient of the Steinberg algebra by its ideal of singular functions and we prove that the essential algebra is simple if and only if the groupoid is minimal and topologically free. When the singular ideal vanishes, one recovers the simplicity result of the previous version of the paper

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space R n . One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap ( A , B ; Ω ) of a capacitor in R n . This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M ( Ω ) of a domain in terms of its local boundary quasiconformal reflection constant H ( Ω ) , a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ ( A , B ; Ω ) of the curve family joining two disjoint continua A and B in a domain Ω.

In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra., The previous version of this paper was broken into two parts: the present version contains the representation-theoretic half (to which we added a number of additional results) and the geometric half has been moved to arXiv:1811.01011

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper, Comment: 35 pages, 4 figures

The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to the recent work [4] joint with C. Chen and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation π of GL 2 n ( C ) with a non-zero Shalika model; (2) construct a new twisted linear period Λ s , χ and give a different expression of the linear model for π; (3) give a necessary and sufficient condition on the character χ such that there exists a uniform cohomological test vector v ∈ V π (which we construct explicitly) for Λ s , χ . As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem in the forthcoming paper ( [11] ).

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W 1 , n ( R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R 4 . The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31] ), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the form S ( α ) = sup ‖ u ‖ H 2 = 1 ⁡ ∫ R 4 ( exp ⁡ ( 32 π 2 | u | 2 ) − 1 − α | u | 2 ) d x , where α ∈ ( − ∞ , 32 π 2 ) . We establish the existence of the threshold α ⁎ , where α ⁎ ≥ ( 32 π 2 ) 2 B 2 2 and B 2 ≥ 1 24 π 2 , such that S ( α ) is attained if 32 π 2 − α α ⁎ , and is not attained if 32 π 2 − α > α ⁎ . This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such ultrapower characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on (suitable) Boolean algebras, which are to simple theories as Keisler's good ultrafilters [15] are to all (first-order) theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.

Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we prove that this space is contractible (in the previous paper it was shown that it is connected)., Comment: 51 pages

The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q-partial differential equations and q-series.

Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations., Comment: 87 pages, comments are welcome. We have extended the previous version

In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative to such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16], which covered the case V was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X,V)(X,V) in the case V is a symplectic normal crossing divisor in X.

In this paper we prove that for the Gromov–Witten theory of P 1 orbifolds of ADE type the genus-2 G-function introduced by B. Dubrovin, S. Liu, and Y. Zhang vanishes. Together with our results in [11] , this completely solves the main conjecture in their paper [4] . In the process, we also found a sufficient condition for the vanishing of the genus-2 G-function which is weaker than the condition given in our previous paper [11] .

In this paper we show how \emph{the metric theory of tensor products} developed by Grothendieck perfectly fits in the study of channel capacities, a central topic in \emph{Shannon's information theory}. Furthermore, in the last years Shannon's theory has been generalized to the quantum setting to let the \emph{quantum information theory} step in. In this paper we consider the classical capacity of quantum channels with restricted assisted entanglement. In particular these capacities include the classical capacity and the unlimited entanglement-assisted classical capacity of a quantum channel. To deal with the quantum case we will use the noncommutative version of $p$-summing maps. More precisely, we prove that the (product state) classical capacity of a quantum channel with restricted assisted entanglement can be expressed as the derivative of a completely $p$-summing norm., Comment: V2: Some proofs have been explained in more detail. New references added. Same results

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincar\'e duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metric. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated to conformal weights on noncommutative tori, and spectral triples associated to duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture., Comment: Final version. 38 pages

In our recent paper, we introduced the notions of relative operator ( α , β ) -entropy and Tsallis relative operator ( α , β ) -entropy as a parameter extensions of relative operator entropy and Tsallis relative operator entropy. In this paper, we give upper and lower bounds of these new notions according to operator ( α , β ) -geometric mean introduced in Nikoufar et al. (2013) [14] .

In the first part of this paper, we obtain mirror formulas for twisted genus 0 two-point Gromov–Witten (GW) invariants of projective spaces and for the genus 0 two-point GW-invariants of Fano and Calabi–Yau complete intersections. This extends previous results for projective hypersurfaces, following the same approach, but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper. In the second and third parts of this paper, we confirm Walcher's mirror symmetry conjectures for the annulus and Klein bottle GW-invariants of Calabi–Yau complete intersection threefolds; these applications are the main results of this paper. In a separate paper, the genus 0 two-point formulas are used to obtain mirror formulas for the genus 1 GW-invariants of all Calabi–Yau complete intersections.

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Cech compactification β N . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Cech compactification of N .

The purpose of this paper is to study the L2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝN × ℝn, satisfying γ0(x) ≡ x, ψ is a C∞ cut-off function supported on a small neighborhood of 0 ∈ ℝn, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝN. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of Lp boundedness, while the third paper deals with the special case when γ is real analytic.

The classical n -logarithm is a multivalued analytic function defined inductively: Li n ( z ) : = ∫ 0 z Li n − 1 ( t ) d log t , Li 1 ( z ) = − log ( 1 − z ) . In this paper we give a simple explicit construction of the Grassmannian n -logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n -dimensional subspaces in C 2 n in generic position to the coordinate hyperplanes by the natural action of the torus ( C ∗ ) 2 n . The classical n -logarithm appears at a certain one dimensional boundary stratum. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms d log f i where f i are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n -logarithm. Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M 0 , n , calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms. Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H • ( X ) . We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H • ( X ) . The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H • ( X ) generated by constant variations, there is a simple way to calculate the coproduct. It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Buckner in two- and three-dimensional Euclidean spaces. We prove that this generalized Buckner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces. The paper also addresses the study of an elegant completion procedure due to Maehara in Euclidean spaces, the natural setting of which are the spaces with a generating unit ball. We prove that, in these spaces, the Maehara completion is also locally Lipschitz continuous, besides establishing other geometric properties of this completion. The paper contains also new estimates of the (local) Lipschitz constants for the wide spherical hull.

We prove a version of the Deligne conjecture for $n$-fold monoidal abelian categories $A$ over a field $k$ of characteristic 0, assuming some compatibility and non-degeneracy conditions for $A$. The output of our construction is a weak Leinster $(n,1)$-algebra over $k$, a relaxed version of the concept of Leinster $n$-algebra in $Alg(k)$. The difference between the Leinster original definition and our relaxed one is apparent when $n>1$, for $n=1$ both concepts coincide. We believe that there exists a functor from weak Leinster $(n,1)$-algebras over $k$ to $C(E_{n+1},k)$-algebras, well-defined when $k=\mathbb{Q}$, and preserving weak equivalences. For the case $n=1$ such a functor is constructed in [Sh4] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster $(n,1)$-algebra over $k$, out of an $n$-fold monoidal $k$-linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for $n>1$) is a passage from weak Leinster $(n,1)$-algebras to $C(E_{n+1},k)$-algebras. As an application, we prove that the Gerstenhaber-Schack complex of a Hopf algebra over a field $k$ of characteristic 0 admits a structure of a weak Leinster (2,1)-algebra over $k$ extending the Yoneda structure. It relies on our earlier construction [Sh1] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra., v6: 49 pages, some inaccuracies in v5 are corrected

This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals., Comment: 49 pages, 7 figures

In this paper, we give a classification of (regular) automorphism groups of relatively minimal rational elliptic surfaces with section over the field C which have non-constant J-maps. The automorphism group Aut ( B ) of such a surface B is the semi-direct product of its Mordell–Weil group MW ( B ) and the subgroup Aut σ ( B ) of the automorphisms preserving the zero section σ of the rational elliptic surface B. The configuration of singular fibers on the surface determines the Mordell–Weil group as has been shown by Oguiso and Shioda (1991), and Aut σ ( B ) also depends on the singular fibers. The classification of automorphism groups in this paper gives the group Aut σ ( B ) in terms of the configuration of singular fibers on the surface. In general, Aut σ ( B ) is a finite group of order less than or equal to 24 which is a Z / 2 Z extension of either Z / n Z , Z / 2 Z × Z / 2 Z , D n (the Dihedral group of order 2n) or A 4 (the Alternating group of order 12). The configuration of singular fibers does not determine the group Aut σ ( B ) uniquely; however we list explicitly all the possible groups Aut σ ( B ) and the configurations of singular fibers for which each group occurs.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers., Comment: 46 pages, 11 figures

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and F 1 -schemes (after Kato, Deitmar and Connes–Consani). Beside this unification, the category of blueprints contains new interesting objects as “improved” cyclotomic field extensions F 1 n of F 1 and “archimedean valuation rings”. It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Titsʼ idea of Chevalley groups over F 1 , congruence schemes, sheaf cohomology, K-theory and a unified view on analytic geometry over F 1 , adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

We exploit the geometric approach to the virtual fundamental class, due to Fukaya–Ono and Li–Tian, to compare Gromov–Witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. Various special cases of the comparison theorem in this paper have long been used in the algebraic category; some of them have also appeared in the symplectic setting. Combined with the inherent flexibility of the symplectic category, the main theorem leads to a confirmation of Pandharipandeʼs Gopakumar–Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. The proof of the main theorem uses deformations of the Cauchy–Riemann equation that respect the submanifold and Carleman Similarity Principle for solutions of perturbed Cauchy–Riemann equations. In a forthcoming paper, we apply a similar approach to relative Gromov–Witten invariants and the absolute/relative correspondence in genus 0.

In this paper, we define a notion of AS-Gorenstein algebra for N -graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves., Comment: This a revised version (with 59 pages now) of our paper on realizations of braided categories, where we have taken into account the referee's report. Indeed, we are much indebted to the referee, whose useful observations greatly improved our manuscript

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen–Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen–Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders., This is a completely reorganized and rewritten version of our earlier paper math/0701313v2, with a number of new results

This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.