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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

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)

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

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

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 focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation., Comment: Submitted to Advances in Mathematics on December 28, 2018

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

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] .

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

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper., 16 pages

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

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

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

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

Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra., Comment: AMSLaTex 67 pages, to appear in Advances in Mathematics

$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ quasi-isomorphism from the dgla $T\poly(M)$ of polyvector fields on a smooth manifold $M$ to the dgla $D\poly(M)$ of polydifferential operators on $M$, which extends the classical Hochschild--Kostant--Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and $\mathfrak{g}$-manifolds. The spaces $\cx{T}$ and $\cx{D}$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to $L_\infty$ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups $\cy{T}$ and $\cy{D}$ admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an $L_\infty$ quasi isomorphism from $\cx{T}$ to $\cx{D}$ whose first Taylor coefficient is equal to $\operatorname{hkr}\circ\td$. Here $\td$ acts on $\cx{T}$ by contraction. Furthermore, we prove a Kontsevich--Duflo type theorem for Lie pairs: the Hochschild--Kostant--Rosenberg map twisted by the square root of the Todd class of the Lie pair $(L,A)$ is an isomorphism of Gerstenhaber algebras from $\cy{T}$ to $\cy{D}$. As applications, we establish formality theorems and Kontsevich--Duflo type theorems for complex manifolds, foliations, and $\mathfrak{g}$-manifolds. In the case of complex manifolds, we recover the Kontsevich--Duflo theorem of complex geometry., Comment: 55 pages, several typos corrected, some references added, some minor cosmetic changes in the presentation

In this paper we characterize the surjective linear variation norm isometries on JB-algebras. Variation norm isometries are precisely the maps that preserve the maximal deviation, the quantum analogue of the standard deviation, which plays an important role in quantum statistics. Consequently, we characterize the Hilbert's metric isometries on cones in JB-algebras., Comment: 21 pages

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra., Comment: This paper has resulted from the author's work as a PhD student in Florianopolis under the advice of professor Ruy Exel. arXiv admin note: substantial text overlap with arXiv:1605.07540 by other authors

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field $k$ of positive characteristic $p$ to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex $WCH_*(A)$ for any associative unital $k$-algebra $A$, with homology groups $WHH_*(A)$. We prove that the group $WHH_0(A)$ coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if $A$ is commutative, finitely generated, and smooth, the groups $WHH_i(A)$ are naturally identified with the terms $W\Omega^i_A$ of the de Rham-Witt complex of the spectrum of $A$., Comment: 64 pages, LaTeX2e

Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a self-dual context which allows one to establish the same theorems in the case of non-abelian group-like structures; the question of whether such a context can be found has been left open for seventy years. We also formulate and prove in our context a universal isomorphism theorem from which all other isomorphism theorems can be deduced.

In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C⁎-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on finite dimensional CAT(0)-cubical spaces.

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions., New Version; with 2D results

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Pl\"ucker functions, and dual pairs, and establish some basic duality results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. For example, if $F$ is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong $F$-matroids coincide. We also give examples of tracts $F$ and weak $F$-matroids which are not strong. Our theory of matroids over tracts is closely related to, but more general than, "matroids over fuzzy rings" in the sense of Dress and Dress-Wenzel., Comment: 35 pages. v2: Final version to appear in Advances in Mathematics. Numerous minor updates and revisions. v1: This paper generalizes and subsumes the results of arXiv:1601.01204. Our treatment now includes partial fields, for example, in addition to hyperfields

The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in $L^2$. This yields the disappearance of any initial vacuum and the exponential convergence of large-amplitude solutions to equilibrium in large time. The isothermal diffuse reflection boundary condition plays a vital role in the analysis. The most key ingredients in our strategy of the proof include: (i) $L^2_{x,v}$--$L^\infty_xL^1_v$--$L^\infty_{x,v}$ estimates along a bootstrap argument; (ii) Pointwise estimates on the upper bound of the gain term by the product of $L^\infty$ norm and $L^2$ norm; (iii) An iterative procedure on the nonlinear term., 48 pages, 1 figure

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group., 32 pages, 19 figures

Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l. In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions., Comment: 21 pages

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^3, [\hat{g}])$ is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is a Berger metric on $S^3$ with $Y(S^3, [\hat{g}])$ close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries., Some changes are made in the references. Comments are welcome!

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem., To appear in Advances in Mathematics

With the intent of laying the groundwork for a program that aims at explicitly describing the space of Cartan (i.e. multiplicative) connections on a general proper Lie groupoid, we begin to investigate the space of such connections in the regular case. We point out that there is a close relationship between Cartan connections on a proper regular groupoid and representations of the groupoid on its own longitudinal bundle (i.e. on the vector distribution tangent to its orbits). This observation enables us to reduce the original problem to a simpler one. We carry out a prospective study of the latter problem, and apply the resulting analysis to produce a number of examples in rank two which serve to illustrate the diversity of the possible obstructions to the existence of multiplicative connections., 38 pages. Completely new paper. Old version (v1) is going to merge into arXiv:1403.2071v3

In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds., to appear in Advances in Mathematics

We study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-character of a finite group., Comment: Version 2: minor corrections. Version 3: major revisions of the first half of the paper. Version 4: minor corrections

We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - \Delta)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3; ii) an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest e.g. in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems., Comment: revised version, typos corrected and new references added

In this paper we give a formula for the homotopy groups of ( n − 1 ) -connected 2n-manifolds as a direct sum of homotopy groups of spheres in the case the n th Betti number is larger than 1. We demonstrate that when the n th Betti number is 1 the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulas for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.

In the paper "Deformation theory of abelian categories", the last two authors proved that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects. We show a generalization of this to pretriangulated dg-categories with a left bounded non-degenerate t-structure with enough derived injectives, the latter being derived enhancements of the injective objects in the heart of the t-structure. Such dg-categories (with an additional hypothesis of closure under suitable products) can be completely described in terms of left bounded twisted complexes of their derived injectives., 51 pages; postprint version

Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics (statistics that depend only on the descent set and length) which has close connections to the theory of $P$-partitions, quasisymmetric functions, and noncommutative symmetric functions. We use our framework to prove that many descent statistics are shuffle-compatible and to give explicit descriptions of their shuffle algebras, thus unifying past results of Stanley, Gessel, Stembridge, Aguiar-Bergeron-Nyman, and Petersen., 52 pages. To appear in Adv. Math

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry., v2: 32 pages, includes a table of contents

Let $M^K_n$ be the moduli space of framed $K$-instantons over $S^4$ with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Let $U^{K}_{n}$ be the Uhlenbeck partial compactification of $M^{K}_{n}$. A scheme structure on $U^{K}_{n}$ is endowed by Donaldson as an algebro-geometric Hamiltonian reduction of ADHM data. In this paper, for $K=SO(N,R)$, $N\ge5$, we prove that $U^{K}_{n}$ is an irreducible normal variety with smooth locus $M^{K}_{n}$. Hence, together with the author's previous result, the K-theoretic Nekrasov partition function for any simple classical group other than $SO(3,R)$, is interpreted as a generating function of Hilbert series of the instanton moduli spaces. Using this approach we also study the case $K=SO(4,R)$ which is the unique semisimple but non-simple classical group., 40 pages, v.2: typos corrected, v.3: dedicatory and acknowledgement messages added, references updated, to appear in Adv. Math

In this paper we deal with free functional inequalities on the circle. There are some interesting changes as opposed to the classical case. For example, the free Poincar\'e inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have a classical counterpart, which does not seem to have been investigated before., Comment: It will appear in Advances of Mathematics

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X ⊂ P d of an arbitrary codimension l with respect to a real l − 1 -dimensional linear subspace V ⊂ P d and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces ( l = 1 ), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in P d hyperbolic with respect to some real d − 2 -dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.

In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise inequality for fractional laplacians as in the work of Constantin and Vicol for the euclidean setting, This work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 and the MTM2011-2281 project of the MCINN (Spain)

The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called $p$-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role of reference measure is played by the $p$-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving $p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure., Comment: Final version, published by Advances in Mathematics

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety., Comment: 40 pages; final version; As pointed out to us by Z. Zhang, we prove quadraticity and not formality of the Kuranishi family. Quadraticity is all we need for our main theorem. The current version reflects this correction. A few other improvements in exposition and correction of typos

This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch-Ciubotaru-Trapa and Ciubotaru. We apply our result to compute the Dirac cohomology of ladder representations for type $A_n$. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one., 33 pages, v2: close to accepted version, typos are corrected

Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem for tangent categories due to Leung to exhibit tangent categories as a certain kind of enriched category., 17 pages. v3: final journal version

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$., Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 2016

An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies., 17 pages. Final version