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We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable., Final version

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience

In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk defined by Huang, Rhodes and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (BPZ equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF and tail expansions of GMC., 27 pages

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein, confirming a conjecture of M. Haiman. We deduce, using Saito's theory of Hodge D-modules, that the scheme X_norm is Cohen-Macaulay and Gorenstein. This confirms a conjecture of M. Haiman. Associated with any principal nilpotent pair in g, there is a finite subscheme of X_norm. The corresponding coordinate ring is a bigraded finite dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a 1-dimensional vector space generated by a remarkable W-harmonic polynomial on h x h. In the special case where g=gl_n the above algebras are closely related to the n!-theorem of Haiman and our W-harmonic polynomial reduces to the Garsia-Haiman polynomial. Furthermore, in the gl_n case, the sheaf gr(M) gives rise to a vector bundle on the Hilbert scheme of n points in C^2 that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka-Macdonald polynomials established earlier by Haiman., The present paper supersedes an earlier paper arXiv:1002.3311

We introduce a framework for a nonlinear potential theory without a kernel on a reexiv e, strictly convex and smooth Banach space of functions. Nonlinear potentials are dened as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic func- tional. The framework allows a development of other main concepts of nonlin- ear potential theory such as capacities, equilibrium potentials and measures of nite energy. x1. Introduction The goal of this paper is to present a fairly general setting which allows a development of basic concepts of nonlinear potential theory. This setting provides a unied approach to several aspects of nonlinear potential theory with kernel, as well as some kernel free potential theory. The concepts that can be developed include capacities of sets and functions, nonlinear poten- tials, equilibrium potentials, reduced functions, balayage, and measures of nite energy. The framework of our approach is a reexiv e, strictly convex and smooth Banach space of functions satisfying two additional hypotheses. Nonlinear potential theory in function spaces has been the subject of re- search in several papers during seventies (e.g., (7), (12), (19)). The goal was to extend the Dirichlet space theory to the nonlinear setting. This was achieved under various hypotheses. The common hypothesis was that the underlying function space is a Banach space with a vector lattice structure. Almost at the same time, a dieren t type of nonlinear potential the- ory began to take shape in the works of Fuglede, Meyers, and Havin and

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with $1$-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -- including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually $\mathbb Z^\ell$ or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered in [2,3] for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25]. We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer., Comment: We significantly improve Theorems 1-4 in v2. The new statements now of Theorems 1&3 allow for arbitrary negatively curved surfaces, as well as much more general classes of geodesic flows in higher dimension. For Theorems 2&4, we classify the centralizer. We also eliminated the global rigidity results in the previous version. These results will appear in a forthcoming paper

The main object considered in this paper is the trace function, defined as a suitably normalized trace of a product of intertwining operators for the Drinfeld-Jimbo quantum group, multiplied by the exponential of an element of the Cartan subalgebra. This function depends of two parameters -- the element of the Cartan subalgebra, and the highest weight of the Verma module in which the trace is taken. The main results of the paper are that the trace function satisfies two systems of difference equations with respect to the first parameter (the quantum Knizhnik-Zamolodchikov-Bernard and Macdonald-Ruijsenaars equations), and that it is symmetric with respect to the two parameters. In particular, this implies that for each of the above two systems of equations there is the dual system with respect to the second parameter, which is also satisfied by the trace function. The paper establishes a connection between the I.Frenkel-Reshetikhin theory of quantum conformal blocks, the work of Felder-Mukhin-Tarasov-Varchenko on the quantum KZB and Ruijsenaars equations, the work of Etingof-I.Frenkel- Kirillov Jr.-Styrkas on traces of intetwining operators, and the Macdonald- Cherednik theory. The methods of the paper are based on the theory of dynamical twists and R-matrices., Comment: 38 pages, amstex; some misprints and small errors were corrected in the new version

Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us denote $c:=i(ab-ba)$, where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012., Comment: LaTeX, 38 pages with 2 figures

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity., 21 pages, AMSTeX. Significantly changed version of paper originally Titled "On minimal isometric embeddings"

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to orbit method and construct nontrivial irreducible Harish-Chandra modules for certain coadjoint orbits. Our examples include new geometric construction of representations associated to certain orbits of real exceptional Lie groups., Comments are welcome

We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue, and a complete classification of the eigenspace dimension depending on the monodromy. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics., Comment: Final version accepted by Illinois J. Math

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty $C^1$-interior., Version 1: 36 pages. Version 2: minor changes, to appear in Duke Mathematical Journal

For each countable ordinal $\alpha$ let $\mathcal{S}_{\alpha}$ be the Schreier set of order $\alpha$ and $X_{\mathcal{S}_\alpha}$ be the corresponding Schreier space of order $\alpha$. In this paper we prove several new properties of these spaces. 1) If $\alpha$ is non-zero then $X_{\mathcal{S}_\alpha}$ possesses the $\lambda$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $\alpha$ is non-zero and $1, Comment: 18 pages. Part of the third author's undergraduate thesis written under the direction of the second author. Submitted

The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free direct summands. In this paper, we provide several methods to compute the (generalized) $F$-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) $F$-signature for some Segre products of polynomial rings., 16 pages, to appear in Illinois J. Math., v2: minor changes, this is the expanded version of Appendix A of arXiv:1702.07058v1

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between $L$ -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between $L$ -groups, for spherical affine homogeneous spaces $X=H\backslash G$ whose dual group is $\operatorname{SL}_{2}$ or $\operatorname{PGL}_{2}$ (with $G$ and $H$ split). More precisely, we construct a transfer operator between orbital integrals for the $(X\times X)/G$ -relative trace formula, and orbital integrals for the Kuznetsov formula of $\operatorname{PGL}_{2}$ or $\operatorname{SL}_{2}$ . Besides the $L$ -group, another invariant attached to $X$ is a certain $L$ -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given $L$ -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank $1$ , of the relations between periods of automorphic forms and special values of $L$ -functions.

This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $\alpha $ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by \begin{equation*} A_{n}x= \frac{1}{m(V^{n})} \int _{V^{n}}\alpha _{g}x\,dm(g),\quad x\in L_{p}( \mathcal{M}),n\in \mathbb{N},1\leq p\leq \infty , \end{equation*} are of weak type $(1,1)$ and of strong type $(p,p)$ for $1 < p, Comment: Final version, to appear in Duke Mathematical Journal

The purpose of this paper is the study of some basic properties of universal induced characters and their applications to the representation theory of the classical groups (for the definition of a universal induced character, see § 3).The starting point was the paper [F] by E. Formanek on matrix invariants. In his paper [F], Formanek has investigated the Hilbert series for the ring of matrix invariants from the point of view of the representation theory of the general linear group and the symmetric group. In this paper we shall study polynomial concomitants of a group from the same point of view.

This paper is a continuation of two previous papers of the author. In the first [4] we discussed a Thompson series associated with the group M24 in which each of the modular forms ηg(τ) attached to elements g ∈ M24 are primitive cusp-forms.

Let $R=K[X_{1},\ldots,X_{n}]$ , where $K$ is a field of characteristic zero, and let $A_{n}(K)$ be the $n$ th Weyl algebra over $K$ . We give standard grading on $R$ and $A_{n}(K)$ . Let $I$ , $J$ be homogeneous ideals of $R$ . Let $M=H^{i}_{I}(R)$ and $N=H^{j}_{J}(R)$ for some $i,j$ . We show that $\operatorname {Ext}_{A_{n}(K)}^{\nu}(M,N)$ is concentrated in degree zero for all $\nu\geq0$ (i.e., $\operatorname {Ext}_{A_{n}(K)}^{\nu}(M,N)_{l}=0$ for $l\neq0$ ). This proves a conjecture stated in Part I of this paper (T. J. Puthenpurakal and J. Singh, On derived functors of graded local cohomology modules, Math. Proc. Cambridge Philos. Soc. 167 (2018), no. 3, 549–565).

In this paper, we show that the expansions of functions from $L^{p}$ -Paley–Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1\lt p\lt \infty $ , even in the cases when they might not converge in $L^{p}$ -norm. We thereby consider the classical Paley–Wiener spaces $PW_{c}^{p}\subset L^{p}({\mathbb{R}})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley–Wiener-like spaces $B_{{\alpha },c}^{p}\subset L^{p}(0,\infty )$ of functions whose Hankel transform ${\mathcal{H}}^{\alpha }$ is supported in $[0,c]$ . As a side product, we show the continuity of the projection operator $P_{c}^{\alpha }f:={\mathcal{H}}^{\alpha }(\chi _{[0,c]}\cdot {\mathcal{H}}^{\alpha }f)$ from $L^{p}(0,\infty )$ to $L^{q}(0,\infty )$ , $1\lt p\leq q\lt \infty $ .

The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,\chi\omega^{1-i}) K )\geq \frac{(1-\epsilon)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function., Comment: 26 pages, final version; Duke Math. J. (accepted)

For each integral homology sphere $Y$, a function $\Gamma_Y$ on the set of integers is constructed. It is established that $\Gamma_Y$ depends only on the homology cobordism of $Y$ and it recovers the Fr{\o}yshov invariant. A relation between $\Gamma_Y$ and Fintushel-Stern's $R$-invariant is stated. It is shown that the value of $\Gamma_Y$ at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of $\Gamma_Y$ are given. In particular, it is shown that if $\Gamma_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself., Comment: 44 pages. Comments are welcome! v2: Remark 1.6 (due to Levin and Lidman) added to the paper, references updated

It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals., Comment: In this version we proved that the grid polyominoes are primes without the use of Groebner basis (see previous version). In particular, we prove that the polyomino ideal is equal to the toric ideal J_P associated to the polyomino as we defined in Section 3

Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are called Teter rings, and studied by Teter, Huneke-Vraciu, Ananthnarayan-Avramov-Moore, and so on. On the positive dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness., 14 pages

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $\Sigma$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, \varphi, L)$ where $(\mathcal E, \varphi)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L \subset \mathcal E$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten, and is an extension of our earlier paper \cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb R)$ case., Comment: 38 pages

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$., Comment: changed title; broadened definition of modular form; added discussion of constant term and Klingen Eisenstein series

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity., To appear in Duke Math

In this paper, we directly show the known equivalence of simple factor dressings of extended frames and the classical Bianchi–Bäcklund transformations for constant mean curvature surfaces. In doing so, we show how the parameters of classical Bianchi–Bäcklund transformations can be incorporated into the simple factor dressings method.

A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenberg's notion of topological boundary for a group., Comment: 43 pages; revised

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank $2$ . Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank $2$ .

FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg/Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper it is shown that not only does this claim prove to be the case, but is a special case of a wider FJRW isomorphism theorem, which in turn allows for a proof of mirror symmetry for a new class of cases in the Landau-Ginzburg setting. We also obtain several interesting geometric applications regarding the Chen-Ruan cohomology of certain Calabi-Yau orbifolds., 28 pages; in Version 2, we have reordered the sections regarding geometry and the Frobenius algebra isomorphism. In order to avoid confusion with supercommutativity issues in FJRW theory, we restrict considerations on the Frobenius algebra to only even classes. Version 3 has corrected one reference. This article has been accepted to the Illinois Journal of Mathematics

In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let $\mathbb{F}_{q}$ be an arbitrary finite field and $A$ be a set in $\mathbb{F}_{q}$ . Suppose $|A\cap (aG)|\le |G|^{1/2}$ for any subfield $G$ and $a\in \mathbb{F}_{q}^{*}$ , then \begin{equation*}\vert \Delta _{\mathbb{F}_{q}}(A^{2})\vert =\vert (A-A)^{2}+(A-A)^{2}\vert \gg \vert A\vert ^{1+\frac{1}{21}}.\end{equation*} Using the same method, we also obtain some results on sum–product type problems.

In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb R^n$ , $1 \leq d \leq n-1$. We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible. The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result of the author concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line., Comment: 28 pages, 2 figures

This paper extends Auslander-Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander-Reiten conjecture.

We present a new tool for the calculation of Denef and Loeser's motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan's theory of motivic volumes of semi-algebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by L\^e Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory., Comment: Grant information completed, no changes to the body of the paper

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano K\"ahler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space., Comment: 41 pages. Final version. Minor change with exposition improved. To appear in Duke Math. J

The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $\Gamma$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $\Gamma$ on $X^{*}\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\Gamma$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\ge3$, the group $\mathrm{Aut}\left(F_{r}\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\mathrm{Aut}\left(F_{2}\right)$ acts on a particular $T_{2}$-system of $\mathrm{PSL}\left(2,p\right)$ as the alternating or symmetric group., Comment: 31 pages, by Chen Meiri and Doron Puder, with an appendix by Dan Carmon. Better exposition than in last version, and some non-accurate statements fixed

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad) where the action is by a (holomorphic) Cartan involution. The main result is that for any complex reductive group, possibly disconnected, there is a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G). As applications we give short proofs of some well known results, including the Sekiguchi correspondence, Matsuki duality, results on Cartan subgroups, the rational Weyl group, and strong real forms. We also compute H^1(Gamma,G) for all simple, simply connected real groups., This paper replaces Galois Cohomology of Real Groups, arXiv:1310.7917 (by the first author). The proof here is more direct and does not require that the complex group G be connected

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2, R) orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space., 50 pages. Updated according to referee suggestions

Let $\iota:(K,\nu)\hookrightarrow(K(x),\mu)$ be a simple purely transcendental extension of valued fields. In order to study such an extension, M. Vaqui\'e, generalizing an earlier construction of S. Mac Lane, introduced the notion of Key polynomials. In this paper we define a related notion of \textbf{abstract key polynomials} associated to $\iota$ and study the relationship between them and key polynomials of Mac Lane -- Vaqui\'e. Associated to each abstract key polynomial $Q$, we define the truncation $\mu_{Q}$ of $\mu$ with respect to $Q$ and we study the properties of those truncations. Roughly speaking, $\mu_{Q}$ is an approximation to $\mu$ defined by the key polynomial $Q$. We also define the notion of an abstract key polynomial $Q'$ being an \textbf{immediate successor} of another abstract key polynomial $Q$ (in this situation we write $Q

Recently, Du, Guth and Li showed that the Schrodinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.

The main result of this paper is that the toric degenerations of flag and Schubert varieties associated to string polytopes and certain Bott–Samelson resolutions of flag and Schubert varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton–Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton–Okounkov bodies of Bott–Samelson varieties with respect to a certain valuation $\nu_{\mathrm{max}}$ coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton–Okounkov bodies of Bott–Samelson varieties with respect to a different valuation $\nu_{\mathrm{min}}$ in terms of Grossberg–Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton–Okounkov bodies coincide.

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maa{\ss} forms of weight one. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight one studied by Kudla, Rapoport and Yang. The results of the paper are much improved in comparison to the 2012 preprint arXiv:1208.2386 which contained partial results in the same direction. Moreover, they are also an improvement of the main result of the authors thesis (CM values of regularized theta lifts, TU Darmstadt, 2013).

Let $F$ be a free group (pro- $p$ group), and let $U$ and $W$ be two finitely generated subgroups (closed subgroups) of $F$ . The Strengthened Hanna Neumann conjecture says that ¶ \[\sum_{x\in U\backslash F/W}\overline{\operatorname{rk}}(U\cap xWx^{-1})\le\overline{\operatorname{rk}}(U)\overline{\mathrm{rk}}(W),\quad \mbox{where }\overline{\operatorname{rk}}(U)=\max\{\operatorname{rk}(U)-1,0\}.\] This conjecture was proved independently in the case of abstract groups by J. Friedman and I. Mineyev in 2011. ¶ In this paper we give the proof of the conjecture in the pro- $p$ context, and we present a new proof in the abstract case. We also show that the Lück approximation conjecture holds for free groups.