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We show that general triple planes with genus and irregularity zero belong to at most 12 families, that we call surfaces of type I to XII, and we prove that the corresponding Tschirnhausen bundle is a direct sum of two line bundles in cases I, II, III, whereas it is a rank 2 Steiner bundle in the remaining cases. We also provide existence results and explicit descriptions for surfaces of type I to VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski. Finally, in the last part of the paper we discuss some moduli problems related to our constructions. [ABSTRACT FROM AUTHOR]

For a finite group G and an arbitrary commutative ring R, Broué has placed a Frobenius exact structure on the category of finitely generated RG-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case R = S/tn, where S is a discrete valuation ring having uniformising parameter t. We prove that the prime ideal spectrum (in the sense of Balmer) of this 'relative' version of the stable module category of RG is a disjoint union of n copies of that for kG, where k is the residue field of S. [ABSTRACT FROM AUTHOR]

In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on S³, we study Lyapunov graphs of nonsingular Smale flows on S¹ ×S². More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on S¹ ×S². We also study the singular type vertices in Lyapunov graphs of nonsingular Smale flows on 3-manifolds. [ABSTRACT FROM AUTHOR]

In this paper we study entire solutions of the Allen-Cahn equation Δu - F'(u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to 2k, k&#8805 2 oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by MM2k. The main result of our paper states that if u ∈ M2k is nondegenerate, then locally near u the set of solutions is a smooth manifold of dimension 2k. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2. [ABSTRACT FROM AUTHOR]

Let A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that A0 is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described. [ABSTRACT FROM AUTHOR]

Among all classes of pseudo-differential operators only the Weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group. In this paper we show that there is, however, a weaker form of symplectic covariance for Shubin's t-dependent operators, in which the intertwiners are no longer metaplectic, but are still invertible non-unitary operators. We also study the case of Born-Jordan operators, which are obtained by averaging the t-operators over the interval [0, 1] (such operators have recently been studied by Boggiatto and his collaborators, and by Toft). We show that covariance still holds for these operators with respect to a subgroup of the metaplectic group. [ABSTRACT FROM AUTHOR]

In an earlier paper, we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras. Here, we consider full quivers (as well as pseudo-quivers) as a combinatoric tool in order to describe PI-varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. [ABSTRACT FROM AUTHOR]

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and Zavarnitsine to groups with triality. [ABSTRACT FROM AUTHOR]

In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori- Yau maximum principle which seems to be interesting in its own. [ABSTRACT FROM AUTHOR]

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call ``Quasisymmetric Schur functions'' (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis. [ABSTRACT FROM AUTHOR]

In this paper, we establish some further results on tropical compactifications. We give an affirmative answer to a conjecture of Tevelev in characteristic 0: any variety contains a Schön very affine open subvariety. Also we show that any fan supported on the tropicalization of a Schön very affine variety produces a Schön compactification. As an application, we show that the moduli space of six points of $ \mathbb{P}^2$ [ABSTRACT FROM AUTHOR]

Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topologically infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting of the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topologically infinite type. [ABSTRACT FROM AUTHOR]

This paper studies the CR-version of Obata theorem on a pseudo-Hermitian CR-manifold $ (M,\theta)$ with contact form $ \theta=\frac{1}{2i} (\partial \rho_A - \bar{\partial} \rho_A)$ $ \rho_A(z)=\vert z\vert^2 + \mathrm{Re} \sum^n_{j=1}A_jz^2_j-1$ $ A_j \in (-1,1).$ [ABSTRACT FROM AUTHOR]

In this paper, we generalize a well-known estimate of Colding-Minicozzi for the extinction time of convex hypersurfaces in Euclidean space evolving by their mean curvature to a much broader class of parabolic curvature flows. [ABSTRACT FROM AUTHOR]

This paper contains two results on the $ L^p$ $ 1

Let $S = K[x_1, ldots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $deg x_i = 1$. Let $Delta $ be a simplicial complex on $[n] = { 1, ldots , n }$ and $I_Delta subset S$ its Stanley--Reisner ideal. We write $Delta ^e$ for the exterior algebraic shifted complex of $Delta $ and $Delta ^c$ for a combinatorial shifted complex of $Delta $. Let $beta _{ii+j}(I_{Delta }) = dim _K mathrm {Tor}_i(K, I_Delta )_{i+j}$ denote the graded Betti numbers of $I_Delta $. In the present paper it will be proved that (i) $beta _{ii+j}(I_{Delta ^e}) leq beta _{ii+j}(I_{Delta ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $beta _{ii+j}(I_{Delta }) leq beta _{ii+j}(I_{Delta ^c})$ for all $i$ and $j$, where the base field is arbitrary. Thus in particular one has $beta _{ii+j}(I_Delta ) leq beta _{ii+j}(I_{Delta ^{lex}})$ for all $i$ and $j$, where $Delta ^{operatorname {lex}}$ is the unique lexsegment simplicial complex with the same $f$-vector as $Delta $ and where the base field is arbitrary. [ABSTRACT FROM AUTHOR]

We consider the non-local operator of variable order as follows: Lf(x) = ... (f(x + z) -- f(x) -- (∇ f(x), z) 1{|z|≤1})n(x, z)/... dz, where α(x) ∈ [α0, α2] for any x ∈ Rd and some constants 0 < α0 ≤ α2 < 2 and n(x, z) is a positive Borel measurable function bounded from above and below. Under mild continuity conditions on α(x) and n(x, z), we establish the H¨older regularity for the associated semigroups. The proof is based on a new construction of the coupling by reflection for non-local operator L, and the results successfully apply to both stable-like processes in the sense of Bass and time-change of symmetric stable processes. [ABSTRACT FROM AUTHOR]

We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in M6 of the locus of smooth plane quintic curves. [ABSTRACT FROM AUTHOR]

We strengthen the volume inequalities for Lp zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case p = ∞ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for Lp zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe's measure transportation proofs of the rank one case of the geometric Brascamp--Lieb and reverse Brascamp--Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp--Lieb and reverse Brascamp--Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky--Rogers lemma. [ABSTRACT FROM AUTHOR]

We prove a variant of Oesterl´e's conjecture describing p-adic periods of the modular curve X0(p), with an additional Γ(2)-structure (and also Γ(3) ∩ Γ0(p) if p ≡ 1 (mod 3)). We use de Shalit's techniques and p-adic uniformization of curves with semi-stable reduction. [ABSTRACT FROM AUTHOR]

Suppose that X is a complex projective variety and L is a pseudoeffective divisor. A numerical reduction map is a quotient of X by all subvarieties along which L is numerically trivial. We construct two variants: the L-trivial reduction map and the pseudo-effective reduction map of Eckl (2005). We show that these maps capture interesting geometric properties of L and use them to analyze abundant divisors. [ABSTRACT FROM AUTHOR]

A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional pencils. Generalized definitions are adopted to construct a projective extension. The main axioms of projective geometry are verified. The methods used are in accordance with Bishop-type modern constructivism. [ABSTRACT FROM AUTHOR]

We construct a new minimal complex surface of general type with pg = 0, K² = 2 and H1 = Z/4Z (in fact, palg 1 = Z/4Z), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a Q-Gorenstein smoothing theory. [ABSTRACT FROM AUTHOR]

We investigate Lp regularity of weighted Bergman projections on the unit disc and Lp regularity of ordinary Bergman projections in higher dimensions. [ABSTRACT FROM AUTHOR]

We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive. [ABSTRACT FROM AUTHOR]

Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally L²-stable, and asymptotically stable in the sense of Martel and Merle. The proof is surprisingly simple and combines the Gardner transform, which links the Gardner and KdV equations, together with the Martel-Merle-Tsai and Martel-Merle recent results on stability and asymptotic stability in the energy space, applied this time to the Gardner equation. As a by-product, the results of Maddocks-Sachs, and Merle-Vega are improved in several directions. [ABSTRACT FROM AUTHOR]

We prove that the KdV-Burgers equation is globally well-posed in H-1(T) with a solution-map that is analytic from H-1(T) to C([0, T];H-1(T)), whereas it is ill-posed in Hs(T), as soon as s < -1, in the sense that the flowmap u0 ↦ u(t) cannot be continuous from Hs(T) to even D'(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows us to lower the C∞ critical index with respect to the KdV equation, it does not permit us to improve the C0 critical index. [ABSTRACT FROM AUTHOR]

If C is a smooth, complete algebraic curve of genus g ≥ 2 over the complex numbers, a point p of C is subcanonical if KC ?OC((2g - 2)p). We study the locus Gg ? Mg,1 of pointed curves (C, p), where p is a subcanonical point of C. Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of Gg and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for g ≤ 6. [ABSTRACT FROM AUTHOR]

We prove that there exist only finitely many commuting squares of finite dimensional *-algebras of fixed dimension, satisfying a "large second relative commutant" condition. We show this by studying the local minima of w → dim(A ∩ wBw*), where A,B are fixed subalgebras of some *-algebra C and w ε C is a unitary. When applied to lattices arising from subfactors satisfying a certain extremality- like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors. [ABSTRACT FROM AUTHOR]

An L2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the ∂-operator is established. This duality is used to study the solution of the ∂-equation with prescribed support. Applications are given to ∂-closed extension of forms, as well as to Bochner-Hartogs type extension of CR functions. [ABSTRACT FROM AUTHOR]

We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in Carnot- Carathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold. [ABSTRACT FROM AUTHOR]

One studies the Zariski tangent cone TX →π X to an affine variety X and the closure &Tmacron;X of ?-1(Reg(X)) in TX. One focuses on the comparison between TX and &Tmacron;X, giving sufficient conditions on X in order that TX = &Tmacron;X. One considers, in particular, the question of when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Another problem considered here is to understand the impact of the Cohen-Macaulayness or normality of &Tmacron;X on the local structure of X. [ABSTRACT FROM AUTHOR]