Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

Geometric flows and differential Harnack estimates for heat equations with potentials

Let $M$ be a closed Riemannian manifold with a Riemannian metric $g_{ij}(t)$ evolving by a geometric flow $\partial _{t}g_{ij} = -2{S}_{ij}$ , where $S_{ij}(t)$ is a symmetric two-tensor on $(M, g(t))$ . Suppose that $S_{ij}$ satisfies the tensor inequality $2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0$ for all vector fields $X$ on $M$ , where ${\mathcal H}(S, X)$ and ${\mathcal E}(S,X)$ are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where $S_{ij} = R_{ij}$ , the Ricci tensor of $M$ , our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols

Let ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ denote the standard weighted Bergman space over the unit ball ${\mathbb{B}^n}$ in ${\mathbb{C}^n}$ . New classes of commutative Banach algebras ${\mathcal{T}(\lambda)}$ which are generated by Toeplitz operators on ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141?C152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of ${\mathbb{B}^n}$ . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n?=?2. We explicitly describe the maximal ideal space and the Gelfand map of ${\mathcal{T}(\lambda)}$ . Since ${\mathcal{T}(\lambda)}$ is not invariant under the *-operation of ${\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}$ its inverse closedness is not obvious and is proved. We remark that the algebra ${\mathcal{T}(\lambda)}$ is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in ${\mathcal{T}(\lambda)}$ is always connected.     

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153?C201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional ${\mathcal{J}}$ associated to a smooth function ${\phi}$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.     

Forbidden Induced Subgraphs for Perfect Matchings

Let ${\mathcal{F}}$ be a family of connected graphs. A graph G is said to be ${\mathcal{F}}$ -free if G is H-free for every graph H in ${\mathcal{F}}$ . We study the problem of characterizing the families of graphs ${\mathcal{F}}$ such that every large enough connected ${\mathcal{F}}$ -free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1–12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315–324, 2006) and Ota et al. (J Graph Theory, 67(3):250–259, 2011), where the authors were able to characterize such graph families ${\mathcal{F}}$ restricted to the cases ${|\mathcal{F}|\leq 1, |\mathcal{F}| \leq 2}$ and ${|\mathcal{F}| \leq 3}$ , respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition ${|\mathcal{F}| \leq k}$ for some k ≥ 4.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Inscribing nonmeasurable sets

Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221?C242, 2001). We will prove that for a partition ${\mathcal{A}}$ of the real line into meager sets and for any sequence ${\mathcal{A}_n}$ of subsets of ${\mathcal{A}}$ one can find a sequence ${\mathcal{B}_n}$ such that ${\mathcal{B}_{n}}$ ??s are pairwise disjoint and have the same ??outer measure with respect to the ideal of meager sets??. We get?also generalization of this result to a class of ??-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let ${\mathcal{M}}$ be a Brakke flow of n-dimensional surfaces in ${\mathbb{R}^N}$ . The singular set ${\mathcal{S} \subset \mathcal{M}}$ has a stratification ${\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}$ , where ${X \in \mathcal{S}^j}$ if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata ${\mathcal{S}^j_{\eta, r}}$ satisfying ${\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}$ . Sharpening the known parabolic Hausdorff dimension bound ${{\rm dim} \mathcal{S}^j \leq j}$ , we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of ${\mathcal{S}^j_{\eta, r}}$ satisfies ${{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}$ . Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by ${\mathcal{B}_r \subset \mathcal{M}}$ the set of points with regularity scale less than r, we prove that ${{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}$ . This gives L ^p -estimates for the second fundamental form for any p < n + 1 ? k. In fact, the estimates are much stronger and give L ^p -estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

The average size of Ramanujan sums over quadratic number fields

This article is concerned with Ramanujan sums ${c_{\mathcal{I}_1}(\mathcal{I}),}$ where ${\mathcal{I},\mathcal{I}_1}$ are integral ideals in an arbitrary quadratic number field ${\mathbb{Q}(\sqrt{d}).}$ In particular, the asymptotic behavior of sums of ${c_{\mathcal{I}_1}(\mathcal{I}),}$ over both ${\mathcal{I}}$ and ${c_{\mathcal{I}_1}(\mathcal{I}),}$ is investigated.     

Weighted estimates for the multisublinear maximal function

A formulation of the Carleson embedding theorem in the multilinear setting is proved which allows obtaining a multilinear analogue of Sawyer’s two weight theorem for the multisublinear maximal function $\mathcal{M }$ introduced by Lerner et al. (Adv Math 220:1222–1264, 2009). A multilinear version of the $B_p$ theorem from Hytönen and Pérez (Anal PDE, 2013) is also obtained and a mixed $A_{\overrightarrow{ P}}-W_{\overrightarrow{ P}}^{\infty }$ bound for $\mathcal{M }$ is proved as well.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.