Electronic Research Announcements

September 2017 , Volume 24

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Desingularization of surface maps
Erica Clay, Boris Hasselblatt and Enrique Pujals
2017, 24: 1-9 doi: 10.3934/era.2017.24.001 +[Abstract](818) +[HTML](433) +[PDF](638.6KB)
Abstract:

We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.

Equational theories of unstable involution semigroups
Edmond W. H. Lee
2017, 24: 10-20 doi: 10.3934/era.2017.24.002 +[Abstract](788) +[HTML](451) +[PDF](389.5KB)
Abstract:

It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.

The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities
Piotr Pokora
2017, 24: 21-27 doi: 10.3934/era.2017.24.003 +[Abstract](857) +[HTML](453) +[PDF](281.0KB)
Abstract:

Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.

Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces
Karina Samvelyan and Frol Zapolsky
2017, 24: 28-37 doi: 10.3934/era.2017.24.004 +[Abstract](701) +[HTML](313) +[PDF](407.9KB)
Abstract:

For a symplectic manifold \begin{document}$(M,ω)$\end{document}, let \begin{document}$\{·,·\}$\end{document} be the corresponding Poisson bracket. In this note we prove that the functional \begin{document}$ (F,G) \mapsto \|\{F,G\}\|_{L^p(M)} $\end{document} is lower-semicontinuous with respect to the \begin{document}$C^0$\end{document}-norm on \begin{document}$C^∞_c(M)$\end{document} when \begin{document}$\dim M = 2$\end{document} and \begin{document}$p < ∞$\end{document}, extending previous rigidity results for \begin{document}$p = ∞$\end{document} in arbitrary dimension.

Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations
Peiying Chen
2017, 24: 38-52 doi: 10.3934/era.2017.24.005 +[Abstract](1129) +[HTML](531) +[PDF](382.7KB)
Abstract:

In this paper, we study the Dirichlet boundary value problem of a class of nonlinear parabolic equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.

Sharpness of the Brascamp–Lieb inequality in Lorentz spaces
Neal Bez, Sanghyuk Lee, Shohei Nakamura and Yoshihiro Sawano
2017, 24: 53-63 doi: 10.3934/era.2017.24.006 +[Abstract](1330) +[HTML](655) +[PDF](407.1KB)
Abstract:

We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.

A note on parallelizable dynamical systems
Josiney A. Souza, Tiago A. Pacifico and Hélio V. M. Tozatti
2017, 24: 64-67 doi: 10.3934/era.2017.24.007 +[Abstract](596) +[HTML](366) +[PDF](261.4KB)
Abstract:

Hájek [3] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber \begin{document}$\mathbb{R}$\end{document}. The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.

Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras
Catarina Carvalho, Victor Nistor and Yu Qiao
2017, 24: 68-77 doi: 10.3934/era.2017.24.008 +[Abstract](607) +[HTML](330) +[PDF](363.3KB)
Abstract:

We characterize the groupoids for which an operator is Fredholm if and only if its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called Fredholm. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many other non-compact manifolds. Moreover, we show that the desingularization of groupoids preserves the class of Fredholm groupoids.

On matrix wreath products of algebras
Adel Alahmadi, Hamed Alsulami, S.K. Jain and Efim Zelmanov
2017, 24: 78-86 doi: 10.3934/era.2017.24.009 +[Abstract](895) +[HTML](478) +[PDF](344.5KB)
Abstract:

We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups introduced by L. Kaloujnine and M. Krasner [17]. We then illustrate its usefulness by proving embedding theorems into finitely generated algebras and constructing nil algebras with prescribed Gelfand-Kirillov dimension.

Real orientations, real Gromov-Witten theory, and real enumerative geometry
Penka Georgieva and Aleksey Zinger
2017, 24: 87-99 doi: 10.3934/era.2017.24.010 +[Abstract](871) +[HTML](461) +[PDF](420.3KB)
Abstract:

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.

Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds
Mikhail Karpukhin
2017, 24: 100-109 doi: 10.3934/era.2017.24.011 +[Abstract](690) +[HTML](378) +[PDF](511.9KB)
Abstract:

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.

Central limit theorems in the geometry of numbers
Michael Björklund and Alexander Gorodnik
2017, 24: 110-122 doi: 10.3934/era.2017.24.012 +[Abstract](556) +[HTML](339) +[PDF](372.83KB)
Abstract:

We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in $\mathbb{R}^d$. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.

The containment problem and a rational simplicial arrangement
Justyna Szpond and Grzegorz Malara
2017, 24: 123-128 doi: 10.3934/era.2017.24.013 +[Abstract](545) +[HTML](376) +[PDF](303.15KB)
Abstract:

Since Dumnicki, Szemberg, and Tutaj-Gasińska gave in 2013 in [11] the first example of a set of points in the complex projective plane such that for its homogeneous ideal I the containment of the third symbolic power in the second ordinary power fails, there has been considerable interest in searching for further examples with this property and investigating the nature of such examples. Many examples, defined over various fields, have been found but so far there has been essentially just one example found of 19 points defined over the rationals, see [18, Theorem A, Problem 1]. In [14, Problem 5.1] the authors asked if there are other rational examples. This has motivated our research. The purpose of this note is to flag the existence of a new example of a set of 49 rational points with the same non-containment property for powers of its homogeneous ideal. Here we establish the existence and justify it computationally. A more conceptual proof, based on Seceleanu's criterion [22] will be published elsewhere [19].

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