Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal–Lacey theorem asserts that the Banach space $C(K)$ of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ${?}_2$. More recently, Ka?kol and Saxon [20] proved that the space $C_p(K)$ endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence $C_p(\beta \mathbb{N})$ lacks infinite-dimensional separable quotient algebras. This motivates the following question: (?) Does$C_p(K)$admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does $C_p(\beta \mathbb{N})$ admit SQ? Our main theorem implies that $C_p(K)$ has SQ for any compact space K containing a copy of $\beta \mathbb{N}$. Consequently, this result reduces problem (?) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of $\beta \mathbb{N}$). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ?), that for some Efimov space K the space $C_p(K)$ has SQ. Some applications of the main result are provided.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

Bi-Sobolev mappings and Kp-distortions in the plane

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms $f:\mathbb{D}\stackrel{onto}{?}\mathbb{D}$, $f\in W_{\mathrm{loc}}^{1,1}(\mathbb{D};\mathbb{D})$, where $\mathbb{D}?{\mathbb{R}}^2$ is the unit disk, such that there exists a constant ${\mathbb{K}}_p\ge 0$ for which the following distortion inequality

$|Df(x){|}^p\le {\mathbb{K}}_p|J_f(x){|}^{p?1}\text{a.e.}x\in \mathbb{D}$

is satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11]. Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with ${\mathbb{K}}_p={\mathbb{K}}_{p,f}(x)$ not necessarily constant, we identify the homeomorphism f whose p-distortion function ${\mathbb{K}}_{p,f}(x)$ is minimal in $L^1$ norm.     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.     

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in $L^2(\mathbb{R})$, we determine a non-countable generalized frame for the non-separable space ${\mathit{AP}}_2(\mathbb{R})$ of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of ${\mathit{AP}}_2(\mathbb{R})$ are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.     

Transcendence degree one function fields over a finite field with many automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field ${\mathbb{F}}_q$ of odd characteristic p. For a positive integer m prime to p, let $F=\mathbb{K}(x,y)$ be the transcendence degree 1 function field defined by $y^q+y=x^m+x^{?m}$. Let $t=x^{m(q?1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus $\mathfrak{g}(K)=(qm?1)(q?1)$, p-rank (Hasse–Witt invariant) $\gamma (K)={(q?1)}^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^{2}m(q?1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of K; more precisely ${\text{Aut}}_{\mathbb{K}}(K)=\mathrm{\Delta }?D$ where Δ is an elementary abelian p-group of order $q^2$ and D has an index 2 cyclic subgroup of order $m(q?1)$. In particular, $\sqrt{m}|{\text{Aut}}_{\mathbb{K}}(K)|>\mathfrak{g}{(K)}^{3/2}$, and if K is ordinary (i.e. $\mathfrak{g}(K)=\gamma (K)$) then $|{\text{Aut}}_{\mathbb{K}}(K)|>{\mathfrak{g}}^{3/2}$. On the other hand, if G is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendence degree 1 function field L of genus $\mathfrak{g}(L)\ge 2$ defined over $\mathbb{K}$, then $|{\text{Aut}}_{\mathbb{K}}(K)|\le 34{(\mathfrak{g}(L)+1)}^{3/2}<68\sqrt{2}\mathfrak{g}{(L)}^{3/2}$; see [15]. This shows that K hits this bound up to the constant $68\sqrt{2}$.Since ${\text{Aut}}_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in ${\text{Aut}}_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of Δ.     

On the trivectors of a 6-dimensional symplectic vector space. II

Let V be a 6-dimensional vector space over a field $\mathbb{F}$, let f be a nondegenerate alternating bilinear form on V and let $\mathit{Sp}(V\text{,}f)?{\mathit{Sp}}_6(\mathbb{F})$ denote the symplectic group associated with $(V\text{,}f)$. The group $\mathit{GL}(V)$ has a natural action on the third exterior power ${?}^{3}V$ of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field $\mathbb{F}$. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of $\mathbb{F}$ that are contained in a fixed algebraic closure $\overline{\mathbb{F}}$ of $\mathbb{F}$. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of $\mathit{Sp}(V\text{,}f)?\mathit{GL}(V)$ on ${?}^{3}V$.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

Regularity in Lp Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:

(*)$Pu+{?}_{t}u=f(x,t),\text{ for }x\in \mathrm{\Omega }?{\mathbb{R}}^n,t\in I?\mathbb{R}.$

1) For strongly elliptic pseudodifferential operators (ψdo's) P on ${\mathbb{R}}^n$ of order $d\in {\mathbb{R}}_{+}$, a symbol calculus on ${\mathbb{R}}^{n+1}$ is introduced that allows showing optimal regularity results, globally over ${\mathbb{R}}^{n+1}$ and locally over $\mathrm{\Omega }\times I$:

$f\in H_{p,\mathrm{loc}}^{(s,s/d)}(\mathrm{\Omega }\times I)?u\in H_{p,\mathrm{loc}}^{(s+d,s/d+1)}(\mathrm{\Omega }\times I),$

for $s\in \mathbb{R}$, $1<p<\infty$. The $H_p^{(s,s/d)}$ are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal ${(?\mathrm{\Delta })}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition $\mathrm{supp}u?\stackrel{￣}{\mathrm{\Omega }}$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I;H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }}))$ with ${?}_{t}u\in L_p(\mathrm{\Omega }\times I)$. Here $H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }})={\dot{H}}_p^{2a}(\stackrel{￣}{\mathrm{\Omega }})$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a?\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ if $a=1/p$, but contains nontrivial elements from $d^a{\stackrel{￣}{H}}_p^a(\mathrm{\Omega })$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,?\mathrm{\Omega })$). The interior regularity of u is lifted when f is more smooth.