Given a complex Banach space $X$ and $2\le q \lt \infty$, we show that $X$ has weak cotype $q$ if and only if there is a constant $c \gt 0$ such that \[ \sum_k \| x_k \| \leq c n^{1-{1/q}} \sup_{\varepsilon_k \pm 1} \left\| \sum_k \varepsilon_k x_k \right\| \] holds for all $n$-dimensional subspaces $E\subset X$ and all vectors $(x_k)_k \subset E$. Moreover, these conditions are equivalent to a decrease rate of order $k^{-1/q}$ for the sequence of eigenvalues of operators on $\ell_\infty$ factoring through $X$. This is an analog of Talagrand's theorem on the equivalence of the cotype $q$ property and the absolutely $(q,1)$-summing property for Banach spaces in the range $q \gt 2$. Surprisingly, this `weak' analog also extends to the case $q=2$. Moreover, we show if $q \gt 2$ and $X$ has weak cotype $q$, then the cotype $q$ constant with $n$ vectors can be estimated by any iterates of the function $L(x)=\max\{1,\log(x)\}$ applied to $(\log n)^{1/q}$.

The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid $C^*$-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of $r$-discrete principal groupoid $C^*$-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite $C^*$-algebras have digraph algebras as their building blocks. The spectrum of almost finite $C^*$-algebras has the structure of an $r$-discrete principal groupoid and this helps in the coordinization of these algebras. Regular canonical subalgebras of almost finite $C^*$-algebras have representations in terms of open subsets of the spectrum for the enveloping $C^*$-algebra. We conclude that regular canonical subalgebras are principal ideal algebras.

A pointwise estimate near a boundary point is obtained for quasiminimizers of the energy integral on a doubling metric measure space admitting a Poincaré inequality. Wiener type conditions sufficient for the (Hölder) continuity of quasiminimizers at a boundary point are also given.

A new simpler proof is given of the theorem of Sedaev-Semenov that the couple $(L^{1}_{w_{0}},L^{1}_{w_{1}})$ of weighted $L^{1}$ spaces on an arbitrary measure space is a Calderón couple, i.e., all interpolation spaces with respect to this couple can be described in terms of the $K$-functional. This theorem has other important consequences. It is a component in an alternative proof of the Brudnyi-Krugljak $K$-divisibility theorem. Also, as shown by Dmitriev, it leads readily to a proof of Sparr's more general result that $(L^{p}_{w_{0}},L^{q}_{w_{1}})$ is a Calderón couple.

We characterise narrow and strong Daugavet operators on $C(K,E)$-spaces; these are in a way the largest sensible classes of operators for which the norm equation $\|\mathrm{Id}+T\| = 1+\|T\|$ is valid. For certain separable range spaces $E$, including all finite-dimensional spaces and all locally uniformly convex spaces, we show that an unconditionally pointwise convergent sum of narrow operators on $C(K,E)$ is narrow. This implies, for instance, the known result that these spaces do not have unconditional FDDs. In a different vein, we construct two narrow operators on $C([0,1],\ell_1)$ whose sum is not narrow.

We study families of linear spaces in projective space whose union is a proper subvariety $X$ of the expected dimension. We establish relations between configurations of focal points and the existence or non-existence of a fixed tangent space to $X$ along a general element of the family. We apply our results to the classification of ruled $3$-dimensional varieties.

We give a skein relation for Jones polynomials among positive, negative, and virtual crossings with some restrictions. We apply this relation to study some properties of virtual knots obtained by replacing a real crossing by a virtual crossing.

Let $T$ be a torsion abelian group and $\lambda$ a cardinal. Among all torsion-free abelian groups $H$ of rank less than or equal to $\lambda$ satisfying $\operatorname{Ext}(H,T)=0$ a group $G$ is called $\lambda$-universal for $T$ if it is universal with respect to group-embedding. We show that in Gödel's constructible universe ($V=L$) there always exists a $\lambda$-universal group for $T$ if $T$ has only finitely many non-trivial bounded $p$-components. This answers a question by Kulikov in the affirmative. Moreover, we prove that in $V=L$ for a large class of torsion-free abelian groups $G$ there exists a completely decomposable group $C$ such that $\operatorname{Ext}(G,T^{\prime})=0$ if and only if $\operatorname{Ext}(C,T^{\prime})=0$ for any torsion abelian group $T^{\prime}$. This is related to a question of Baer.

We show that a continuous map or a continuous flow on $\mathbb{R}^{n}$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set $W$ with the property that the forward orbit of every point in $\mathbb{R}^{n}$ intersects $W$, then there is a fixed point in $W$. Consequently, if the omega limit set of every point is nonempty and uniformly bounded, then there is a fixed point.

The purpose of this paper is to prove a conjecture of Conway, which asserts that the class of combinatorial games constitutes a ``universally embedding" partially ordered abelian group.

Vanishing Carleson-type measures defined with additional logarithmic terms are characterized by using functions in $BMOA$ and the Bloch space. The results are applied to Cesáro type operators on $BMOA$ and the Bloch space.

We show that hulls of closed prime ideals in $H^\infty$ are ``small'': they are contained in the $k$-hulls of single points. Combined with our notion of $p$-parts, this will prove Alling's conjecture for a large class of prime ideals.

We prove that a complex unital uniform algebra is isomorphic to the disk algebra if and only if every closed subalgebra with one generator is isomorphic to the whole algebra. Moreover, every such subalgebra of the disk algebra is isometrically isomorphic to the disk algebra. On the way we prove: (1) for a function $f$ in the disk algebra the interior of the polynomial hull of the set $f(\overline U)$, where $\overline U$ is the closed unit disk, is a Jordan domain; (2) if a uniform algebra $A$ on a compact Hausdorff set $X$ containing the Cantor set separates points of $X$, then there is $f\in A$ such that $f(X)=\overline U$.

We give sharp gradient estimates for harmonic functions of rotation invariant stable Lévy processes near the boundary of Lipschitz domains. We also obtain sharp gradient estimates for harmonic functions of corresponding Feynman-Kac semigroups under some assumptions on the potential $q$.

Extending Malý's geometric definition of absolutely continuous functions of $n$ variables (in a sense equivalent to that of Rado-Reichelderfer), we define classes of $p$-absolutely continuous functions $(1\leq p \lt n)$ and show that this weaker notion of absolute continuity still implies differentiability almost everywhere, although it does not imply continuity or Lusin's condition (N).

We investigate the connection between the degree of an algebraic number over a field of characteristic zero and the degree of a linear form in its conjugates. Special attention is given to the case of linear forms in two distinct conjugates. In the process, we show that certain relations with dominant term are impossible, generalizing a result obtained by Smyth for the field of rational numbers. We also prove analogous multiplicative results. As an application, we describe algebraic numbers of prime degree which can be expressed as sums of two distinct conjugates of an algebraic number of the same degree.

We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles.

In this paper we establish a connection between Jacobi forms over a totally real field $k=\mathbb{Q}(\zeta+\zeta^{-1}) $, $\zeta=e^{2 \pi i/p}$, and codes over the field ${\mathbb F}_p$. In particular, we derive a theta series, which is a Jacobi form, from the complete weight enumerator or the Lee weight enumerator of a self-dual code over ${\mathbb F}_p$.

We study the isometry groups of Aleksandrov spaces with curvature bounded above. We prove that the metric of any finitely compact geodesically complete $\operatorname{CAT}(K)$-space $(K \lt 0)$, all of whose spheres are arcwise connected, can be recovered from the family of all closed balls of a given positive radius. As a corollary, we obtain that every bijection of such a space onto itself which preserves this family is an isometry. In particular, these results hold for any simply connected Riemannian space with sectional curvature at most $K$, where $K \lt 0$, and of dimension greater than $1$.