We classify Artin–Schreier extensions of valued fields with nontrivial defect according to whether they are connected with purely inseparable extensions with nontrivial defect, or not. We use this classification to show that in positive characteristic, a valued field is algebraically complete if and only if it has no proper immediate algebraic extension and every finite purely inseparable extension is defectless. This result is an important tool for the construction of algebraically complete fields. We also consider extremal fields (=fields for which the values of the elements in the images of arbitrary polynomials always assume a maximum). We characterize inseparably defectless, algebraically maximal and separable-algebraically maximal fields in terms of extremality, restricted to certain classes of polynomials. We give a second characterization of algebraically complete fields, in terms of their completion. Finally, a variety of examples for Artin–Schreier extensions of valued fields with nontrivial defect is presented.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring and $R=S/I$ where $I \subset S$ is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Söderberg theory states that the multiplicity of $R$ is bounded above by a function of the maximal shifts in the minimal graded free resolution of $R$ over $S$ as well as bounded below by a function of the minimal shifts if $R$ is Cohen-Macaulay. In this paper, we study the related problem to show that the total Betti-numbers of $R$ are also bounded above by a function of the shifts in the minimal graded free resolution of $R$ as well as bounded below by another function of the shifts if $R$ is Cohen-Macaulay. We also discuss the cases when these bounds are sharp.

We introduce quantum semigroup compactifications and study the universal quantum semigroup compactification of a coamenable locally compact quantum group. If $G$ is a classical locally compact group, the universal semigroup compactification corresponds to the C*-algebra of the bounded left uniformly continuous functions on $G$, so we study the analogous C*-algebra associated with a locally compact quantum group.

The article obtains some lower bounds for the sectional category of a map based on the cohomology of the base space and the total space. We also obtain geometric results on the multiplicity of maps and show applications to equilibria on the boundary of stability regions (basins of attraction) of dynamical systems on differentiable manifolds. We consider a number of generalizations of Lusternik–Schnirelmann’s theorem which states that if a covering of an $n$ dimensional sphere consists of $n + 1$ closed sets, then at least one of the sets contains antipodal points. An elementary proof is given for a generalization of this result to packings of Euclidean spaces.

In this paper we present a method to find the regularity of projective monomial curves in terms of an ordering of monoids associated to them. We use this result to find the regularity of certain monomial curves and investigate where regularity is attained in their minimal graded free resolutions.

We consider holomorphic functions on a non-singular subvariety of a smoothly bounded convex domain of finite type. A sufficient and necessary condition is proved for such a function to have an extension to a $p$-integrable holomorphic function on the whole domain. This is shown under transversallity assumption and certain non-degeneracy condition of the subvariety.

We study characterizations of ergodicity, weak mixing and strong mixing of W*-dynamical systems in terms of joinings and subsystems of such systems. Ergodic joinings and Ornstein’s criterion for strong mixing are also discussed in this context.

Relativizing an idea from multiplicity theory, we say that an element of a von Neumann algebra $\mathcal{M}$ is $n$-divisible if it commutes with a type $\mathrm{I}_n$ subfactor. We decide the density of the $n$-divisible operators, for various $n$, $\mathcal{M}$, and operator topologies. The most sensitive case is $\sigma$-strong density in $\mathrm{II}_1$ factors, which is closely related to the McDuff property. We also make use of Voiculescu's noncommutative Weyl-von Neumann theorem to obtain several descriptions of the norm closure of the $n$-divisible operators in $\mathcal{B}(\ell^2)$. Here are two consequences: (1) in contrast to the larger class of reducible operators, the divisible operators are nowhere dense; (2) if an operator is a norm limit of divisible operators, it is actually a norm limit of unitary conjugates of a single divisible operator. The following application is new even for $\mathcal{B}(\ell^2)$: if an element of a von Neumann algebra belongs to the norm closure of the $\aleph_0$-divisible operators, then the weak* closure of its unitary orbit is convex.

We show that for any $d\times d$ expansive matrix $A$ with integer entries and $|\det(A)|=2$, the set of all $A$-dilation MRA wavelets is path-connected under the $L^2(\mathbb{R}^d)$ norm topology. We do this through the application of $A$-dilation wavelet multipliers, namely measurable functions $f$ with the property that the inverse Fourier transform of $(f\widehat{\psi})$ is an $A$-dilation wavelet for any $A$-dilation wavelet $\psi$ (where $\widehat{\psi}$ is the Fourier transform of $\psi$). In this process, we have completely characterized all $A$-dilation wavelet multipliers for any integral expansive matrix $A$ with $|\det(A)|=2$.

We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int(1+|x|) |V(x)|\,dx$ is finite and $H$ has no resonance at zero. In the resonance case, we assume $\int(1+|x|^2) |V(x)|\, dx$ is finite.

In this paper, we will use the splitting of the Frobenius direct image of line bundles on toric varieties to explicitly construct an orthogonal basis of line bundles in the derived category $D^b(X)$ where $X$ is a Fano toric variety with (almost) maximal Picard number.

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions defined over bounded domains of the hyperbolic plane.

Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B and C. In this paper, we show that type A submanifolds correspond exactly to real Kähler submanifolds, we construct new examples of such submanifolds in $\mathbb{R}^6$ and $\mathbb{R}^{10}$, and we obtain classification results on submanifolds with second fundamental forms of maximal type.

We show that every CR-automorphism of the closure of a Levi degenerate hyperquadric in the projective space extends to a holomorphic automorphism of the projective space.

We characterize fractal chordarc curves in Euclidean space by the fact that they remain bilipschitz homogeneous under inversion. We illustrate this result by constructing two examples. The techniques used in these constructions provide a means of calculating various dimensions of bilipschitz homogeneous Jordan curves.

We establish several results yielding linear independence of the affine system generated by $\psi$ in exchange for conditions on the space $V(\psi)$ of negative dilates. A typical assumption yielding linear independence is that the space $V(\psi)$ is shift-invariant. In particular, the affine system generated by a Parseval wavelet is linearly independent. As an illustration of our techniques, we give an alternative proof of the theorem of Linnell (see Proc. Amer. Math. Soc. 127 (1999), 3269–3277) on linear independence of Gabor systems.

We correct the statement of a theorem in Illinois J. Math. 53 (2009), 1157–1172 concerning the Betti numbers of a compact $(κ, μ)$-spaces.