We provide estimates for suitable normalizations of the affine and dual affine quermassintegrals of a convex body $K$ in $\mathbb{R}^{n}$. These follow by a more general study of normalized $p$-means of projection and section functions of $K$.

Given an automorphism $\tau$ of a compact group $G$, we study the factorization of $C(\tau,K)$, the contraction group of $\tau$ modulo a closed $\tau$-invariant subgroup $K$, into the product $C(\tau)K$, of the contraction group $C(\tau)$ of $\tau$, and $K$. We prove that the factorization $C(\tau,K)=C(\tau)K$ holds for every closed $\tau$-invariant subgroup $K$ if and only if $G$ contains arbitrarily small closed normal $\tau$-invariant subgroups $N$ with finite-dimensional quotients $G/N$. For metrizable groups, we obtain that $C(\tau)K$ is a dense subgroup of $C(\tau,K)$, for every closed $\tau$-invariant subgroup $K$. These results are used to link the contraction group to the properties of the dynamical system $(G,\tau)$. It follows that $\tau$ is distal if and only if $C(\tau)$ is trivial, while ergodicity of $\tau$ implies that $C(\tau)$ is nontrivial. When $G$ is metrizable, the closure of $C(\tau)$ is the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically and, at the same time, it is the smallest among closed normal $\tau$-invariant subgroups $N$ such that $\tau$ acts distally on $G/N$. If $\tau$ is ergodic, then its restriction to any closed connected normal $\tau$-invariant subgroup $N$ with finite-dimensional quotient $G/N$ is also ergodic. Moreover, when $G$ is connected, the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically is necessarily connected.

In this paper, we establish the validity of the Chow group problem for complete regular local rings $R$ of dimension up to 4. For dimension $n$ ($>4$) over ramified regular local ring $R$, we have two results: (1) When $I$ is an ideal of height 3 such that $R/I$ is a Gorenstein ring, then $[I]=0$ in $A_{n-3}(R)$. (2) We reduce any prime ideal of height $i$ to an almost complete intersection ideal of height $i$ and in some special cases of almost complete intersection ideal of height $i$, we show that all Chow groups except the top one vanish. A necessary and sufficient condition for the vanishing of Chow groups is also derived using Eisenstein extension.

Let $(R,\mathfrak{m})$ be an equidimensional excellent local ring of prime characteristic $p>0$. We give an alternate proof of the existence of a uniform test exponent for any given $c\in R^{\circ}$ and all ideals generated by (full or partial) systems of parameters. This follows from a more general result about the existence of a test exponent for any given Artinian $R$-module. If we further assume $R$ is Cohen–Macaulay, then there exists a test exponent for any given $c\in R^{\circ}$ and all perfect modules with finite projective dimension.

Following the work of Weaver, we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for $k=1,2$ we show that measures on $\mathbb{R}^{k}$ that induce rank-$k$ modules of derivations must be absolutely continuous to Lebesgue measure. An analogous result holds true for measures concentrated on $k$-rectifiable sets with respect to $k$-dimensional Hausdorff measure.

These rigidity results also apply to the metric space setting and specifically, to spaces that support a doubling measure and a $p$-Poincaré inequality. Using our results for the Euclidean plane, we prove the $2$-dimensional case of a conjecture of Cheeger, which concerns the non-degeneracy of Lipschitz images of such spaces.

We introduce the notion of the action of a group on a labeled graph and the quotient object, also a labeled graph. We define a skew product labeled graph and use it to prove a version of the Gross–Tucker theorem for labeled graphs. We then apply these results to the $C^{*}$-algebra associated to a labeled graph and provide some applications in non-Abelian duality.

Several important results for $p$-summing operators, such as Pietsch’s composition formula and Grothendieck’s theorem, share the following form: there is an operator $T$ such that $S\circ T$ is $p$-summing whenever $S$ is $q$-summing. Such operators were called $(q,p)$-mixing by Pietsch, who studied them systematically. In the operator space setting, G. Pisier’s completely $p$-summing maps correspond to the $p$-summing operators between Banach spaces. A natural modification of the definition yields the notion of completely $(q,p)$-mixing maps, already introduced by K. L. Yew, which is the subject of this paper. Some basic properties of these maps are proved, as well as a couple of characterizations. A generalization of Yew’s operator space version of the Extrapolation theorem is obtained, via an interpolation-style theorem relating different completely $(q,p)$-mixing norms. Finally, some composition theorems for completely $p$-summing maps are proved.

In this work, we give a presentation of the prefix expansion ${\operatorname{\mathbf {Pr}} (G)}$ of an inverse semigroup $G$ as recently introduced by Lawson, Margolis and Steinberg which is similar to the universal inverse semigroup defined by the second named author in case $G$ is a group. The inverse semigroup ${\operatorname{\mathbf {Pr}} (G)}$ classifies the partial actions of $G$ on spaces. We extend this result and prove that Fell bundles over $G$ correspond bijectively to saturated Fell bundles over ${\operatorname{\mathbf {Pr}} (G)}$. In particular, this shows that twisted partial actions of $G$ (on $C^{*}$-algebras) correspond to twisted (global) actions of ${\operatorname{\mathbf {Pr}} (G)}$. Furthermore, we show that this correspondence preserves $C^{*}$-algebra crossed products.

We call “flippable tilings” of a constant curvature surface a tiling by “black” and “white” faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to “flip” the tiling by pushing all black faces forward on the left-hand side and backward on the right-hand side. Among those tilings, we distinguish the “symmetric” ones, for which the metric on the surface does not change under the flip. We provide some existence statements, and explain how to parameterize the space of those tilings (with a fixed number of black faces) in different ways. For instance, one can glue the white faces only, and obtain a metric with cone singularities which, in the hyperbolic and spherical case, uniquely determines a symmetric tiling. The proofs are based on the geometry of polyhedral surfaces in 3-dimensional spaces modeled either on the sphere or on the anti-de Sitter space.

We introduce a family of rigid, linearly independent classes in $H^{*}(\operatorname{WU}_{q})$. The family is different from the one studied by Hurder in (Invent. Math. 66 (1982), 313–323), and some of the classes are decomposed into products of elements of $H^{*}(\operatorname{WU}_{q})$. We will show the independence by examining a complexification of Baker’s example in (Comment. Math. Helv. 53 (1978), 334–363).

In this paper, we show that either, a Euclidean space $\mathbb{R}^{n}$, or a standard sphere $\mathbb{S}^{n}$, is the unique manifold with nonnegative scalar curvature which carries a structure of a gradient almost Ricci soliton, provided this gradient is a non trivial conformal vector field. Moreover, in the spherical case the field is given by the first eigenfunction of the Laplacian. Finally, we shall show that a compact locally conformally flat almost Ricci soliton is isometric to Euclidean sphere $\mathbb{S}^{n}$ provided an integral condition holds.

We prove sharp $L^{3}$ bounds for a variable coefficient generalization of the Wolff circular maximal function $M^{\delta}f$. For each fixed radius $r$, $M^{\delta}f(r)$ is the maximal average of $f$ over the $\delta$-neighborhood of a circle of radius $r$ and arbitrary center. In this paper, we consider maximal averages over families of curves satisfying the cinematic curvature condition, which was first introduced by Sogge to generalize the Bourgain circular maximal function. Our proof manages to avoid a key technical lemma in Wolff’s original argument, and thus our arguments also yield a shorter proof of the boundedness of the (conventional) Wolff circular maximal function. At the heart of the proof is an induction argument that employs an efficient partitioning of $\mathbb{R}^{3}$ into cells using the discrete polynomial ham sandwich theorem.

Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius $t\geq1$ there exists a full circle of radius either $t$ or $2t$ with discrepancy greater than $c\sqrt{t}$ for some numerical constant $c>0$. We also show that for every $t\geq1$ there exists a circular arc of radius exactly $t$ with discrepancy greater than $c\sqrt{t}$. Finally, we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.

Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur–Horn problem, where matrix algebras and diagonals are replaced respectively, by finite factors and maximal Abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa $A$ in a finite factor $M$ with conditional expectation $\mathbb{E}_{A}$, can each $x\in A$ with $0\leq x\leq1$ be expressed as $\mathbb{E}_{A}(p)$ for a projection $p\in M$?

In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur–Horm problem for the Cartan masa in the hyperfinite factor.

Consider the complex linear space endowed with the canonical pseudo-Hermitian form of arbitrary signature. This yields both a pseudo-Riemannian and a symplectic structure. We prove that those submanifolds which are both Lagrangian and minimal with respect to these structures minimize the volume in their Lagrangian homology class. We also describe several families of minimal Lagrangian submanifolds. In particular, we characterize the minimal Lagrangian surfaces in pseudo-Euclidean complex plane endowed with its natural neutral metric and the equivariant minimal Lagrangian submanifolds of indefinite complex space with arbitrary signature.