Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices and $\|\cdot \|_{(p,k)}$ the $(p,k)$ norm on $M_{mn}$ with a positive integer $k\leq mn$ and a real number $p>2$. We show that a linear map $\phi :M_{mn}\rightarrow M_{mn}$ satisfies

$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$

if and only if there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$. The result is also extended to multipartite systems.

The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement of the “complexified unit circle” $\{(z,w) \in \widehat {{\mathbb C}}^2 \colon z \cdot w = 1\}$. We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric ${}_2F_1$ functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed Möbius invariant subspaces of eigenspaces of the canonical Laplacian of $\Omega $. Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.

Let A be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let

$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$

$f: X \to {\Bbb R}$

$\mu $

$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$

${\mathcal E}(G)$

$\Lambda _G(p_1, \ldots , p_n)$

$C>0$

(0.1)

$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$

$f_i$

$1\le i\le n$

$T_K$

$X={\Bbb F}_q^d$

$K(x^i,x^j)$

$x^i-x^j$

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analog of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

Building on work of Furstenberg and Tzkoni, we introduce $\mathbf {r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

We prove some conditions for higher-dimensional algebraic fibering of pro-p group extensions, and we establish corollaries about incoherence of pro-p groups. In particular, if $1 \to K \to G \to \Gamma \to 1$ is a short exact sequence of pro-p groups, such that $\Gamma $ contains a finitely generated, non-abelian, free pro-p subgroup, K a finitely presented pro-p group with N a normal pro-p subgroup of K such that $K/ N \simeq \mathbb {Z}_p$ and N not finitely generated as a pro-p group, then G is incoherent (in the category of pro-p groups). Furthermore, we show that if K is a finitely generated, free pro-p group with $d(K) \geq 2$, then either $\mathrm{Aut}_0(K)$ is incoherent (in the category of pro-p groups) or there is a finitely presented pro-p group, without non-procyclic free pro-p subgroups, that has a metabelian pro-p quotient that is not finitely presented, i.e., a pro-p version of a result of Bieri–Strebel does not hold.

Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan–Lusztig cells for W. Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W. Low elements in W were introduced to study the word problem of the corresponding Artin–Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W. In this article, we show, in the case of an affine Coxeter group, that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a by-product of our proof, we show that the descent walls – the walls that separate a region from the fundamental alcove – of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter $\varepsilon $ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon \log (1/\varepsilon )+O(\varepsilon )$ for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semiconcave costs for a finer estimate, and lower bounds for $\mathscr {C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

The proof of Theorem 3.14 contains an unsubstantiated claim. To overcome this problem, we add a hypothesis to the statement of 3.14 and we provide a new valid proof. We adjust Theorem 3.15, Corollary 3.16, Proposition 4.23, Theorem 4.26, Corollary 4.29, and Corollary 4.32 accordingly.