In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

Given a finite abelian group $G$ and $t\in \mathbb{N}$, there are two natural types of subsets of the Cartesian power $G^t$; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$ and (cosets of) subgroups $H$ of $G^t$. A basic question is whether two such sets intersect. In this paper, we show that this decision problem is NP-complete. Furthermore, for fixed $G$ and $S$, we give a complete classification: we determine conditions for when the problem is NP-complete and show that in all other cases the problem is solvable in polynomial time. These theorems play a key role in the classification of algebraic decision problems in finitely generated rings developed in later work of the author.

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$, we recover the conditions given in [7] for the existence of a unital partial action of $G$ on $R$.

Let $f \in \mathbb{Q}[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in \mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively, $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.

Twisted arrow $\infty$-categories of $(\infty ,1)$-categories were introduced by Lurie, and they have various applications in higher category theory. Abellán García and Stern gave a generalization to twisted arrow $\infty$-categories of $(\infty ,2)$-categories. In this paper, we introduce another simple model for twisted arrow $\infty$-categories of $(\infty ,2)$-categories.

Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$, with good ordinary reduction at $p$. We compare the $\Lambda$-ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$-extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$. This generalizes and complements previous work over ${\mathbb{Z}}_p$-extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$-extensions of $K$ which are contained in $\mathbb{L}_\infty$.

We classify almost Ricci–Bourguignon solitons on three-dimensional almost $\alpha$-cosymplectic manifolds. We study almost Ricci-Bourguignon solitons on almost $\alpha$-cosymplectic manifolds, with an emphasis on their classification and geometric properties. Key results include soliton type characterization (shrinking, steady, expanding) via the parameter $\rho$ and conditions under which these solitons become Einstein. We also show that Ricci semi-symmetric manifolds with $\eta$-parallel tensors reduce to almost cosymplectic structures. A five-dimensional example of an almost contact manifold admitting a Ricci-Bourguignon soliton has been constructed. Also, Lie-group classifications in dimension three are obtained, which are almost RB transversal solitons on almost $\alpha$-cosymplectic manifolds.