Okounkov (2003, Progr. Math. 213, 329–347) conjectured the log-concavity about the structure constants for many interesting basis from representation theory. For the cluster algebra, Gross et al. (2018, J. Amer. Math. Soc. 31, 497–608) introduced the atomic theta basis. We prove that the coefficients of the exponents of any cluster variable of type $A_n$ are log-concave. We show that the structure constants for the theta basis of type $A_2$ are log-concave. As for larger generality, we conjecture the log-concavity of the structure constants for the theta basis of the cluster algebra.

For a positive braid $\beta \in \mathrm {Br}^{+}_{k}$, we consider the braid variety $X(\beta )$. We define a family of open sets $\mathcal {U}_{r, w}$ in $X(\beta )$, where $w \in S_k$ is a permutation and r is a positive integer no greater than the length of $\beta $. For fixed r, the sets $\mathcal {U}_{r, w}$ form an open cover of $X(\beta )$. We conjecture that $\mathcal {U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb {C}[X(\beta )]$ constructed in Casals et al. (2025, J. Amer. Math. Soc. 38, 369–479), Galashin et al. (2026, Invent. Math. 243, 1079–1127), and Galashin et al. (2022, Braid variety cluster structures, I: 3D plabic graphs) and that $\mathcal {U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal {U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures.

We prove the correctness of the AKS algorithm [1] within the bounded arithmetic theory $T^{\text {count}}_2$ or, equivalently, the first-order consequences of the theory $\text {VTC}^0$ expanded by the smash function, which we denote by $\text {VTC}^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + \mathrm {iWPHP}$ augmented by two algebraic axioms and then shows that they are provable in $\text {VTC}^0_2$. The two axioms are: a generalized version of Fermat’s Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra:

• • In $\mathrm {PV}_1$: We formalize Legendre’s Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $\mathbb {Z}/p$.

• • In $S^1_2$: We prove the inequality $\text {lcm}(1,\dots , 2n) \geq 2^n$.

• • In $\text {VTC}^0$: We verify the correctness of the Kung–Sieveking algorithm for polynomial division.

This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph $\Gamma (R)$ for the Boolean ring $R = (\mathbb {Z}_2)^n$. The unique structure of this graph, characterised by its diameter of $3$ and lack of common neighbours, has hindered all standard approaches. We introduce a novel combinatorial method that constructs an explicit resolving set. Consequently, we provide a precise formula for $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$, closing a notable gap in the literature on metric dimensions of zero-divisor graphs. As an application, we compute the metric dimension of a zero-divisor graph of a ring with a Boolean factor.

We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of a transfer homomorphism implies that the order and the associated Dedekind domain share the same arithmetic properties. This is not the case for arbitrary orders in Dedekind domains.

We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced $\mathbb {F}_p$-algebra has a perfectoid tower.

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of $\mathsf {D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take $G=\mathcal {O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

The study of Koszul binomial edge ideals was initiated by V. Ene, J. Herzog, and T. Hibi in 2014, who found necessary conditions for Koszulness. The binomial edge ideal $J_G$ associated to a finite simple graph G is always generated by quadrics. It has a quadratic Gröbner basis if and only if the graph G is closed. However, there are many known nonclosed graphs G where $J_G$ is Koszul. We characterize the Koszul binomial edge ideals by a simple combinatorial property of the graph G.

In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t-path ideals of rooted trees. In addition, we study the regularity of powers of t-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\Delta $ is a d-dimensional simplicial tree connected in codimension one, then reg$(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg$(I(\Delta ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.

By extending some basic results about cohomological dimension of tensor products to non-positive DG-rings, the Intersection Theorem for DG-modules is examined over commutative noetherian local DG-rings with bounded cohomology. Some applications are provided. The first is to improve the DG-setting of the amplitude inequality in [Forum Math. 22 (2010) 941–948]. The second is to show Minamoto’s conjecture in [Israel J. Math. 242 (2021) 1–36]. The third is to obtain the DG-version of the Vasconcelos conjecture about Gorenstein rings.

In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components indexed by some equivalence classes of cocharacters of a maximal torus. This decomposition enables the definition of new enumerative invariants associated with the stack, which we begin to explore.

For each three-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent three-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current article, we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then, we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this article, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha _p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen–Macaulay, and F-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

Let X be a zero-dimensional reduced subscheme of a multiprojective space $\mathbb {V} $. Let $s_i$ be the length of the projection of X onto the ith component of $\mathbb {V}$. A result of Van Tuyl states that the Hilbert function of X is completely determined by its restriction to the product of the intervals $[0, s_i - 1]$. We extend this result to arbitrary zero-dimensional subschemes of $\mathbb {V}$.

We study the transfer of (co)silting objects in derived categories of module categories via the extension functors induced by a morphism of commutative rings. It is proved that the extension functors preserve (co)silting objects of (co)finite type. In many cases the bounded silting property descends along faithfully flat ring extensions. In particular, the notion of bounded silting complex is Zariski local.

Let n be a positive integer and f belong to the smallest ring of functions $\mathbb R^n\to \mathbb R$ that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists $d\in \mathbb N$ such that for all $m\in \{0,\dots , n\}$ and $c\in \mathbb R^{m}$, if $x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ is harmonic, then it is polynomial of degree at most d. In particular, f is polynomial if it is harmonic.

Let d be a positive integer, and let $\mathfrak {a}$ be an ideal of a commutative Noetherian ring R. We answer Hartshorne’s question on cofiniteness of complexes posed in Hartshorne (1970, Invent. Math. 9, 145–164) in the cases $\mathrm {dim}R=d$ or $\mathrm {dim}R/\mathfrak {a}=d-1$ or $\mathrm {ara}(\mathfrak {a})=d-1$, show that if $d\leqslant 2$, then a complex $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite if and only if each homology module $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if R is regular local, $\mathfrak {a}$ is perfect and $d\leqslant 2$, then $X\in \mathrm {D}(R)$ is $\mathfrak {a}$-cofinite if and only if every $\mathrm {H}_i(X)$ is $\mathfrak {a}$-cofinite; if $d\geqslant 3$, then $X\in \mathrm {D}_\sqsubset (R)$ is $\mathfrak {a}$-cofinite and $\mathrm {Ext}^j_R(R/\mathfrak {a},\mathrm {H}_i(X))$ is finitely generated for $j\leqslant d-2$ and $i\in \mathbb {Z}$ if and only if every $\mathrm {H}_{i}(X)$ is $\mathfrak {a}$-cofinite.

In this article, we show that any $\mathbb {A}^2$-fibration over a discrete valuation ring which is also an $\mathbb {A}^2$-form is necessarily a polynomial ring. Further, we show that separable $\mathbb {A}^2$-forms over principal ideal domains are trivial.