Related papers: Stillman's question for twisted commutative algebr…

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the…

Consider an unknot $c$ in $S^3$ and a knot $K$ in ${S^3-N(c)}$. Twisting the knot $K$ along $c$, or equivalently applying $\frac{1}{m}$-surgery on $c$, produces a family of knots $\{K_m\}_{m \in \mathbb{Z}}$. We use bordered Floer homology…

We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a…

We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It…

Given an ample Hausdorff groupoid $G$, a unital commutative ring $R$, and a discrete twist $(\Sigma,i,q)$, we establish a generalised uniqueness theorem for the twisted Steinberg algebra $A_R(G;\Sigma)$. By applying this theorem when $G$ is…

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability…

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all…

This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers…

There is a longstanding conjecture by Fr\"oberg about the Hilbert series of the ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by generic forms. We prove this conjecture true in the case when $I$ is generated by a…

In this paper, we introduce the notion of excellent extension of rings. Let $\Gamma$ be an excellent extension of an artin algebra $\Lambda$, we prove that $\Lambda$ satisfies the Gorenstein symmetry conjecture (resp. finitistic dimension…

In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\tilde{\mathbb{A}}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $\Lambda$ of…

This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $\alpha…

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of…

We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one…

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the…

In this paper, we use inductive methods similar to those employed in a 2025 paper by Alberts, Lemke Oliver, Wang and Wood in order to prove many new cases of the Twisted Malle's Conjecture. Previously, this conjecture had only been proven…

We give a number of results about families of Ulam sets. Generalizing behavior of Ulam sets U(1,n), we prove using an novel model theoretic approach that there is a rigidity phenomenon for Ulam sets U(a,b) as b increases. Based on this, we…

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a conjecture by Blackadar and Handelman on…

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the…

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings $(A,\bar{\mathfrak{m}})$ such that the maximal ideal $\bar{\mathfrak{m}} \subseteq…