Related papers: The squaring operartion and the Singer algebraic t…

The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of…

Given a bounded sequence \omega of positive numbers and its associated unilateral weighted shift W_{\omega} acting on the Hilbert space \ell^2(\mathbb{Z}_+), we consider natural representations of W_{\omega} as a 2-variable weighted shift,…

In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's…

In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homology groups $H_*(BV_s, \F_2)$, where $V_s = \F_2^s$. We find a relationship between the intersection of kernels of $Sq^{2^i}$ and the intersection of…

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We…

Let $\Bbbk$ be a field and let $I$ be a monomial ideal in the polynomial ring $Q=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex which provides a finite free resolution for $Q/I$ as a $Q$-module. Later, Gemeda constructed…

We derive a Lorentzian OPE inversion formula for the principal series of $sl(2,\mathbb{R})$. Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric…

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a…

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between L-groups of spherical varieties should give rise to a functorial transfer of…

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space,…

The operator-valued Schur-class is defined to be the set of holomorphic functions $S$ mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator…

In these short notes, we will show the following. Let F_q be a finite field and let E/\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E. Under an assumption, we show that it is NP-complete to check if S_r…

Let $S_\alpha = k\langle x_1,\dots,x_n\rangle /(x_i x_j - \alpha_{ij} x_j x_i)$ be a standard graded skew polynomial algebra over an algebraically closed field $k$ of characteristic not equal to $2$. We show the following results. When $n$…

We construct a $k[[Q]]$-linear predifferential graded Lie algebra $L^*_{X/S}$ associated to a log smooth and saturated morphism $f: X \rightarrow S$ and prove that it controls the log smooth deformation functor. This provides a geometric…

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…

We construct a 2-colored operad G^+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that…

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned…

Let G=SU(2) and let \Omega G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(\Omega G) is a divided polynomial algebra…

The goal of this article and its precursor is to demonstrate, by example, the existence of "transfer operators" betweeen relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all…