Related papers: Sheaves and $K$-theory for $\mathbb{F}_1$-schemes
The main goal of this thesis is to study $\mathbb{K}$-uniruled sets that appear in affine geometry. At the beginning we discuss the property of $\mathbb{K}$-uniruledness and its equivalent conditions. Then we bound from above the degree of…
We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its $\pi_0$. We prove this as a consequence of a more general devissage result for stable infinity…
Let $\mathscr{M}$ be a monoidal model category that is also combinatorial and left proper. If $\mathscr{O}$ is a monad, operad, properad, or a PROP; following Segal's ideas we develop a theory of Quillen-Segal $\mathscr{O}$-algebras and…
We give an a priori construction of the two-dimensional reduction of three-dimensional quantum Chern-Simons theory. This reduction is a two-dimensional topological quantum field theory and so determines to a Frobenius ring, which here is…
This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's…
We display a symmetric monoidal equivalence between the stable $\infty$-category of filtered spectra, and quasi-coherent sheaves on $\mathbb{A}^1 / \mathbb{G}_m$, the quotient in the setting of spectral algebraic geometry, of the flat…
We show the equivalence between Deitmar's and Toen-Vaquie's notions of schemes over F_1 (the 'field with one element'), establishing a symmetry with the classical case of schemes, seen either as spaces with a structure sheaf, or functors of…
This paper investigates the $\mathrm{K}$-theory of twisted groupoid $\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism…
Consider the diagonal action of $SL_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$ where $V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a "standard monomial" basis for the…
A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories…
Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the $\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper we study local properties of…
We review Deitmar's theory of monoidal schemes to start with, and have a detailed look at the standard examples. It is explained how one can combinatorially study such schemes through a generalization of graph theory. In a more general…
This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a…
The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $SL_2$ character variety of a topological surface. We realize the skein algebra of the $4$-punctured sphere as the output of a mirror symmetry…
By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the…
We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{\pi_1(X)}])$…
Let $R$ be a discrete valuation ring with fraction field $K$ and $X$ a flat $R$-scheme. Given a faithful action of a $K$-group scheme $G_K$ over the generic fibre $X_K$, we study models $G$ of $G_K$ acting on $X$. In various situations, we…
The purpose of this paper is to present a mathematical theory of the half-twisted $(0,2)$ gauged linear sigma model and its correlation functions that agrees with and extends results from physics. The theory is associated to a smooth…
Quillen's algebraic K-theory is reconstructed via Voevodsky's algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a…
Quantum sheaf cohomology is a deformation of the cohomology ring of a sheaf. In recent years, this subject had an impetuous development in connection with the $(0; 2)$ non-linear sigma model from super-strings theory. The basic piece in…