Related papers: The $\mathbb C$-motivic Adams-Novikov spectral seq…

These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…

We show that algebraic K-theory KGL, the motivic Adams summand ML and their connective covers acquire unique E-infinity structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine…

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category…

The structure set $\ST^{TOP}(M)$ of an $n$-dimensional topological manifold $M$ for $n \geqslant 5$ has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory.…

We use categorification of monoid actions to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by To\"{e}n and Vaqui\'{e}, along…

We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full…

In this paper, we develop an enhancement of derived algebraic geometry to apply to $\mathbb{A}^1$-homotopy theory introduced by Morel and Voevodsky. We call the enhancement "motivic derived algebraic geometry". We shall actually formulate…

We show that the bounded derived category of regular holonomic D-modules on a smooth variety is equivalent to the homotopy catgory of compact (or constructible) modules over the motivic ring spectrum $H_{dR}$ representing algebraic de Rham…

We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…

We compute the homotopy groups of the {\eta}-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that…

We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice…

We study natural D-modules on the moduli stack of elliptic curves over a field of characteristic zero. We use this to produce an algebro-geometric version of the algebra of higher depth mock modular forms, studied from a Conformal Field…

We study the moduli space of A-infinity structures on a topological space as well as the moduli space of A-infinity-ring structures on a fixed module spectrum. In each case we show that the moduli space sits in a homotopy fiber sequence in…

Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C…

Let M be a von Neumann algebra, f a faithful normal state and denote by M^f the fixed point algebra of the modular group of f. Let U_M and U_{M^f} be the unitary groups of M and M^f. In this paper we study the quotient U_M/U_{M^f} endowed…

This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and…

The main result of this paper is a computation of the motivic cohomology of varieties of n \times m-matrices of of rank m, including both the ring structure and the action of the reduced power operations. The argument proceeds by a…

We consider a nontrivial action of $\mathrm{C}_2$ on the type $1$ spectrum $\mathcal{Y} := \mathrm{M}_2(1) \wedge \mathrm{C}(\eta)$, which is well-known for admitting a $1$-periodic $v_1-$self-map. The resultant finite…

We introduce a family of modules, called Markoff modules, generated by a cluster-mutation-like iterative process. We show that these modules are combinatorially similar to Christoffel words. Furthermore, we construct a bijective map between…

We study sequences of modular representations of the symplectic and special linear groups over finite fields obtained from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, and the…