Related papers: Lefschetz properties through a topological lens
Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, defined by invariance under the…
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…
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…
Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf=gu. We give some conditions on the square which imply that u is formally smooth. This result encapsulates a variety of (apparently unrelated)…
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are…
These notes are an introduction to basic properties of Andre-Quillen homology for commutative algebras. They are an expanded version of my lectures at the summer school: Interactions between homotopy theory and algebra, University of…
This paper gives an overview of some basic properties of Leibniz algebras. Some of the results were known earlier, but in the article they are accompanied by new simple proofs. Some of the results are new. The article can be viewed as a…
We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a…
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta…
Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is…
For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple…
We study G-graded Artinian algebras having Poincar\'e duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and…
MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more…
These are expanded notes from a four lecture mini-course given by the author at the Spring School on Non-archimedean geometry and Eigenvarieties, held at the University of Heidelberg in March 2023. The course discusses coherent sheaves,…
We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by…
Motivated by our attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the…
In the recent paper "The Nakayama functor and its completion for Gorenstein algebras", a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were…
We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust.…
We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing…
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral…