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We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of…

This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…

This is an introduction to the algebraic aspect of Teichm\"uller dynamics, with a focus on its interplay with the geometry of moduli spaces of curves as well as recent advances in the field.

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…

Several definitions of differential operators on modules over noncommutative rings are discussed.

We investigate some general machinery for describing semidualizing modules over generic constructions like ladder determinantal rings with coefficients in a normal domain. We also pose and investigate natural localization questions that…

The paper mostly collects material on generic rank of $A$--modules with respect to differential geometric applications. Our research was motivated by geometry of $A$--structures. In particular, we discuss the case where $A$ is an unitary…

The well-known geometric approach to field theory is based on description of classical fields as sections of fibred manifolds, e.g. bundles with a structure group in gauge theory. In this approach, Lagrangian and Hamiltonian formalisms…

These are lecture notes for the course "MATS4120 Geometry of geodesics" given at the University of Jyv\"askyl\"a in Spring 2020. Basic differential geometry or Riemannian geometry is useful background but is not strictly necessary. Exercise…

The problem of equivalency for linear differential operators of the first order is discussed.

An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra homomorphism $\zeta:\frak g\to \frak X(M)$. We define orbits for such an action. In general the space of orbits $M/\frak g$ is not a manifold and even has a bad…

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple…

We demonstrate a method of associating the principal symbol at a $K$-point with a linear differential operator acting between modules over a commutative algebra, and we use it to define the ellipticity of a linear differential operator in a…

Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical…

Let $R$ be a commutative noetherian ring. The $n$-semidualizing modules of $R$ are generalizations of its semidualizing modules. We will prove some basic properties of $n$-semidualizing modules. Our main result and example shows that the…

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To…

In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that…

These are some notes on the basic properties of algebraic K-theory and G-theory of derived algebraic spaces and stacks, and the theory of fundamental classes in this setting.

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be…