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This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

We investigate P. Halmos' two projections theorem, (or two subspaces theorem) in the context of a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra).

Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.

Hypergeometric solutions to seven q-Painlev\'e equations in Sakai's classification are constructed. Geometry of plane curves is used to reduce the q-Painlev\'e equations to the three-term recurrence relations for q-hypergeometric functions.

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study…

We propose a general definition of unprojection, and prove that it indeed generalizes previous efforts.

To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already…

In this paper we give a brief overview of the geometry of involute gears, from a mathematical more than an engineering perspective. We also list some of the many variant geared mechanisms and discuss some of our 3D printed mechanisms.

This is a technical introduction to the paper "Extension of twisted Hodge metrics for Kahler morphisms" by the authors.

We present a systematic means to impose Galilean invariance within field theory. We begin by defining the most general background geometries consistent with Galilean invariance and then turn to applications within effective field theory,…

The authors establish a relation of the theory of varieties with degenerate Gauss maps in projective spaces with the theory of congruences and pseudocongruences of subspaces and show how these two theories can be applied to the construction…

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…

In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…

We generalize Hrushovski's group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically stable types, which we recall in…

We show that one can construct two equivalent gauge theories from a linking theory and give a general construction principle for linking theories which we use to construct a linking theory that proves the equivalence of General Relativity…

Using Gillet's technique of projective envelopes, we prove a homological descent theorem for the connective K-homology of schemes.

We give an a geometric interpretation of the Hasse-Arf theorem for function fields using the recently proved Oort conjecture.

A generalized translational invariant noncommutative field theory is analyzed in detail, and a complete description of translational invariant noncommutative structures is worked out. The relevant gauge theory is described, and the planar…

General structure of the multivariate plain and q-hypergeometric terms and univariate elliptic hypergeometric terms is described. Some explicit examples of the totally elliptic hypergeometric terms leading to multidimensional integrals on…

This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe…