Related papers: Rationality in Differential Algebraic Geometry
We study rationality properties of real singular cubic threefolds.
In this note, we give a motivic characterization of the integral cohomology of dual boundary complexes of smooth quasi-projective complex algebraic varieties. As a corollary, the dual boundary complex of any stably affine space (of positive…
Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum…
We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the…
A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…
The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the stan- dard parametric form, but it can also be in the implicit form which is commonly used in…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric…
CFTs are naturally defined on Riemann surfaces. The rational ones can be solved using methods from algebraic geometry. One particular feature is the covariance of the partition function under the mapping class group. In genus $g=1$, this…
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
We give a new computation of Hochschild (co)homology of the exterior algebra, together with algebraic structures, by direct comparison with the symmetric algebra. The Hochschild cohomology is determined to be essentially the algebra of…
The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which…
This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…
We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on…
A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…
In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…
A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…
We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…