Bakı Dövlət Universiteti Rəqəmsal Tədqiqat Laboratoriyası

Çıxış üçün Keegan Jonathan Flood'a xüsusi təşəkkürümüzü bildiririk!
Çıxışın məzmunu aşağıdakı kimidir :

Classically, jet bundles provide the framework for variational calculus as well as for both the Lagrangian and the Hamiltonian formalism in physics. Further, the geometric theory of differential equations is formulated in the context of jet bundles. Indeed, all of classical differential geometry can be encoded in jet constructions and operations since all geometric objects (in a sense that can be made precise) arise as sections of associated bundles to jet groups and every natural operation between such geometric objects can be seen to depend only on the infinity jet of the corresponding section. For these reasons, there is naturally a great deal of interest in extending the notion of jet to settings generalizing classical differential geometry (e.g. higher geometry, noncommutative geometry, etc.). In this talk, we will be concerned with the extension of jet theory to the noncommutative setting. Noncommutative differential geometry generalizes classical differential geometry by replacing the commutative algebra $C^{\infty}(M)$ of smooth functions on a smooth manifold $M$ with an arbitrary unital associative algebra $A$ over a commutative ring $\mathbbm{k}$. One then considers a dg-algebra over $A$, which plays the role of a generalized notion of exterior algebra and imposes a notion of ‘‘geometry’’ on $A$. We will see that this data is sufficient to construct, via homological algebra and category theory, a generalization of the classical notion of jet to this noncommutative setting. Further, we will see that this allows for the extension of a considerable amount of differential geometric machinery to the noncommutative setting. In particular, we will discuss differential operators and their corresponding principal symbols and see how they can be used to provide a partial answer to the open question of determining the ‘‘total space'' of a given $A$-module.