Date: April 14, 2025
Time: 13.39 AST, 9.30 GMT
Location:
Advanced concepts in Mathematical Physics: an operator algebraic approach
Christiaan J.F. van de Ven, TNO
Abstract:
In this talk, I will give an overview of my research interests and current projects within the field of modern Mathematical Physics. A central focus of my work is the mathematical derivation of the foundations of quantum mechanics. In particular, I highlight the concept of asymptotic emergence, a rigorous mathematical framework for understanding the classical and macroscopic limits of quantum systems.
Key topics of my interest in this context include the abstract formulation of natural phenomena such as spontaneous symmetry breaking (SSB), phase transitions, and entropy. I explore these ideas through the lens of strict deformation quantization of Poisson manifolds, C*-algebras, analysis, and large deviation theory. These themes will be central to the seminar.
Date: March 28, 2025
Time: 15.00 AZT 11.00 am GMT
Location:
Jets, differential operators, and principal symbols in noncommutative geometry
Keegan Jonathan Flood, UniDistance Suisse
Abstract:
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.