In this talk, I will present a new formalism for studying the dynamics of field theory on manifolds with boundary. Based on the ideas of relative cohomology [Margalef-Bentabol & Villaseñor, 2021], I will define what it means for a manifold and its boundary to be multisymplectic and I will motivate this definition from the variational principle. I will explain how to generalize observables, graded Poisson brackets, and conserved charges to manifolds with boundary. I will present the Lagrangian formalism for holonomic variations, deriving the Euler–Lagrange equations and the Poincaré–Cartan form. Finally, I will illustrate the formalism with examples in mechanics and scalar field theory. 

Degree-2 QP manifolds are the standard geometric home for Courant algebroids. Classically, they do a great job at the bottom of the ladder: degree 0 and 1 functions perfectly capture the scalars and gauge parameters of generalised geometry. But to build the full kinematics of Double Field Theory, we need the physical fields, field strengths, and Bianchi identities. These live at higher QP degrees, and the classical framework is incomplete.

In this talk, I’ll show how deformation quantisation fixes this. By deforming the graded exterior algebra into a Clifford algebra via a star product, the deformed master equation algebraically forces the generalised dilaton into existence, completing the tensor hierarchy to all higher degrees.

From there, we move to dynamics: I'll show how reducing the structure group with a generalised metric uniquely fixes the DFT action.

The Frobenius theorem is an irreplaceable tool in the toolbox of every differential geometrist. It is sometimes useful to distinguish between two versions of the theorem - a local and a global one. The local Frobenius theorem states that for every involutive regular distribution there locally exists a flat coordinate chart. It is known to hold for graded manifolds of many different gradings. On the other hand, the global Frobenius theorem equates involutive regular distributions with foliations by integral submanifolds. In this short talk, we would like to give some theorems and examples which show that in the realm of Z-graded geometry things may not be so simple. For instance, one may have a non-involutive distribution which admits a foliation by integral submanifolds. 

This talk explores the relationship between $k$-contact geometry and the Lie symmetries of finite-order partial differential equations. Building on the geometric formulation of PDEs through contact manifolds and jet spaces, we show that every finite-order jet bundle equipped with its Cartan distribution can be viewed as an $N_k$-contact manifold, locally represented in adapted jet coordinates by natural $N_k$-contact forms. Using this approach, we also recover standard objects from the literature, such as the characteristics of Lie symmetries and $\lambda$-symmetries of PDEs. If time permits, the presented results will be illustrated with several examples. 

Interpreting Morita equivalence of noncommutative spacetime manifolds as a gauge theory duality dates back to the seminal work of Connes, Douglas, and Schwarz from 1997 on noncommutative tori. In this talk, we propose a pathway to vastly generalise this framework utilising the theory of quantum groups and their homogeneous spaces.

We will first establish the appropriate notion of equivariant Morita equivalence for quantum homogeneous spaces. To demonstrate the power of this framework, we will explore a highly surprising example: the Morita equivalence between deformed versions of 2D de Sitter and Anti-de Sitter spaces.

Finally, we will outline work-in-progress on formulating gauge theories over (some) quantum homogeneous spaces, and discuss how Morita equivalence should manifest as a duality in this new setting.

This work investigates the (p, q)−deformed Airy equation, which generalizes the classical Airy differential equation using two deformation parameters within the framework of (p, q)−calculus. An analytical solution is derived by constructing a power series expansion involving (p, q)−derivatives, leading to a deformed version of the Airy function that reflects quantum or discrete effects. Due to the complexity of the full infinite series, a numerical solution is also developed by truncating the series after a finite number of terms. This numerical approximation allows for practical evaluation and graphical representation of the solution’s behavior for specific values of p and q, where 0 < p < q < 1. Comparison between the exact and numerical methods reveals strong agreement in regions where the series converges rapidly, validating the effectiveness of the truncated model. The study emphasizes the importance of (p, q)−deformation in capturing non-classical dynamics and provides a useful framework for further applications in quantum mechanics and discrete systems.