February 9th 2024 - Paolo Meda (University of Pavia), "The Semiclassical Einstein Equations and the Stability of Linearized Solutions".
Abstract: The semiclassical formulation of gravity is discussed in the framework of algebraic quantum field theory in curved spacetimes. The main topic of the talk is the Semiclassical Einstein Equations, which describe the backreaction of a quantum scalar field on spacetime geometry. In the first part of the talk, it is shown that an initial-value problem for local solutions of the Semiclassical Einstein Equations can be formulated in cosmological spacetimes, after fixing four initial data on the scale factor of the Universe. In the second part of the talk, it is studied the problem of stability of linearized solutions, using a toy model which mimics the semiclassical equations in cosmological spacetimes. In this case, it is proved that, if the quantum field driving the backreaction is massive, then there are choices of renormalization constants for which linear perturbations with compact spatial support decay for large times, thus indicating stability of the underlying theory [arXiv:2007.14665, 2201.10288]. The content of the talk is based on two papers I wrote during my PhD. I will try to provide a lay introduction to the topic (Semiclassical Einstein Equations + algebraic methods) before discussing the results, to make the topic as accessible to a broad audience as possible.
February 2nd 2024 - David Kern (University of Göttingen) "Constraint Geometry and Reduction of Dirac Geometry".
Abstract: In this talk we first motivate constraint geometry by recalling Poisson geometry. Afterwards we introduce constraint manifolds as well as constraint vector bundles and prove a constraint Cartan calculus. Finally, we apply it to Dirac manifolds showing their reduction.
January 26th 2024 - Dominique Manchon (Clermont-Ferrand), "The free tracial pre-Lie-Rinehart algebra".
Abstract: Tracial Lie-Rinehart algebras are a purely algebraic version of finite-dimensional Lie algebroids, for which the trace is given fibrewise. A tracial Lie-Rinehart algebra is pre-Lie if moreover both torsion and curvature vanish. The simplest example is given by the tangent bundle the smooth manifold R^d, the trace being given by the divergence of vector fields. We shall describe in this talk the free tracial pre-Lie-Rinehart algebra in terms of aromatic rooted trees. Joint work with Gunnar Fløystad and Hans Z. Munthe-Kaas.
December 1st 2023 - Jean-David Jacques (University of Potsdam), "Post-Lie algebra of derivations and regularity structures".
Abstract: Post-Lie algebra structures are a generalization of Pre-Lie algebras. They have their roots in geometry and correspond to the algebraic properties satisfied by the covariant derivative in the case of a flat and constant torsion connection. In my talk, I will provide a brief overview of the new theory of regularity structures developed by F. Otto and colleagues, and discuss how post-Lie algebra structures arise in this framework.
November 24th 2023 - Fabrizio Zanello (University of Potsdam), "Higher currents for the sine-Gordon model in perturbative Algebraic Quantum Field Theory".
Abstract: First, we review the 2-dimensional sine-Gordon model in Classical Field Theory and derive recursive formulas for the components of an infinite number of conserved currents. Then, we describe the formulation of the sine-Gordon model in perturbative Algebraic Quantum Field Theory and show, in the framework of Epstein-Glaser renormalization, that the interacting components of the currents are super-renormalizable by power counting. Finally, we describe how, under suitable conditions, the formal power series describing the interacting components of the currents are in fact summable.
November 17th 2023 - Umida Baltaeva (Uzbekistan Academy of Sciences), "Boundary value problems for loaded partial differential equations with the classical and nonclassical operators".
Abstract: I will introduce loaded equations and their applications to applied problems. Next, I will discuss the investigation of the boundary value problem for the linear loaded partial differential equations and their relation with nonlocal boundary value problems for classical partial differential equations. Further, I investigate the Cauchy-type problem for a loaded parabolic-hyperbolic type equation with Riemann-Liouville fractional differential operator.
October 27th 2023 - Pablo Linares (Imperial College, London), "Rough paths based on multi-indices and their algebraic renormalization".
Abstract: Following the approach of Otto et. al., we derive a notion of rough path based on multi-indices suitable for real-valued rough differential equations, and build its associated algebraic renormalization group. The construction is based on well-known results on pre-Lie algebras (Guin-Oudom construction), applied to pre-Lie products that arise from shifting the affine spaces of solutions and equations. We show that these products correspond in the tree-based language to grafting and insertion, respectively.
