In the first lecture I will present the basics of algebraic quantum field theory (AQFT), which is a mathematically rigorous framework to study foundations of QFT. I will also talk about its perturbative version (pAQFT) that allows us to treat physically interesting models like quantum electrodynamics (QED) and the Standard Model of particle physics in 4 spacetime dimensions, which are currently not accessible to non-perturbative methods. In the second lecture, I will present my own contributions, which include: quantization of gauge theories and effective quantum gravity in pAQFT, formulation of functional renormalization group equations in Lorentzian signature and incorporation of quantum reference frames (QRFs) into the (p)AQFT setting.

In the first lecture I will present the basics of algebraic quantum field theory (AQFT), which is a mathematically rigorous framework to study foundations of QFT. I will also talk about its perturbative version (pAQFT) that allows us to treat physically interesting models like quantum electrodynamics (QED) and the Standard Model of particle physics in 4 spacetime dimensions, which are currently not accessible to non-perturbative methods. In the second lecture, I will present my own contributions, which include: quantization of gauge theories and effective quantum gravity in pAQFT, formulation of functional renormalization group equations in Lorentzian signature and incorporation of quantum reference frames (QRFs) into the (p)AQFT setting.

Denoising diffusion models have become a central tool in modern generative modelling, yet our theoretical understanding of them remains incomplete. We discuss diffusion models from a statistical perspective, with an emphasis on provable guarantees for high-dimensional data generation. The first talk provides a self-contained introduction to diffusion models and reviews recent results on convergence guarantees under smoothness assumptions, while highlighting the gap between idealised theoretical constructions and practical implementations. The second talk focuses on reflected diffusion models, which have been proposed as a principled approach to stabilising diffusion-based generation on bounded domains. We present recent results establishing minimax-optimal rates of convergence in total variation, up to polylogarithmic factors, and introduce refined score approximation techniques based on spectral methods and neural network analysis. Together, the two talks introduce recent theoretical developments in diffusion models and illustrate current research directions.

In the first part of the talk, we consider general stochastic interacting systems evolving on discrete spaces, such as large finite or infinite graphs. Even when the local interaction rules are simple, their interplay with the spatial structure can generate a wide range of macroscopic behaviors over long space and time scales. Understanding phase transitions between these behaviors is of particular interest. We describe such stochastic interacting systems via their ancestral structure, which is particularly amenable to analysis in the (non-spatial) mean-field case, often considered as a first step.

We then specialize to interacting particle systems, which have a finite local state space that in many cases consists of just two states, interpretable, for instance, as the presence or absence of a particle. We give an overview of results for several classical interacting particle systems and highlight the methods used to obtain them.

In the second part of the talk, we focus on variants of the contact process, a classical interacting particle systems that models the spread of an infection in space. Here, the particle at a particular site models the presence of an infection that can be passed on to its connected (uninfected) neighbours at a certain rate. The model also includes spontaneous recovery from the infection and can thus be viewed as an SIS (susceptible-infected-susceptible) epidemic model. Macroscopic properties of intererst for these models are for example the conditions needed for global (or local) long-term survival of the infection.

While the classical contact process on lattices (such as ℤ^d) has been well studied we focus in particular on more recent work regarding the contact process on a larger variety of graphs (such as trees), random graphs and random graphs that change their connective properties dynamically and randomly over time.

Parts of the talk are based on joint work with Natalia Cardona-Tobon, Marcel Ortgiese, Marco Seiler and Jan Swart.

We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type and show that, once the population approaches the carrying capacity, its evolution converges to a Gillespie-Wright-Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals. We prove that, provided that the advantageous and disadvantageous branching measures are ordered, this ancestral line process converges to the moment dual of the limiting diffusion. This talk is based on joint work with Julian Kern.

In the first part of the talk, we consider general stochastic interacting systems evolving on discrete spaces, such as large finite or infinite graphs. Even when the local interaction rules are simple, their interplay with the spatial structure can generate a wide range of macroscopic behaviors over long space and time scales. Understanding phase transitions between these behaviors is of particular interest. We describe such stochastic interacting systems via their ancestral structure, which is particularly amenable to analysis in the (non-spatial) mean-field case, often considered as a first step.

We then specialize to interacting particle systems, which have a finite local state space that in many cases consists of just two states, interpretable, for instance, as the presence or absence of a particle. We give an overview of results for several classical interacting particle systems and highlight the methods used to obtain them.

In the second part of the talk, we focus on variants of the contact process, a classical interacting particle systems that models the spread of an infection in space. Here, the particle at a particular site models the presence of an infection that can be passed on to its connected (uninfected) neighbours at a certain rate. The model also includes spontaneous recovery from the infection and can thus be viewed as an SIS (susceptible-infected-susceptible) epidemic model. Macroscopic properties of intererst for these models are for example the conditions needed for global (or local) long-term survival of the infection.

While the classical contact process on lattices (such as ℤ^d) has been well studied we focus in particular on more recent work regarding the contact process on a larger variety of graphs (such as trees), random graphs and random graphs that change their connective properties dynamically and randomly over time.

Parts of the talk are based on joint work with Natalia Cardona-Tobon, Marcel Ortgiese, Marco Seiler and Jan Swart.

Denoising diffusion models have become a central tool in modern generative modelling, yet our theoretical understanding of them remains incomplete. We discuss diffusion models from a statistical perspective, with an emphasis on provable guarantees for high-dimensional data generation. The first talk provides a self-contained introduction to diffusion models and reviews recent results on convergence guarantees under smoothness assumptions, while highlighting the gap between idealised theoretical constructions and practical implementations. The second talk focuses on reflected diffusion models, which have been proposed as a principled approach to stabilising diffusion-based generation on bounded domains. We present recent results establishing minimax-optimal rates of convergence in total variation, up to polylogarithmic factors, and introduce refined score approximation techniques based on spectral methods and neural network analysis. Together, the two talks introduce recent theoretical developments in diffusion models and illustrate current research directions.