We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker--Planck type equations with a non-local diffusion operator for the full range of the non-locality exponents in (0,1).  This implies Hölder continuity.  We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities.  Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

Topological defects in quantum field theory can be understood as a generalised notion of symmetry, where the operation is not required to be invertible. Duality transformations are an important example of this. By considering defects of various dimensions, one is naturally led to more complicated algebraic structures than just groups. So-called 2-groups are a first instance, which arise from invertible defects of codimension 1 and 2. Without invertibility one arrives at so-called "fusion categories”. I would like to explain how one can "gauge" such non-invertible symmetries in the case of topological field theories, and I will focus on results in two and three dimensions. This talk is based on joint work with Nils Carqueville, Vincentas Mulevicius, Gregor Schaumann, and Daniel Scherl.

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. To help with the investigation of related questions, we introduce holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points, all of which are illustrated on explicit examples.

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.    

In the early 2010s, Riche and Williamson proposed new character formulas for simple and indecomposable tilting modules over reductive algebraic groups $G$ in positive characteristic. Even better, they showed their formulas would follow from a conceptually satisfying "categorical conjecture", which they were able to prove for $G = GL_n$. Our first goal in this talk will be to explain the statement of the categorical conjecture, indicating its connection to representation theory and assuming minimal background knowledge. Subsequently, we will introduce Smith–Treumann theory and outline how it may be applied to prove the categorical conjecture in general. Time permitting, we will conclude with remarks on future directions of study.

Combining physics with machine learning is a rapidly growing field of research. Thereby, most work focuses on leveraging machine learning methods to solve problems in physics. Here, however, we focus on the reverse direction of leveraging structure of physical systems (e.g. dynamical systems modeled by ODEs or PDEs) to construct novel machine learning algorithms, where the existence of highly desirable properties of the underlying method can be rigorously proved. In particular, we propose several physics-inspired deep learning architectures for sequence modelling as well as for graph representation learning. The proposed architectures mitigate central problems in each corresponding domain, such as the vanishing and exploding gradients problem for recurrent neural networks or the oversmoothing problem for graph neural networks. Finally, we show that this leads to state-of-the-art performance on several widely used benchmark problems.

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states and incoherence-mediated remote synchronization.

Synchronization, epidemic processes and information spreading are natural processes that emerge from the interaction between people. Mathematically, all of them can be described by models that, despite their simplicity, they can predict collective behaviors. In addition, they have in common a very interesting feature: a phase transition from an active state to an absorbing state. For example, the spread of an epidemic is characterized by the infection rate, the control parameter, which basically determines whether the epidemic will spread in the network or, if this rate is very low, few people become infected and the system falls into an absorbing state where there are no more infected people. In this presentation we will present the analytical and computational approaches used to investigate these classical models of statistical physics that present phase transitions and we will also show how the network topology influences such dynamical processes. The behavior of such dynamics can be much richer than we imagine.