Between 1969 and 1974, Doplicher, Haag and Roberts published a series of papers, studying the structure of the algebra of observables of general QFTs. Only very recently did those ideas get adapted to the study of discrete systems, or quantum lattice systems.

In this talk, mostly based on Corey Jones' original paper (arXiv 2304.00068), I will give an overview of the mathematical machinery behind what he called "discrete DHR theory". I will also present some of the main results that have been developed in this formalism: a new tool for the study of Quantum Cellular Automata, and a SymTFT-like construction for discrete systems.

We will begin with an overview of Stark's conjectures before discussing the case of imaginary quadratic fields, covering both the limit formula and the existence of elliptic units. The classical expositions of these are at times lacking in intuition, but thanks to Kato's deep insights 20 years ago, we can present more geometric and illuminating proofs of both results.

Fix an Artin representation rho. Work in progress by Emmanuel Lecouturier and Loïc Merel claims that the special values L(f,rho,1) for certain modular forms f see some global data related to the L-function attached to rho. We first give a brief exposition on Mazur’s Eisenstein ideal, which lies at the heart of their work. We then describe this conjectural phenomenon in a few simple cases, the last being related to a conjecture of Harris and Venkatesh.

Quantum field theory is hard for several reasons, for example one can rarely compute perturbation series (=Feynman integrals) at large loop order, and even if you can, the series diverges. Conversely, intrinsically non-perturbative approaches like the functional renormalization group require approximations that are often not easy to control, or have unclear relations to perturbative computations.
Tropical field theory is a new approach for solving these issues for a generic theory without restricting to unphysical boundary cases. It keeps almost all qualitative and combinatorial features of perturbative QFT (in particular all non-planar diagrams, renormalization, relative numerical importance of Feynman integrals, and divergence of perturbation series), while at the same time reducing the analytic complexity, and establishing a rigorous connection to non-perturbative functional/path integral methods of QFT. Based on 2512.21091 with Erik.

Negative-form symmetries arise when one extends the usual p-form dictionary below ordinary zero-form symmetries. Conceptually, however, they are different: the action of (-n)-form symmetries on a QFT modifies the parameters or background data that defines the QFT, as opposed to acting on the extended operators of the theory. For example, (-1)-form symmetries are implemented by spacetime-filling topological operators that act on a theory by shifting its theta-angle. I will review recent work arxiv:2606.05543 that has begun to develop the machinery of (-2)-form symmetries, which act of a QFT by modifying the anomaly inflow data – equivalently the SymTFT action – thereby relating QFTs whose ordinary global symmetries differ by anomaly data.

Global symmetries have been generalized to non-invertible ones. For finite symmetries in $(1+1)$d, these are known as unitary fusion category symmetries. One natural question is: which fusion categories can arise as symmetries on a lattice? 
Progress has been made including the anyon chains, which realizes any fusion category symmetries. However, their Hilbert spaces do not admit the usual tensor product structure (tensor product of local Hilbert spaces over each site).
In [arxiv:2507.05185], Evans and Jones introduced an operator-algebraic framework and showed that a fusion category symmetry can be realized on a tensor product quasi-local algebra if and only if it is "integral". After reviewing this result, I will discuss a recent extension by Bunner and Jones [arxiv:2605.21327], who showed that this constraint disappears after stabilization with infinite-dimensional ancilla spaces on anyon chains. As a consequence, every unitary fusion category can be realized on tensor product Hilbert spaces.

Quantum magic quantifies the computational resources needed for quantum operations that cannot be easily performed classically. This requires unitaries, known as "Non-Clifford gates", that map Pauli operators to outside the Pauli group. I will first provide a pedagogical introduction to these concepts following [arXiv:quant-ph/9807006] and then summarise the recent results of [arXiv:2604.14271] constructing non-Clifford gates from path integrals in Chern-Simons theories, whose magic-generating properties are determined by the algebraic data of the topological field theory.