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Bergman's Diamond Lemma for ring theory gives an algorithm to produce a (non-canonical) basis for a ring presented by generators and relations. After demonstrating this algorithm in concrete, geometrically-minded examples, I'll turn to preprojective algebras and their multiplicative counterparts. Using the Diamond Lemma, I'll reprove a few classical results for preprojective algebras. Then I'll propose a conjectural basis for multiplicative preprojective algebras. Finally I'll explain why the set is a basis in the case of multiplicative preprojective algebras for quivers containing a cycle, the subject of joint work with Travis Schedler.

Results from the first digital quantum simulation of an SU(2) gauge theory are presented. This was done by analytically constructing gauge-invariant states and implementing a Trotterized time evolution operator for that basis on superconducting hardware. By using error mitigation techniques, electric energy measurements could be reliably extracted following one Trotter-Suzuki time step. This work is a small but important step toward determining what field-theoretic calculations will be possible using near-term devices.

In quantum spin systems, the existence of a spectral gap above the ground state has strong implications for the low-energy physics. We survey recent results establishing spectral gaps in various frustration-free spin systems by verifying finite-size criteria. The talk is based on collaborations with Abdul-Rahman, Lucia, Mozgunov, Nachtergaele, Sandvik, Yang, Young, and Wang.

I will explain how dark energy in cosmology could arise from the

noisy diffusion of energy from the low energy degrees of freedom of matter (described in terms of QFT)

to the fundamental Planckian granularity (expected from quantum gravity). This

perspective leads to a natural model resolving the fine tuning problem associated to the small

value of the cosmological constant. However, recent observations suggest that the dark energy

component in our universe might not be constant and should  instead have grown from the recombination time to the present.

I will present new ideas suggesting a possible explanation within the same framework.

Analytic renormalisation "à la Speer" using a multivariable approach typically leads to meromorphic germs in several variables whose poles are linear.  In particular,  Feynman integrals, multizeta functions and their generalisations, namely  discrete sums on cones and discrete sums associated with trees give rise to meromorphic germs at zero with linear poles. We shall present  a multivariable renormalisation scheme which amounts to a minimal subtraction scheme in several variables. It preserves locality in so far as the  evaluation at poles is expected to factor  on functions with independent sets of variables. Inspired by Speer, we shall discuss a class of generalised evaluators that do the job. Using a theory of  Laurent expansions  on meromorphic germs with linear poles at zero, we shall relate these generalised evaluators to the geometry of cones.

This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang.