tbranzov

With next lecture we are continuing the mini-course started in November. We shall discuss O. Viro’s method of patchworking. For that we need some staff from real algebraic curves, which will be discussed first, as much as the time permits.

The emergence of continuum spacetime from a quantum substrate remains a key challenge in theoretical physics. Continuum geometry may arise dynamically through renormalization group (RG) flows in theory space. This talk reviews conceptual aspects of continuum emergence within quantum gravity, emphasizing non-perturbative RG methods in analyzing the global structure of theory space and its phase transitions. It then proposes, in a collaborative framework, an extension of this analysis to discrete quantum gravity approaches, investigating whether topological invariants of RG trajectories—such as fixed-point indices, flow connectivity, or higher-order topological characteristics—can serve as structural probes of the theory space, particularly within tensorial group field theory (TGFT). Further, we propose developing a computational framework based on Monte Carlo RG and lattice field theory techniques to construct alternative formulations not always accessible by analytical methods. The broader aim is to position quantum gravity research at the intersection of complexity science, computational physics, and pure mathematics.

In this talk I will introduce valuation-theoretic tools, such as the graded algebra grν(R) of a general valuation ν on a commutative ring R. This will serve as a tool to algebraically encode the leading terms of our analytic solutions. The graded algebra construction presents certain advantages, such as functoriality, which is essential for proving the following lifting theorem.