Honestly, I hate watching podcasts and video recorded lectures, and this one is a little more than an hour long. But, I've had my eye on conformal symmetry papers for a while now, as a potentially promising direction for new breakthroughs, and this one was definitely eye catching.

So, even if I don't watch it, I may chase down some of Shaposhnikov's papers on the topics, at least, to see what progress he's made and to better grasp the gist of his arguments.

UPDATE January 3, 2022: Upon reading several of the papers, I am very impressed that Shaposhnikov and his colleagues are really onto ground breaking Copernican revolution class insights, and I eagerly await what more there is to come. The first two papers below are very impressive and complementary.

I very much recommend taking a look at the talk from earlier this year by Mikhail Shaposhnikov, Conformal symmetry: towards the link between the Fermi and the Planck scales. Shaposhnikov has done a lot of fascinating work over the years, developing in detail a point of view which hasn’t got a lot of attention, but that seems to me very compelling.He argues that the SM and GR make a perfectly consistent theory up to the Planck scale, with the “naturalness problem” disappearing when you don’t assume something like a GUT scale with new heavy particles.Watching the discussion after the talk, one sees how many people find it hard to envision such a possibility, even though all experimental evidence shows no signs of such particles. For more about what he is in mind, see the talk or some of the many papers he’s been writing about this.

From Woit at Not Even Wrong. Some of the recent papers developing these ideas are:

The standard way to do computations in Quantum Field Theory (QFT) often results in the requirement of dramatic cancellations between contributions induced by a "heavy" sector into the physical observables of the "light'' (or low energy) sector - the phenomenon known as "hierarchy problem''. This procedure uses divergent multi-loop Feynman diagrams, their regularisation to handle the UV divergences, and then renormalisation to remove them. At the same time, the ultimate outcome of the renormalisation is the mapping of several finite parameters defining the renormalisable field theory into different observables (e.g. all kinds of particle cross-sections).

In this paper, we first demonstrate how to relate the parameters of the theory to observables without running into intermediate UV divergences. Then we go one step further: we show how in theories with different mass scales, all physics of the "light" sector can be computed in a way which does not require dramatic cancellations induced by physics of the "heavy" sector.The existence of such a technique suggests that the "hierarchy problem'' in renormalisable theories is not really physical, but rather an artefact of the conventional procedure to compute correlation functions.If the QFT is defined by the "divergencies-free'' method all fine-tunings in theories with well separated energy scales may be avoided.

We show how Einstein-Cartan gravity can accommodate both global scale and local scale (Weyl) invariance. To this end, we construct a wide class of models with nonpropagaing torsion and a nonminimally coupled scalar field. In phenomenological applications the scalar field is associated with the Higgs boson. For global scale invariance, an additional field -- dilaton -- is needed to make the theory phenomenologically viable.In the case of the Weyl symmetry, the dilaton is spurious and the theory reduces to a sub-class of one-field models.In both scenarios of scale invariance, we derive an equivalent metric theory and discuss possible implications for phenomenology.

We study scalar, fermionic and gauge fields coupled nonminimally to gravity in the Einstein-Cartan formulation.We construct a wide class of models with nondynamical torsion whose gravitational spectra comprise only the massless graviton. Eliminating non-propagating degrees of freedom, we derive an equivalent theory in the metric formulation of gravity.It features contact interactions of a certain form between and among the matter and gauge currents. We also discuss briefly the inclusion of curvature-squared terms.

Quantum field theories with exact but spontaneously broken conformal invariance have an intriguing feature: their vacuum energy (cosmological constant) is equal to zero.Up to now, the only known ultraviolet complete theories where conformal symmetry can be spontaneously broken were associated with supersymmetry (SUSY), with the most prominent example being the N=4 SUSY Yang-Mills. In this Letter we show thatthe recently proposed conformal "fishnet" theory supports at the classical level a rich set of flat directions (moduli) along which conformal symmetry is spontaneously broken.We demonstrate that, at least perturbatively, some of these vacua survive in the full quantum theory (in the planar limit, at the leading order of 1/N(c) expansion) without any fine tuning. The vacuum energy is equal to zero along these flat directions, providing the first non-SUSY example of a four-dimensional quantum field theory with "natural" breaking of conformal symmetry.