This talk reflects on my work on several key problems in Quantum Field Theory (QFT).

• The dynamical mechanism of mass generation of gauge fields (1963-1965), now known as the Higgs mechanism. I will discuss the history of the struggle to publish this paper (co-authored with Sasha Polyakov) against the objections of the conservative Landau School.

• The bootstrap approach to anomalous dimensions in scale- and conformal-invariant QFT (1967-1971). I will describe the origin of the bootstrap idea, where anomalous dimensions are determined from bootstrap equations.

• The loop equations in gauge QFT and turbulence problems (1981-1993). This involves the nonperturbative formulation of Gauge QFT without gauge fixing, the area law in QFT, and turbulence.

• The exact solution to the decaying turbulence problem in the 3 + 1 dimensional Navier-Stokes equation (2022-2024). I will present the exact duality of the turbulent velocity field in 3+1 dimensions to the one-dimensional quantum system of Fermi particles on a ring. This includes the exact solution of this dual Fermi ring system, the refutation of Kolmogorov scaling laws, and a new infinite spectrum of anomalous dimensions in turbulence related to the zeros of the Riemann zeta function.

This will be an informal Zoom talk, closely following the account of the chiral anomaly discovery and related calculations given in Chapter 3 of my Selecta, Adventures in Theoretical Physics: Selected Papers with Commentaries, Vol. 37 in the World Scientific Series in 20th Century Physics. This chapter is entitled, 'Chiral Anomalies and Their Nonrenormalization, Perturbative Corrections to Scaling, and Trace Anomalies to all Orders', and reviews both the chiral anomaly discovery and some  topics growing out of it.

I will describe basic principles of inflationary cosmology, its history and its present status. Some of its most unexpected results include the possibility that our universe was born from less than one milligram of matter, that galaxies are children of quantum fluctuations, and that our universe can be a self-reproducing fractal, a multiverse consisting of exponentially large parts with different properties. I will describe a large set of observational results testing various predictions of inflationary theory. I will also introduce a large class of inflationary models which can describe all presently available inflation-related observational data using a single parameter.

The year 2024 marks the fiftieth anniversary of a phase transition in our perception of Fundamental Physics. From phenomenological models and specific theories, each one applied to a restricted set of experimental data, we had to think in terms of a fundamental theory of universal validity. This revolution had a theoretical, rather an aesthetic motivation. It was a triumph of abstract theoretical thought which brought geometry into physics. For most of us this transition was a revelation, for some others it was a painful experience. In this talk I will attempt to narrate a part of this story, although I do not consider myself to be an unbiased observer.

Topological integrals of motion, magnetic monopoles in grand unification theories. Alice strings and Cheshire charge. Instantons. Atiyah-Singer index theorem, dimension of the space of instantons and Adler-Bell-Jackiw anomaly. Topological quantum field theories, Ray-Singer invariants, Chern-Simons theory, invariants of knots.