Roman Fedorov has been recently funded by the US National Science Foundation (award DMS 2001516) for the proposal “Principal Bundles and Higgs Bundles in Algebraic Geometry” for the period of August 1, 2020 – July 31, 2020. This project is directed at the study of principal bundles began in the early XXth century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that principal bundles became a basis for the so-called standard model -- a physical theory describing three out of four fundamental interactions. In mathematics, principal bundles penetrate many areas: geometry, number theory, mathematical physics, and others. In 1950's Fields Medalist Jean-Pierre Serre recognized the importance of principal bundles in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a principal bundle and formulated a certain deep conjecture. This conjecture, as well as some remaining questions, are among the oldest unsolved foundational questions in mathematics. The first part of the project is aimed at proving some of these conjectures. The remaining parts of the projects are related to the so-called Higgs principal bundles, which can be thought of as mathematical incarnations of the Higgs boson -- a recently found elementary particle. These parts of the project belong to the famous Langlands program unifying number theory, algebraic geometry, harmonic analysis, and mathematical physics.