Proofs and Computations: A Journey Through Mathematical Theory

This lecture delves into the historical milestones of mathematical proofs and computations, starting from Euclid's 'The Elements' to the shattered dreams of Hilbert in the 20th century with Gödel's Incompleteness Theorem and Turing's Undecidability Theorem. It explores decision problems, deductive proof systems like Peano Arithmetic, and the essentials of proof systems. The lecture also covers probabilistic proofs, interactive proofs, quantum computation, and quantum proof systems, highlighting the impact of randomness and errors in proofs. It concludes with the discussion on zero-knowledge interactive proofs, two-prover interactive proofs, and the conceptual implications of quantum mechanics in proof systems.