Explore advanced mathematical concepts in this 58-minute lecture by Maxim Kontsevich from the Institut des Hautes Etudes Scientifiques. Delve into calculations with Getzler-Gauss-Manin connections for non-commutative algebras, examining the effects of inverting elements and the resulting periodic cyclic homology. Investigate meaningful examples including quantum tori, (micro)-differential operators, and noncommutative Laurent polynomials. Progress through topics such as integration cycles, complex numbers, frame maps, quantum periods, and matrix models, gaining insights into algebraic structures and their applications in advanced mathematics.