Explore the fundamental concepts of invariants in manifolds from six dimensions in this lecture by Pavel PUTROV from ICTP. Delve into the intricate problem description, vector spaces, and complex numbers. Examine grading systems in vector spaces and their applications in lower dimensions. Investigate multi-multiple environments and their significance. Understand the crucial definition of state field correspondence and its relation to vertex operator algebra. This comprehensive talk, part of the School and Workshop on Gauge Theories and Differential Invariants, provides a deep dive into advanced mathematical concepts essential for understanding high-dimensional manifolds and their invariant properties.