Explore a cutting-edge application of topological data analysis in this 51-minute conference talk on topological optimization. Delve into the novel approach of using persistent homology to guide optimization processes. Discover how existing methods treat persistence calculation as a black box and learn about the limitations of backpropagating gradients only onto simplices involved in particular pairs. Examine the innovative technique of utilizing cycles and chains from persistence calculations to prescribe gradients to larger subsets of the domain. Understand the special case that serves as a building block for general losses, which can be solved exactly in linear time. Analyze empirical experiments demonstrating the practical benefits of this algorithm, including a significant reduction in the number of steps required for optimization by an order of magnitude. Gain insights from the joint work of Dmitriy Morozov and Arnur Nigmetov, presented at the Applied Algebraic Topology Network.