Trunov, Stanislav A.2023-05-052023-05-052023-08-01https://hdl.handle.net/2097/43306In this dissertation, we discuss the Lusternik-Schnirelmann category and relevant results. We will then introduce variants for a closed manifold M of the Lusternik-Schnirelmann category using Singhof-Takens fillings as well as a variant for the minimum number of critical points for any smooth function on M. We will show that the category of a Singhof-Takens filling by topological balls with corners is related to the minimum number of critical points for any smooth function on M such that the critical points admit a gradient convex ball neighborhood. We will also show that the category of a Singhof-Takens filling by smooth balls with corners is related to the minimum number of critical points for any smooth function on M.en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Lusternik-Schnirelmann categoryCritical pointsFillingDissertation