The talk discusses the concept of positive semidefinite rank and its connection to semidefinite programming (SDP) representations of convex sets. It begins with a review of Euclidean distance matrices (EDMs) and their characterization via Schoenberg's theorem. EDMs can be represented as SDP problems, which allows for the modeling of distance geometry problems. The concept of semidefinite representation is introduced, where a convex set can be expressed as the projection of a positive semidefinite matrix in a higher-dimensional space. Examples include EDMs and the unit disk in 2D, which can be represented using SDP of size n and 2, respectively. The talk contrasts existential and complexity questions in SDP representations. The Helton-Nie conjecture suggests that any convex set defined by polynomial inequalities has an SDP representation. The complexity question asks for the smallest SDP representation of a convex set, which is defined as the positive semidefinite rank. The importance of lifting is emphasized, with examples showing how regular polygons can be described with few inequalities. The talk connects lifting to the rank of slack matrices of polytopes, where the positive semidefinite rank determines the minimal SDP representation size. The positive semidefinite rank of a matrix is defined as the smallest size of a positive semidefinite factorization. An example is given where a matrix with entries (i-j)^2 has a positive semidefinite rank of 2, independent of the matrix size. The talk concludes with a theorem by Gouveia, Parrilo, and Thomas (2011) stating that the smallest SDP representation of a polytope is determined by the positive semidefinite rank of its slack matrix. It also discusses the relationship between LP and SDP lifts, showing that SDP lifts can be more powerful than LP lifts. A family of polytopes is presented where the ratio of SDP rank to LP rank tends to zero, highlighting the efficiency of SDP in certain cases. The talk concludes with a summary of the connections between SDP representations, matrix factorizations, and the relative power of SDP and LP lifts in representing convex sets.The talk discusses the concept of positive semidefinite rank and its connection to semidefinite programming (SDP) representations of convex sets. It begins with a review of Euclidean distance matrices (EDMs) and their characterization via Schoenberg's theorem. EDMs can be represented as SDP problems, which allows for the modeling of distance geometry problems. The concept of semidefinite representation is introduced, where a convex set can be expressed as the projection of a positive semidefinite matrix in a higher-dimensional space. Examples include EDMs and the unit disk in 2D, which can be represented using SDP of size n and 2, respectively. The talk contrasts existential and complexity questions in SDP representations. The Helton-Nie conjecture suggests that any convex set defined by polynomial inequalities has an SDP representation. The complexity question asks for the smallest SDP representation of a convex set, which is defined as the positive semidefinite rank. The importance of lifting is emphasized, with examples showing how regular polygons can be described with few inequalities. The talk connects lifting to the rank of slack matrices of polytopes, where the positive semidefinite rank determines the minimal SDP representation size. The positive semidefinite rank of a matrix is defined as the smallest size of a positive semidefinite factorization. An example is given where a matrix with entries (i-j)^2 has a positive semidefinite rank of 2, independent of the matrix size. The talk concludes with a theorem by Gouveia, Parrilo, and Thomas (2011) stating that the smallest SDP representation of a polytope is determined by the positive semidefinite rank of its slack matrix. It also discusses the relationship between LP and SDP lifts, showing that SDP lifts can be more powerful than LP lifts. A family of polytopes is presented where the ratio of SDP rank to LP rank tends to zero, highlighting the efficiency of SDP in certain cases. The talk concludes with a summary of the connections between SDP representations, matrix factorizations, and the relative power of SDP and LP lifts in representing convex sets.