The second eigenvalue of the Laplacian matrix of a graph measures its connectivity: the graph is connected if and only if the second eigenvalue is positive with larger values corresponding to higher connectivity. By varying the weights of the edges on the graph, one can typically increase the second eigenvalue to improve desirable properties of the graph, such as robustness and mixing times of random walks. In this talk, I will introduce conformally rigid graphs, which are undirected graphs in which uniform edge weights maximize the second eigenvalue or minimize the largest eigenvalue amongst all normalized edge weights. This property serves as a filter for many interesting extremal graphs, but conformally rigid graphs are rare and even proving graphs are conformally rigid is surprisingly difficult. In this talk, I will introduce a new framework for conformal rigidity connecting this optimization problem to the geometry of graph embeddings, subdifferential analysis, representation theory, and semidefinite programming.