Advisor: Jon Wellner We consider the problem of forming confidence intervals and tests for the location of the mode in the setting of nonparametric estimation of a log-concave density. We thus study the class of log-concave densities with fixed and known mode. We find the maximum likelihood estimator for this class, give a characterization of it, and, under the null hypothesis, show our estimator is uniformly consistent and is $n^{2/5}$-tight at the mode. We also show uniqueness of the analogous limiting "estimator" of a quadratic function with white noise. This sets the stage for us to find the joint asymptotic distribution of the unconstrained and constrained estimators and then the limiting distribution of the likelihood ratio statistic for the mode.