Diffusion maps, spectral clustering, and reaction coordinates of dynamical systems are explored in this paper. The authors present a unifying view of these tasks by considering diffusion maps, which embed high-dimensional data into a low-dimensional space using eigenvectors of random walks. Assuming data is sampled from a probability distribution $ p(\mathbf{x}) = e^{-U(\mathbf{x})} $, the eigenvectors of diffusion maps converge to eigenfunctions of a differential operator defined on the support of the distribution. Different normalizations of the Markov chain lead to different differential operators. For example, the normalized graph Laplacian corresponds to a backward Fokker-Planck operator with potential $ 2U(\mathbf{x}) $, suitable for spectral clustering. Another normalization leads to a backward Fokker-Planck operator with potential $ U(\mathbf{x}) $, suitable for analyzing long-time dynamics of stochastic systems. A third normalization leads to the Laplace-Beltrami operator, suitable for analyzing the geometry of the dataset. The paper discusses the connection between these eigenvalues and eigenvectors to the underlying geometry and probability distribution of the dataset. It also presents examples of diffusion maps applied to various datasets, including a double well potential and the iris dataset. The authors conclude that diffusion maps provide a powerful tool for analyzing the geometry and probability distribution of empirical data, with applications in data analysis, spectral clustering, and the study of dynamical systems.Diffusion maps, spectral clustering, and reaction coordinates of dynamical systems are explored in this paper. The authors present a unifying view of these tasks by considering diffusion maps, which embed high-dimensional data into a low-dimensional space using eigenvectors of random walks. Assuming data is sampled from a probability distribution $ p(\mathbf{x}) = e^{-U(\mathbf{x})} $, the eigenvectors of diffusion maps converge to eigenfunctions of a differential operator defined on the support of the distribution. Different normalizations of the Markov chain lead to different differential operators. For example, the normalized graph Laplacian corresponds to a backward Fokker-Planck operator with potential $ 2U(\mathbf{x}) $, suitable for spectral clustering. Another normalization leads to a backward Fokker-Planck operator with potential $ U(\mathbf{x}) $, suitable for analyzing long-time dynamics of stochastic systems. A third normalization leads to the Laplace-Beltrami operator, suitable for analyzing the geometry of the dataset. The paper discusses the connection between these eigenvalues and eigenvectors to the underlying geometry and probability distribution of the dataset. It also presents examples of diffusion maps applied to various datasets, including a double well potential and the iris dataset. The authors conclude that diffusion maps provide a powerful tool for analyzing the geometry and probability distribution of empirical data, with applications in data analysis, spectral clustering, and the study of dynamical systems.