The paper "Finite Element Exterior Calculus, Homological Techniques, and Applications" by Douglas N. Arnold, Richard S. Falk, and Ragnar Winther introduces a new theoretical approach to designing and understanding finite element discretizations for various systems of partial differential equations (PDEs). This approach leverages tools from differential geometry, algebraic topology, and homological algebra to develop discretizations that are compatible with the geometric, topological, and algebraic structures underlying well-posed PDE problems. The authors focus on finite element spaces of piecewise polynomial differential forms, which connect to each other in discrete subcomplexes of elliptic differential complexes and are related to continuous elliptic complexes through commuting projections. The paper covers several applications, including the finite element discretization of the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, as well as the construction of preconditioners. It addresses computational challenges such as the stability of finite element methods for the Poisson equation, the convergence to incorrect solutions for vector Poisson problems, and the computation of spectra for elliptic eigenvalue problems. The authors also present new results, such as the development of bases and degrees of freedom for finite element spaces of differential forms and the use of group representation theory to characterize these spaces. The paper is structured into two main parts. The first part develops the finite element exterior calculus, covering exterior algebra, exterior calculus, and Hodge theory. The second part applies these tools to concrete problems, including discretization of the Hodge Laplacian, eigenvalue problems, Maxwell's equations, and preconditioning. A significant application is the mixed discretization of elasticity equations, where the authors use vector-valued differential forms and the BGG resolution. The authors emphasize the importance of compatibility with the underlying geometric, topological, and algebraic structures to ensure stability and accuracy in numerical solutions. They also discuss the unification of different problems under a common framework, such as the treatment of the Hodge Laplacian, which unifies many important second-order differential operators.The paper "Finite Element Exterior Calculus, Homological Techniques, and Applications" by Douglas N. Arnold, Richard S. Falk, and Ragnar Winther introduces a new theoretical approach to designing and understanding finite element discretizations for various systems of partial differential equations (PDEs). This approach leverages tools from differential geometry, algebraic topology, and homological algebra to develop discretizations that are compatible with the geometric, topological, and algebraic structures underlying well-posed PDE problems. The authors focus on finite element spaces of piecewise polynomial differential forms, which connect to each other in discrete subcomplexes of elliptic differential complexes and are related to continuous elliptic complexes through commuting projections. The paper covers several applications, including the finite element discretization of the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, as well as the construction of preconditioners. It addresses computational challenges such as the stability of finite element methods for the Poisson equation, the convergence to incorrect solutions for vector Poisson problems, and the computation of spectra for elliptic eigenvalue problems. The authors also present new results, such as the development of bases and degrees of freedom for finite element spaces of differential forms and the use of group representation theory to characterize these spaces. The paper is structured into two main parts. The first part develops the finite element exterior calculus, covering exterior algebra, exterior calculus, and Hodge theory. The second part applies these tools to concrete problems, including discretization of the Hodge Laplacian, eigenvalue problems, Maxwell's equations, and preconditioning. A significant application is the mixed discretization of elasticity equations, where the authors use vector-valued differential forms and the BGG resolution. The authors emphasize the importance of compatibility with the underlying geometric, topological, and algebraic structures to ensure stability and accuracy in numerical solutions. They also discuss the unification of different problems under a common framework, such as the treatment of the Hodge Laplacian, which unifies many important second-order differential operators.