I will introduce several optimality problems in integer homology and real cohomology that we have studied recently. For integer homology, I will first discuss our STOC 2010 result for the Optimal Homologous Chain Problem (OHCP). We showed that for relatively torsion-free complexes, OHCP can be solved in polynomial time. This was surprising because with mod 2 coefficients OHCP is NP-hard, as shown earlier by Chen and Freedman. Next I will describe our SoCG 2011 result by introducing the Optimal Bounding Chain Problem (OBCP) and its relative version. We used these to show that the Least Spanning Area Problem for knots embedded in 3-manifolds with trivial homology can be solved in polynomial time. An earlier result of Agol, Hass, and Thurston is that for general 3-manifolds the problem is NP-hard. Both OHCP and OBCP discussed above were formulated as 1-norm minimization while satisfying some constraint involving boundary matrices. If instead, one uses 2-norm minimization, real coefficients, and coboundary matrices, the OHCP problem transforms into one that is needed as a fundamental step in finite element exterior calculus for solving elliptic partial differential equations. I will discuss our recent results from our arXiv eprint on Cohomologous Harmonic Cochains where we proved a discrete Hodge-deRham isomorphism theorem. Different parts of the above results are joint work with T. Dey, N. Dunfield, K. Kalyanaraman, B. Krishnamoorthy, S. Watts, and H. Wang.