Strongly correlated systems provide a fertile ground for discovering exotic states of matter, such as those with topologically non-trivial properties. Among these are geometrically frustrated magnets, which harbor spin liquid phases with fractional excitations.
On the experimental front, this has motivated the search for new low dimensional quantum materials and on the theoretical front, this area of research has led to analytical and numerical advances in the study of quantum many-body systems.
I present aspects of our theoretical and numerical work in the area of frustrated magnetism, focusing on the frustrated kagome geometry, which has seen a flurry of research activity owing to several near-ideal material realizations. On the theoretical front, the kagome problem has a rich history and poses multiple theoretical puzzles which continue to baffle the community. First, I present a study of the spin-1 antiferromagnet, where our numerical calculations indicate that the ground state is a trimerized valence bond (simplex) solid with a spin gap , contrary to previous proposals. I show evidence from recent experiments that support our findings but also pose new questions. The second part of the talk follows from an unexpected outcome of my general investigations in the area for the well-studied spin-1/2 case . I explain the existence of an exactly solvable point in the XXZ-Heisenberg model for the ratio of Ising to transverse coupling $J_z/J=-1/2$ . This point in the phase diagram, previously unreported in the literature, has "three-coloring" states as its exact quantum ground states and is macroscopically degenerate. It exists for all magnetizations and is the origin or "mother" of many of the observed phases of the kagome antiferromagnet. I revisit aspects of the contentious and experimentally relevant Heisenberg case and discuss its relationship to the newly discovered point [3,4].
 H. J. Changlani, A.M. Lauchli, Phys. Rev. B 91, 100407(R) (2015).
 K. Kumar, H. J. Changlani, B. K. Clark, E. Fradkin, Phys. Rev. B 94, 134410 (2016).
 H. J. Changlani, D. Kochkov, K. Kumar, B. K. Clark, E. Fradkin, Phys. Rev. Lett. 120, 117202 (2018).
 H. J. Changlani, S. Pujari, C.M. Chung, B. K. Clark, under preparation.