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The arealaw conjecture states that ground states of gapped local Hamiltonians with n.n. interactions on a Ddimensional lattice satisfy an arealaw for the entanglement entropy: the entanglement entropy of any contiguous region with respect to the reset of the system scales like its boundary area, rather than its volume. Such states hold much less entanglement than typical quantum states, which have important implications to our ability to represent them efficiently using tensor networks and simulate them classically. So far, the conjecture has only been fully proven in the 1D case (Hastings' 2007). However, a sequence of improvements of Hastings' result in 1D has recently led to a breakthrough towards a possible proof of the 2D case. In this talk I will introduce this conjecture and give an overview of research that led to the recent breakthroughs. This will include the machinery of approximate ground state projectors (AGSPs), and its relation to questions from theory of polynomial approximations. Host: Zoe Holmes 