Here's a question about orthogonal matrices. Consider orthogonal matrices U of size NxN with the following "energy": E(U) = -sqrt(N) sum_{i j} |U_i j| (the negative sum of the absolute values of all the elements, appropriately scaled). The lowest energy matrices are Hadamard matrices, as these have equal-magnitude elements. We'll consider only N where Hadamard matrices exist. Let U_0 and U_1 be any two distinct Hadamard matrices, so two ground states of our energy function. Let U(t) be a continuous curve between these two matrices, i.e. U(0)=U_0 and U(1)=U_1. Here's the question: find a good lower bound on the "energy barriers" between ground states: the maximum over t, of E(U(t))-E(U(0)). You can approach this by first thinking about which kinds of Hadamard pairs have low barriers, and then bounding the barrier for that type of pair. The factor sqrt(N) is just so that the scaled orthogonal matrix elements are +/-1 in the Hadamard ground states (the usual convention). A simple case is where U_0 and U_1 differ just by the swap of two rows (or columns). A curve connecting them could be a Givens rotation; that has barrier energy of order N. Can you come up with a pair/curve that has a lower barrier? -Veit