Abstract

Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of any other subspace of V to be the sum of the weights of all the 1-dimensional subspaces it contains. What is the minimum possible number of k-dimensional subspaces of V with non-negative weight? Together with Ameera Chowdhury and Ghassan Sarkis, we prove that if n >= 3k, then this number is no less than the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988. The talk will discuss this conjecture and its proof as well as the related conjecture and results in the Boolean Lattices.



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