The MMS conjecture for vector spaces

10/29/2013 - 12:15pm
10/29/2013 - 1:10pm
Shahriar Shahriari (Pomona College)

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.

Mudd Science Library 126, Pomona College