Oct. 31, 3:30pm, Wean 8220

Ameerah Chowdhury, Carnegie Mellon University

A proof of the Manickam–Miklós–Singhi conjecture for vector spaces

Ameerah Chowdhury, Carnegie Mellon University

A proof of the Manickam–Miklós–Singhi conjecture for vector spaces

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 a subspace *S*
of *V* to be the sum of the weights of all the *1*-dimensional subspaces it contains. We prove that if *n* ≥ *3k*, then the number of *k*-dimensional subspaces in
*V* with nonnegative weight is at least 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.

Joint work with Ghassan Sarkis (Pomona College) and Shahriar Shahriari (Pomona College).