The principal investigator (PI) and his collaborators are interested in a diverse set of problems concerning incidence matrices, additive combinatorics, difference sets and Hadamard matrices. Incidence matrices arise whenever one considers relations between two finite sets. Many questions in finite geometry, combinatorics and representation theory of finite groups reduce to the computation of ranks of certain incidence matrices over finite fields. The PI and his collaborators have had some successes in the past few years investigating the modular ranks and the Smith normal forms of classes of incidence matrices arising from finite geometry and combinatorics. The PI intends to continue his work in this direction, aiming at solving several open problems in this area. Secondly, the PI will continue his recent work on Snevily's conjecture on Latin transversals of addition tables of finite abelian groups and apply the exterior algebra method to other problems in additive combinatorics. In the third part, the PI intends to study relative difference sets and further explore their connections to Hadamard matrices and mutually unbiased bases.
Incidence matrices are basic mathematical objects which are frequently encountered in various branches of mathematics, computer science and engineering. Many of the incidence matrices considered in this proposal can be used to generate efficient error-correcting codes, which are used nowadays in our daily life, for example, in CD players, high speed modems, and cellular phones. Difference sets and Hadamard matrices are important objects in combinatorial designs theory, which have found many applications in radar, spread-spectrum communications and cryptography.
