Some combinatorial problems in vector spaces over finite fields

This thesis is devoted to study some combinatorial problems in vector spaces over finite fields. It is divided into three chapters.;In the first chapter, we study the solvability of systems of certain equations over finite fields via spectra of graphs. We give a unified proof of several results on the solvability of systems of equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems of norm, bilinear or quadratic equations over finite fields are solvable in any large subset of a vector space over finite field. We will see that after appropriate graph theoretic results are developed, many old and new results immediately follow.;In the second chapter, we study the distribution of volumes of d-dimensional parallelepipeds determined by a large subset of d-dimensional vector spaces over finite fields. We also obtain similar results for distribution of permanents of matrices with restricted entries over finite fields.;The third chapter is based on the results obtained in several papers written (some in collaboration with coauthors) at various time during my graduate studies at Harvard. More precisely, we will discuss the following topics: maximal sets of pairwise orthogonal vectors in finite fields, sum of products and Erdos distance problem in random sets over finite fields, the spectrum of unitary Euclidean graphs, and explicit constructions of N-e.c. graphs.