This research concerns the arithmetic theory of quadratic forms. In particular there will be special emphasis on the classification and representation problems of positive definite forms. With respect to the classification theory the Principal Investigator will investigate both the explicit and complete determination of certain genera as well as the relationships between even unimodular quadratic forms over integers of a number field and their algebraic descents. In the area of representation of quadratic forms he will study primitive representations of n-ary forms by positive definite m-ary quadratic forms and representations of binary forms by certain positive quaternary quadratic forms. Quadratic forms are homogeneous polynomials in many variables of degree two. This research concerns the number theory problems which arise in determining which integers can be values of them when integer are substituted for their variables. Classifying quadratic forms in classes that all have the same values and representing the forms by simpler forms are valuable tools in answering this problem.
