Conics are curves obtained as the intersection of the surface of a cone with a plane giving rise to ellipses,
parabolas and hyperbolas. In projective geometry, they can be described as the zeros of a homogeneous
polynomial of degree two in three variables. Such homogeneous quadratic polynomials in several variables are
called quadratic forms, and just as for conics, they give rise to geometric objects called quadrics and they
can be defined over any field. There are well developed theories for quadratic forms and for quadrics using
algebraic and geometric methods, respectively, that interact in subtle ways giving rise to a rich theory
combining both algebraic and geometric aspects. One part of the project deals with the classification of
quadrics according to various ways of comparing them geometrically, such as isomorphism, (stable) birational
equivalence, or motivic equivalence. Motivic equivalence of quadrics is a fairly recent concept that grew out
of Voevodsky's Fields Medal winning work on motivic homotopy theory. Depending on the equivalence relation,
the classification can be finer or coarser. We plan to compare these different classification methods,
focusing in particular on the case where the base field has characteristic 2, i.e. where 2=0. In this case,
the known results are far less complete than in characteristic not 2. An important algebraic tool for
classifying quadratic forms in characteristic 2 is the theory of certain algebraic objects called Kato's
cohomology groups that can be defined for any base field of positive characteristic. The second part of the
project deals with properties of these cohomology groups, in particular, how they behave under extension of
the base field and when certain elements in such a cohomology group annihilate other elements. A different
aspect of quadratic forms concerns the study of representations of elements by quadratic forms over any
commutative base ring. A classic example is Lagrange's theorem that states that four is the least positive
integer n such that each positive integer can be written as a sum of n squares of integers. Here, the
quadratic form is given by the sum of four squares over the ring of integers. One defines the Pythagoras
number p of a ring as the least positive integer n such that each sum of squares in that ring can be written
as a sum of n squares (or infinity if no such n exists). Lagrange's theorem then states that the Pythagoras
number of the ring of integers is 4. The level s of a ring is the least positive integer n such that -1 can
be written as a sum of n squares (or infinity if no such n exists). If s is finite, s and p are related in a
subtle way: p is at least s and at most s+2. We study which values for s, p and other invariants related to
the notion of level can be realized by rings with finite s, and to determine these values explicitly for
certain types of rings.