Quadratic forms, quadrics, sums of squares and Kato's cohomology in positive characteristic

圆锥曲线是锥面与平面交点得到的曲线，可以得到椭圆、抛物线和双曲线。在射影几何中，它们可以被描述为二阶三变量齐次多项式的零点。这种有几个变量的齐次二次多项式被称为二次型，就像圆锥曲线一样，它们产生了称为二次型的几何对象，它们可以在任何域上定义。二次型和二次型的理论分别使用代数和几何方法，它们以微妙的方式相互作用，产生了结合代数和几何方面的丰富理论。该项目的一部分是根据不同的几何比较方法来处理二次曲面的分类，如同构、（稳定的）双向等价或动机等价。二次元的动机等价是一个相当新的概念，它产生于Voevodsky获得菲尔兹奖的关于动机同伦理论的工作。根据等价关系，分类可以更精细或更粗糙。我们计划比较这些不同的分类方法，特别关注基础域具有特征2的情况，即2=0。在这种情况下，已知的结果远不如特征不2完整。对特征2中的二次型进行分类的一个重要代数工具是某些代数对象的理论，这些代数对象被称为Kato上同群，它可以被定义为任何具有正特征的基域。本课题的第二部分研究了这些上同群的性质，特别是它们在基域扩展下的行为，以及当这些上同群中的某些元素湮灭其他元素时的行为。二次型的另一个方面是研究元素在任意可交换基环上的二次型表示。一个经典的例子是拉格朗日定理，该定理指出4是最小的正整数n，因此每个正整数都可以写成整数n的平方和。这里，二次形式是由整数环上的四个平方和给出的。人们将环的毕达哥拉斯数p定义为最小的正整数n，使得环上的每个平方和可以写成n的平方和（如果不存在n，则为无穷大）。拉格朗日定理则指出整数环的毕达哥拉斯数为4。环的层s是最小的正整数n，使得-1可以写成n平方的和（如果不存在n则为无穷大）。如果s是有限的，那么s和p以一种微妙的方式联系在一起：p至少是s，最多是s+2。研究了具有有限s的环可以实现s， p和其他与水平概念相关的不变量的哪些值，并明确地确定了某些类型环的这些值。

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.