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

二次曲线是圆锥体的表面与产生椭圆、抛物线和双曲线的平面的交点而得到的曲线。在射影几何中，它们可以描述为三个变量的二次齐次多项式的零点。这种多元齐次二次多项式被称为二次形式，就像二次曲线一样，它们产生了被称为二次曲线的几何对象，它们可以在任何域上定义。对于二次型和分别使用代数和几何方法的二次曲线，有很好的理论，它们以微妙的方式相互作用，产生了结合了代数和几何方面的丰富的理论。该项目的一部分涉及根据几何比较的不同方法对四次曲线进行分类，例如同构、(稳定的)二元等价或动机等价。二次曲面的动机等价性是一个较新的概念，它源于沃沃茨基关于动机同伦理论的菲尔兹奖获奖著作。根据等价关系，分类可以更细，也可以更粗。我们计划比较这些不同的分类方法，特别是在基场具有特征2的情况下，即其中2=0。在这种情况下，已知的结果远不如特征2中的结果完整。特征2中用于分类二次型的一个重要的代数工具是称为加藤上同调群的某些代数对象的理论，它可以对任何正特征基域定义。该项目的第二部分涉及这些上同调群的性质，特别是它们在基场的扩张下的行为，以及当这样的上同调群中的某些元素湮灭其他元素时。二次型的另一个方面涉及研究任意交换基环上的二次型对元素的表示。一个经典的例子是拉格朗日定理，它指出4是最小的正整数n，使得每个正整数可以写成n个整数平方的和。这里，二次型由整数环上的四个平方和给出。人们将环的毕达哥拉斯数p定义为最小正整数n，使得该环中的每个平方和可以写成n个平方的和(或者，如果不存在这样的n，则写成无穷大)。拉格朗日定理指出整数环的毕达哥拉斯数为4。环的水平S是最小的正整数n，使得-1可以记为n的平方之和(如果不存在这样的n，则记为无穷大)。如果S是有限的，则S和p以一种微妙的方式联系在一起：p至少是S，至多是S 2。我们研究了S，p和其他与水平概念有关的不变量的哪些值可以由具有有限S的环实现，并对某些类型的环明确地确定了这些值。

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.