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

二次曲线是由圆锥表面与平面相交而得到的曲线，从而产生椭圆、抛物线和双曲线。在射影几何中，它们可以被描述为三元二次齐次多项式的零点。这样的齐次二次多项式在几个变量被称为二次形式，就像二次曲线，他们产生了几何对象称为二次曲线，他们可以定义在任何领域。有发达的理论二次形式和二次使用代数和几何方法，分别相互作用的微妙方式产生了丰富的理论结合代数和几何方面。该项目的一部分涉及二次曲面的分类，根据几何上比较它们的各种方法，如同构，（稳定）双有理等价或motivic等价。二次曲面的动机等价是一个相当新的概念，它源于Voevodsky的菲尔兹奖获奖作品动机同伦理论。根据等价关系，分类可以更细或更粗。我们计划比较这些不同的分类方法，特别关注基场具有特征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中的完整。 一个重要的代数工具分类二次型的特点2是理论的某些代数对象称为加藤的上同调群，可以定义为任何基地领域的积极特点。该项目的第二部分涉及这些上同调群的性质，特别是，它们如何在基场的扩展下表现，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是研究在任何交换基环上用二次型表示元素。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，使得每个正整数都可以写成整数的n平方和。在这里，二次型由整数环上的四个平方和给出。一个定义的毕达哥拉斯数p的一个环作为最小的正整数n，使每一个平方和在该环可以被写为一个总和n平方（或无穷大，如果没有这样的n存在）。拉格朗日的定理，然后指出，毕达哥拉斯数的环的整数是4。环的水平s是最小的正整数n，使得-1可以写成n的平方和（或无穷大，如果不存在这样的n）。如果s是有限的，则s和p以一种微妙的方式相关：p至少是s，至多是s+2。 我们研究的值s，p和其他不变量的概念有关的水平可以实现环有限s，并确定这些值明确为某些类型的环。

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.