K-theory and motivic cohomology of singular schemes

研究奇异方案的代数k理论和动机理论。第一个课题是证明特征p为负k理论的Weibel消失猜想和Vorst正则性猜想；这个典型的案例之前由Cortinas、Haesemeyer、Schlichting和Weibel处理过。第二个项目涉及Suslin的奇异同源性。通过Suslin和Voevoedsky的工作，这些群是对偶的，它们在代数闭域上的特征是素数。Geisser提出研究其在特征p（特别是在有限域上）上的有理部和p-初等部的行为，以及与非代数闭域上的上同调的关系。在这个项目中，Geisser将研究代数k理论和动机上同理论。这些理论形成的不变量有助于理解多项式方程系统的解集结构，该系统具有系数和域中的解，如有理数、实数或复数。如果解集是光滑的，那么这些不变量就很好理解了（例如，如果曲线没有交叉点或顶点，那么它们就是光滑的）。建议者将尝试在一般情况下推进理解，使用最近发展起来的方法。这可能会导致密码学中的应用，因为一些密码系统依赖于（困难的）找到多项式方程系统的解。

The proposer will study the algebraic K-theory and motivic theories of singular schemes. The first project is to prove Weibel's vanishing conjecture and Vorst's regularity conjecture for negative K-theory in characteristic p; the characteristic 0 case has been previously handled by Cortinas, Haesemeyer, Schlichting and Weibel. The second project deals with Suslin's singular homology. With coefficients prime to the characteristic over an algebraically closed field these groups are dual to etale cohomology by work of Suslin and Voevoedsky. Geisser proposes to study the behavior of its rational and p-primary part in characteristic p (especially over finite fields), as well as the relationship to etale cohomolgy over non-algebraically closed fields.In this project, Geisser will study algebraic K-theory and motivic cohomology theories. These theories form invariants which help to understand the structure of the set of solutions to a system of polynomial equations with coefficients and solutions in fields, like the rational numbers, real numbers or complex numbers. These invariants are fairly well understood if the solution set is smooth (for example, curves are smooth if they have no intersections or cusps).The proposer will try to advance the understanding in the general case, using methods which have been developed recently. This could lead to applications in cryptography, because several cryptosystems rely on (the difficulty in) finding solutions to systems of polynomial equations.