Algebraic Cycles, Motives, and K-Theory

PI将从事几个关于代数循环，动机和K理论研究的项目。特别是，他打算使用形态阿贝尔-雅可比映射，使用劳森同调定义，以更好地理解一个光滑的，复杂的品种，代数等价于零的循环群，并更好地理解格里菲斯的阿贝尔-雅可比映射在这样的循环上的行为。(Some这项研究可能与Andreas Rosenschon合作进行。例如，PI试图使用形态的阿贝尔-雅可比映射来寻找格里菲斯阿贝尔-雅可比映射的所谓“普适性”的反例，并更好地理解周群的挠子群。 一个相关的项目将使用劳森同调，形态上同调，和形态阿贝尔-雅可比映射，以更好地理解（仍然是假设的）类别的混合动机。 PI还将研究周期和K理论的真实的代数簇。在这个方向上的一个目标是定义和研究高代数K-群的真实的品种。这些群应该是真实的代数几何不变量，只依赖于由方程组切出的真实的点的集合。 另一个项目涉及开发1-motives和2-motives的形态上同调类似物。最后，PI与Christian Haesemeyer和Eric Friedlander合作，希望将形态上同调和劳森同调的定义扩展到真实的和复数以外的基础领域。代数簇是由多项式方程集合定义的几何对象。例如，我们可以用一个包含两个变量的方程来定义一个圆，但是，一般来说，代数簇是由可能有很多变量的任何数量的方程给出的，由此产生的几何对象可能不适合我们通常的三维世界。代数几何中的一个基本问题是两个给定的多项式方程组是否实际上确定“相同”（即，同构）代数簇。理想情况下，人们希望有一个不变量列表，可以唯一地确定两个变种何时同构，但尽管它明显合理，这个目标是完全不现实的。相反，人们希望找到以某种几何上有意义的方式对品种进行分类的不变量。代数循环、K-理论和动机上同调就是这样的不变量的例子，近年来我们对这些基本不变量的理解有了巨大的进步。也许最著名的是Voevodsky在素数2时证明了Beilinson-Lichtenbaum猜想，为此他在2002年获得了菲尔兹奖。 尽管最近取得了许多成功，但对于这些基本不变量，甚至对于光滑的、射影的复变量，我们仍然有许多不了解。Hodge猜想（这是七个千禧年问题之一，由克莱数学研究所主办）和布洛赫猜想关于零循环的表面，这两个仍然没有解决，是两个这样的例子。 PI将进行几个研究项目，旨在进一步了解代数簇的基本不变量，通常侧重于光滑，射影复杂的品种。他的大部分项目都集中在研究此类簇的圈，通常使用相对较新的Lawson同调不变量、形态上同调不变量和形态Abel-Jacobi映射。

The PI will pursue several projects concerning the study of algebraic cycles, motives, and K-theory. In particular, he intends to employ the morphic Abel-Jacobi map, defined using Lawson homology, to better understand the group of cycles on a smooth, complex variety that are algebraically equivalent to zero and to better understand the behavior of Griffiths' Abel-Jacobi map on such cycles. (Some of this research may be carried out in collaboration with Andreas Rosenschon.) For example, the PI seeks to use the morphic Abel-Jacobi map to find counter-examples to the so-called "universality" of Griffiths' Abel-Jacobi map and to better understand the torsion subgroup of the Chow group. A related project will employ Lawson homology, morphic cohomology, and the morphic Abel-Jacobi map to better understand the (still hypothetical) category of mixed motives. The PI will also study cycles on and the K-theory of real algebraic varieties. One goal in this direction is to define and study higher algebraic K-groups for real varieties. These groups ought to be real algebro-geometric invariants, depending only on the set of real points cut out by a system of equations. Another project concerns developing a morphic cohomology analogue of 1-motives and 2-motives. Finally, the PI, in collaboration with Christian Haesemeyer and Eric Friedlander, hopes to extend the definitions of morphic cohomology and Lawson homology to ground fields other than the real and complex numbers.An algebraic variety is a geometric object defined by a collection of polynomial equations. For example, one can define a circle using one such equation involving two variables, but, in general, an algebraic variety is given by any number of equations in possibly a great many variables, and the resulting geometric object might not fit inside our usual three-dimensional world. A basic question in algebraic geometry is whether two given systems of polynomials equations actually determine the "same" (i.e., isomorphic) algebraic variety. Ideally, one would like a list of invariants that uniquely determine when two varieties are isomorphic, but despite its apparent reasonableness, this goal is completely unrealistic. Rather, one hopes to find invariants that classify broad categories of varieties in some geometrically meaningful way. Algebraic cycles, K-theory, and motivic cohomology are examples of such invariants, and there have been tremendous advances in recent years in our understanding of these fundamental invariants. Perhaps the most famous one is Voevodsky's proof of the Beilinson-Lichtenbaum conjecture at the prime 2, for which he was awarded the Fields Medal in 2002. For all the recent successes, there is still much we do not understand about these basic invariants, even for smooth, projective complex varieties. The Hodgeconjecture (which is one of the seven Millennium Problems sponsored by the Clay Mathematics Institute) and Bloch's conjecture concerning zero-cycles on surfaces, both of which remain unsolved, are two such examples. The PI will pursue several research projects aimed at further understanding the fundamental invariants of algebraic varieties, typically focusing on smooth, projective complex varieties. Most of his projects focus on the study of cycles on suchvarieties, often by using the relatively new invariants of Lawson homology, morphic cohomology, and the morphic Abel-Jacobi map.