Algebraic Cycles, Motives, and K-Theory

PI将进行几个有关代数循环、动机和k理论研究的项目。特别是，他打算使用使用Lawson同调定义的morphic Abel-Jacobi映射，以更好地理解在代数上等同于零的光滑复杂变量上的环群，并更好地理解Griffiths的Abel-Jacobi映射在这些环上的行为。（其中一些研究可能是与安德烈亚斯·罗森schon合作进行的。）例如，PI试图使用morphic Abel-Jacobi映射来找到Griffiths的Abel-Jacobi映射的所谓“普适”的反例，并更好地理解Chow群的扭转子群。一个相关的项目将采用Lawson同调、形态上同调和形态Abel-Jacobi映射来更好地理解（仍然是假设的）混合动机的类别。PI还将研究上的循环和实代数变量的k理论。这个方向的一个目标是定义和研究实变量的高代数k群。这些群应该是真正的代数几何不变量，只依赖于由方程组切出的实数点的集合。另一个项目是发展1动机和2动机的同态上同调类比。最后，PI与Christian Haesemeyer和Eric Friedlander合作，希望将形态上同调和Lawson同调的定义扩展到实数和复数以外的地域。代数变量是由多项式方程的集合定义的几何对象。例如，一个人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，一个代数变化是由任意数量的方程给出的，可能有很多变量，结果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确定“相同”（即同构）的代数变体。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量何时是同构的，但是尽管它看起来很合理，但这个目标是完全不现实的。相反，人们希望找到不变量，以某种几何上有意义的方式对大类的变量进行分类。代数循环、k理论和动机上同都是这些不变量的例子，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在质数为2时证明了Beilinson-Lichtenbaum猜想，并因此获得了2002年的菲尔兹奖。尽管最近取得了成功，但对于这些基本不变量，甚至对于光滑的射影复变，我们仍然有很多不了解的地方。霍奇猜想（由克莱数学研究所赞助的七个千年问题之一）和布洛赫关于曲面上零环的猜想就是两个这样的例子，这两个猜想都没有得到解决。PI将进行几个研究项目，旨在进一步了解代数变量的基本不变量，通常侧重于光滑，射影复变量。他的大部分项目都集中在这些变体上的循环研究上，通常使用相对较新的不变量Lawson同调、态上同调和态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.