Open Algebraic Varieties and Group Actions

我的研究领域是代数几何，研究由多项式方程定义的对象。一个例子是圆，由方程x^2+y^2=1定义。这和双曲线xy=1是同一个物体吗？答案并不像看起来那么明显，当我们允许复数时，答案是肯定的，当我们只允许真实的数时，答案是否定的。我特别感兴趣的是平坦的仿射空间和一些非常基本的问题。如果我们在平面上看一条直线和一条抛物线，那么它们实际上是代数几何中的相同（技术上：同构）对象，并且存在一个平面的对称性（技术上：自同构）。在更高的维度中完全理解这种现象是一项极其困难的任务，需要来自不同和高度发达的数学学科（如代数，几何，拓扑和逻辑）的复杂技术。 我过去、现在和未来的研究旨在深化这一尚未完成的理解。已在高质量期刊上发表 *。* 从科学的角度来看，我的工作是对基础纯数学的贡献。在实践方面，我培养了许多研究生和博士后研究员，他们现在在世界各地都有自己成功的职业生涯。此外，我可以自豪地说，我经常在年轻同事职业生涯的早期阶段，在任何正式的监督安排之外，担任他们的导师。我组织并参加了许多国际会议。我有一个很大的合作者圈子，他们经常访问蒙特利尔。 我的研究计划和它所产生的活动对加拿大数学的高国际地位做出了重大贡献。

My field of research is Algebraic Geometry, the study of objects defined by polynomial equations. An example is the circle, defined by the equation x^2+y^2=1. Is this the same object as the hyperbola xy=1? The answer is less obvious than it may look, it is yes when we allow complex numbers in the solution, no when we allow only real numbers. Of particular interest to me are the flat affine spaces and some very fundamental questions about them. If in the plane we look at a line and a parabola, then they are actually the same (technically: isomorphic) objects in Algebraic Geometry and there is a symmetry (technically: an automorphism) of the plane carrying one into the other. Understanding such phenomena fully in higher dimensions is an extremely difficult task requiring sophisticated techniques from diverse and highly developed mathematical disciplines such as Algebra, Geometry, Topology and Logic. My past, present and future research aims at deepening this as yet incomplete understanding. It has been published in high quality jornals.*** ***From a scientific point of view my work is a contribution to basic Pure Mathematics. On the practical side, I have trained numerous graduate students and postdoctoral fellows who now have successful careers of their own the world over. In addition, I am proud to say, I have frequently had a mentor's role for young colleagues at an early stage of their careers outside any formal supervisory arrangements. I have organized, and participated in, numerous international conferences. I have a large circle of collaborators that frequently visit Montreal. My research program and the activity that it has generated have contributed materially to the high international standing of Canadian Mathematics.********