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。圆就是一个例子，它由方程x^2+y^2=1定义。这个物体和双曲线xy=1是同一个物体吗？答案并不像看起来那么明显，当我们在解决方案中允许复数时，答案是肯定的，当我们只允许实数时，答案是否定的。我特别感兴趣的是平坦仿射空间及其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相同的(技术上：同构)对象，并且存在一个对称(技术上：自同构)的平面将一个带入另一个。在更高的维度上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的复杂技术，如代数、几何、拓扑学和逻辑。我的过去、现在和未来的研究旨在加深对这一尚未完全理解的认识。它已经在高质量的期刊上发表了。 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，我培养了无数的研究生和博士后研究员，他们现在在世界各地都有自己的成功职业生涯。此外，我可以自豪地说，我经常在任何正式的监督安排之外，为处于职业生涯早期阶段的年轻同事担任导师的角色。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，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国际地位做出了实质性的贡献。

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.