CBMS Regional Conference in the Mathematical Sciences - "Finite Morse Index Solutions and Related Topics" -Winter 2007

讲座将涉及有界域上的非线性椭圆方程的边值问题。他们将包括当前感兴趣的几个重要课题，如分叉理论，域形状，莫尔斯指数的解决方案，和峰值解决方案;这发生在各种各样的数学应用，如数学生物学，催化理论，材料科学和燃烧。也将讨论这些主题之间的相互关系。讲座将显示之间的密切联系理论的峰值解决方案和理论的有限莫尔斯指数的解决方案。有限莫尔斯指数解是研究非线性增长（比临界增长更快）的非线性椭圆方程的非常有用的工具。 会谈还将包括一些有趣的开放问题，可能是该地区的研究人员感兴趣的。

The lectures will involve boundary-value problems for nonlinear elliptic equations on bounded domains. They will encompass several important topics of current interest such as bifurcation theory, domain shape, Morse index solutions, and peak solutions; which occur in a great variety of applications of mathematics such as mathematical biology, catalysis theory, material science, and combustion. There will also be a discussion of interrelationships between these topics.The lectures will show a close connection between the theory of peak solutions and the theory of finite Morse index solutions. It is becoming increasingly apparent that finite Morse index solutions are very useful tools for studying nonlinear elliptic equations with rapid growth in the nonlinearity (faster than the critical growth). The talks will also include a number of interesting open problems that may be of interest to researchers in the area.