Heegaard Floer homology: algebraic curves, knot genera, and double null-concordance

与求解多项式方程有关的问题出现在古希腊数学中。坐标几何的出现使人们对这类问题的本质有了更好的理解，而复数的引入提供了一个重要的新视角。这样，确定平面上定义为二元多项式方程解集的曲线的性质就涉及到四维空间中曲面的理解问题。首席调查员正在调查出现的曲面的拓扑属性。据观察，这些表面的局部性质与纽结有关；该项目包括对经典纽结理论的四维方面的持续研究。除了他在纽结理论方面的研究，首席研究员还维护着一个网站，致力于为学生和研究人员提供关于纽结的最新信息。除了帮助从事纽结理论工作的人们(包括纯数学家和应用学家)，该网站还作为一个重要的教育工具，接触到世界各地的学生。三变量齐次多项式定义了二维复射影空间中的复曲线。首席调查员最近与Borodzik和Borodzik-Hedden共同完成的工作限制了在曲线在拓扑上是球面或环面的情况下出现奇点的可能性。方法是理解曲线邻域边界的Heegaard Floer同调。由曲线的补集引起的对这些同调群的约束反过来又约束了奇点的产生。进一步发展这一点是首席调查员的研究目标之一。在研究复杂曲线的奇点时，人们自然而然地会研究纽结的四个亏格。当局限于圆环结时，这个主题是很容易理解的。与Van Cott合作，首席研究员正在应用经典技术和Heegaard Floer理论的组合来理解环结的连通和的四个亏格，从一对环结的线性组合开始。除了这项工作之外，首席研究员还试图扩展他与Gilmer之前关于不可定向四球亏格的工作；一个目标是应用Heegaard Floer理论来建立具有大的不可定向四球亏格的纽结的新例子。

Problems related to finding solutions to polynomials equations arose in ancient Greek mathematics. The advent of coordinate geometry led to a better understanding of the nature of such problems, and the introduction of complex numbers provided an important new perspective. In this way, determining the nature of a curve in the plane defined as the solution set of a polynomial equation with two variables is related to the problem of understanding surfaces in four dimensional space. The Principal Investigator is investigating topological properties of the surfaces that arise. It has been observed that the local properties of these surfaces relate to knots; this project includes continuing research on four-dimensional aspects of classical knot theory. Coupled with his research in knot theory, the Principal Investigator maintains a website devoted to providing students and researchers access to current information about knots. In addition to assisting people working in knot theory (both pure mathematicians and applied), the website serves as an important educational tool, reaching students around the world.A homogenous polynomial equation with three variables defines a complex curve in two-dimensional complex projective space. Recent work of the Principal Investigator, done jointly with Borodzik and Borodzik-Hedden, has restricted the possibilities for singularities in the case that the curve is topologically a sphere or torus. The approach was to understand the Heegaard Floer homology of the boundary of a neighborhood of the curve. Constraints on these homology groups arising from the complement of the curve in turn constrain the singularities the arise. Developing this further is one goal of the Principal Investigator's research. In studying singularities of complex curves, one is led naturally to studying the four-genus of knots. When restricted to torus knots, the topic is well understood. Working with Van Cott, the Principal Investigator is applying a combination of classical techniques and Heegaard Floer theory to understand the four-genus of connected sums of torus knots, beginning with linear combinations of a pair of torus knots. Independent of this work, the Principal Investigator is also trying to extend his previous work with Gilmer concerning the non-orientable four-genus of knots; one goal is to apply Heegaard Floer theory to build new examples of knots with large nonorientable four-ball genus.