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

有关多项式方程的求解问题出现在古希腊数学中。 坐标几何的出现使人们更好地理解了这些问题的本质，而复数的引入提供了一个重要的新视角。 在这种方式中，确定定义为具有两个变量的多项式方程的解集的平面中的曲线的性质与理解四维空间中的曲面的问题有关。首席研究员正在研究出现的表面的拓扑特性。 据观察，这些表面的局部性质与结有关;该项目包括继续研究经典结理论的四维方面。加上他在结理论的研究，首席研究员保持一个网站，致力于为学生和研究人员提供有关结的最新信息。 除了帮助人们在纽结理论工作（包括纯数学家和应用），该网站作为一个重要的教育工具，达到世界各地的学生。一个齐次多项式方程与三个变量定义一个复杂的曲线在二维复杂的射影空间。 最近的工作的主要研究者，共同完成与Borodzik和Borodzik，Hedden，限制了可能性的奇异的情况下，曲线是拓扑的一个领域或环面。 该方法是了解Heegaard Floer同源的边界附近的曲线。 由曲线的补而产生的对这些同调群的约束反过来又约束了所产生的奇点。 进一步发展这一点是首席研究员研究的目标之一。在研究复杂曲线的奇异性时，人们很自然地会研究纽结的四个亏格。 当仅限于环面结时，该主题很容易理解。 与货车科特，首席研究员正在应用经典技术和Heegaard Floer理论相结合，以了解环面结的连接和的四个亏格，从一对环面结的线性组合开始。 独立于这项工作，首席研究员还试图扩展他以前与吉尔默关于不可定向的四球纽结的工作;一个目标是应用Heegaard Floer理论来构建具有大型不可定向的四球纽结的新例子。

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.