Knots and surfaces in three- and four-manifolds: Applications of symplectic topology and quantum algebra to low dimensional topology

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The theme of this project is to study topological and geometric objects in low-dimensions. One of the primary ways the project will accomplish this is by using tools from symplectic geometry, gauge theory, and quantum algebra. Many of these tools come packaged as algebraic invariants e.g. Floer homology theories or combinatorial knot invariants. A byproduct of the project will be a deepened understanding of the invariants themselves, clarifying the relationships between them and the internal structure of their underlying theories. Specific goals include understanding the knotted curves in the three-dimensional sphere on which a surgery procedure produces simple manifolds, a general exploration of the knots which arise from algebraic curves in projective surfaces, and understanding the extent of, and method by which, combinatorial invariants detect geometric objects e.g. whether Khovanov homology detects the unknot.The study of three- and four-dimensional spaces, and knotted curves and surfaces within them, is a central task to our understanding of both large and small scale aspects of the universe. Determining the shape of the universe depends upon a mathematical understanding of the possible shapes that could occur and the properties these shapes have. These properties are known as invariants, and the project furthers our understanding of space by the discovery of new invariants and the study of existing invariants. In this pursuit it has been quite fruitful to examine the way in which curves and surfaces can be tied in knots. This kind of knotting is not only relevant to understanding the shape of space, but has recently become significant in the study of DNA. Confined to a small space, long strands of DNA naturally become knotted, and certain processes depend upon an understanding of the complexity of these knots. Applications of this project include effective ways of measuring different types of complexity of knots.