This project seeks to resolve specific questions in the area of 3- dimensional manifolds, knot theory, mapping class groups, and contact topology using a combination of geometric, topological and algebraic techniques. Tools from functional analysis and von Neumann algebras are also being used. The PI proposes to undertake the following projects. (1) Make significant advances toward the classification of the knot concordance group. In particular, classify the successive quotients of its (n)-solvable filtration and find new structure in the group. (2) Establish a higher-order Heegaard Floer homology theory that categorifies the higher-order Alexander polynomials defined by the PI. Use this to show that certain classical families of topologically slice knots are not smoothly slice. (3) Define new interesting canonical subgroups of the mapping class group related to the generalized Johnson subgroups and show their homology groups are infinitely generated. (3) Determine the precise relationship between certain subgroups of the mapping class group of a surface and the topology of their mapping tori (which are 3-manifolds). (4) Understand a precise relationship between transverse knots in S^3 and contact structures of arbitrary 3-manifolds that arise as cyclic and simple branched covers. Use this relationship to better understand the geometric invariants of a contact structure such as the support genus and binding number.
Understanding the geometric structure of objects in 3-dimensional space is of crucial scientific importance. From cancer treatments based on the knotting of cellular DNA, to antiviral drugs based on the geometrical shapes of proteins, to non-invasive visualization of the shape of the heart, to contemplating the ``shape'' of space-time itself, we seek precise mathematical descriptions of 3-dimensional objects. When one thinks of a precise mathematical description, one often thinks in terms of numbers, but ordinary numbers are insufficient to capture the complexities of our world. Multiplication of ordinary numbers is ``commutative.'' However, the physics of the twentieth century has taught us that matter and energy cannot be described merely by numbers. Rather, vectors and matrices are required, and multiplication of matrices is not commutative, that is AB does not usually equal BA. Every physical interaction is thus based on noncommutative algebra. This project is investigating how this noncommutative algebra yields a mathematical description of the geometric structure of 3-dimensional space and of objects in 3- dimensional space. Of particular importance is the manner in which closed strings in 3-dimensional space are knotted in 3- and in 4- dimensions. The PI will use non-commutative mathematical objects to better understand the knotting of strings and 3-dimensional spaces in general.
The fundamental problem in low-dimensional topology is to understand the shape of 3- and 4-dimensional spaces, also called manifolds. One can think of a 3-manifold as the space we live in and a 4-manifold as space plus time. Such spaces not only arise in topology but also in many other fields like algebraic geometry and number theory. A important aspect of this theory is to understand the way that strings (called knots) can be tied or knotted up in 3- and 4-dimensions, called knot theory (in 3-dimensions) or knot concordance (in 4-dimensions). Knots have been studied for over a century, ever since Lord Kelvin (incorrectly) hypothesized that atoms were made of knotted tubes of ether. However, we are still far from a complete classification of them or how they are related to one another. Knot theory also plays an role in the study of DNA and cancer research since one can view circular DNA as a knot and certain enzymes, called topoisomerases, manipulate DNA (this manipulation can be viewed as certain simple moves on knots). In this project, the PI investigated many question about the behavior of knots in 4-dimensions. In particular, the PI studied a mathematical structure that can be built out of knots in 4-dimensions called the knot concordance group. The PI and collaborators showed that this structure is extremely complicated. In fact, the PI gave evidence that it may have a fractal-like behavior, meaning that you can find arbitrarily small copies of it inside itself. To do this, the PI used non-commutative algebraic techniques. In addition, the PI produced results concerning certain knots with geometric constraints called transverse knots, as well as the structure of self-maps of surfaces, and knotted up graphs. As part of this project, the PI proposed to establish a two-week summer mathematics program for high-school girls in the Houston area. The goal of the program was to introduce high level abstract mathematics to the young women in such a way that they will view mathematics as useful, fun and exciting. In 2008, the PI started up the Rice University Mathematical Institute for Young Women (RUMIYW) (http://math.rice.edu/RMI/). This ran for 5 summers (2008-2012) and was extremely successful. Each summer, the program accepted 20 gifted female rising 10th or 11th graders from the Houston area. The teaching staff consisted entirely of females in mathematics: the PI, a postdoctoral assistant and two graduate students. During the two weeks, the program covered the following material: a two-week course in knot theory as well as individual modules on combinatorics, modular arithmetic, group theory, cryptography, countable and uncountable sets, probability, logic and proofs. It also built several connections between knot theory and the material developed in the modules. Nearly all of these topics are not covered in a high school mathematics syllabus. The students also investigated some unsolved problems in knot theory which introduced them to the concept of research in mathematics and demonstrated that, unlike the homework that they do in their typical math classes, not all mathematics problems are already solved. In addition to this material, women were invited from outside academia (Baylor College of Medicine, oil industry, etc.) to discuss their backgrounds, careers and how mathematics is used in their careers. In addition, during this period the PI was the Ph.D. advisor of 5 graduate students (all women) and was the mentor or co-mentor of 5 postdocs (4 of which were women). She also organized 5 special sessions at meetings of the American Mathematical Society, was a mentor for the American Women in Mathematics workshop at the 2012 AMS/MAA Joint Math Meetings, gave several presentations to high school teachers about knots and tangles, was a Plenary lecturer at the Sonya Kovalevsky High School Mathematics Day held at Indiana University, was a member of the Project NExT panel (professional development program for new or recent Ph.D.s) at the 2012 AMS/MAA Joint Math Meetings, and was an Invited lecturer at the NSF sponsored summer school for graduate students "More Examples of Groups."
