This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The area of research of the present proposal is low dimensional topology. The problems discussed in this proposal find origin in the study of the connections between three-dimensional topology and symplectic topology of certain families of four-manifolds. The study of symplectic manifolds has become a core component of four dimensional topology, that might lead to a better understanding, and possibly a classification scheme, for four-manifolds. This research has origin in the attempt to determine which four-manifolds admitting a free circle action can be endowed with a symplectic form. In collaboration with Stefan Friedl we have solved this problem for the product case, and in the case where the canonical class of the symplectic structure is trivial. In both cases the result shows that the orbit space admits a fibration over the circle. The proof relates Seiberg-Witten theory in dimension four with various tools of three-manifold topology, as the study of twisted Alexander polynomials and their role in detecting fibrations. Refinement of these techniques will hopefully allow us to completely solve the problem. Other problems that will be the focus of this research concern the relation between twisted Alexander polynomials and topological properties of three-manifolds, as well as the investigation of other classes of four-manifolds that are closely connected with three-manifolds, as four-manifolds that fiber over the circle.
Low dimensional topology is an area of mathematics where the use of rigorous techniques sheds light, but is also enriched, by our "visual" perception and intuition of objects living in our three- (and four-) dimensional world. The prototype of this type of study, knot theory, originated more than a hundred year ago from the need to understand atoms. While the original models have proved too naive, the study of knots finds now more sophisticated applications in natural sciences and physics. Similarly, the study of symplectic manifolds finds in origin in physics, and its connections with theoretical physics has enriched both fields. My research aims at better understanding the connection between symplectic four-manifolds and the three-dimensional topology, and its results will hopefully motivate further research in these fields also by other scholars. Besides the intellectual merit in better understanding their connection, these subjects lend themselves very naturally for training purposes of students at both undergraduate and graduate level. Far-reaching courses on these subjected can be given with prerequisites limited to standard lower-division courses. This has an impact in the training future mathematicians and teachers of mathematics, as well as in providing an understanding of cutting edge research in mathematics for students interested in a career in applied science and technology.
