Conference on Categorical Methods in Topology and Quantum Geometry; Fall 2009, Dallas, Texas

Harvey Mudd College, Claremont CA

Investigators

Abstract

A scientific symposium titled "Categorical Methods in Topology and Quantum Geometry" will take place at the National Meeting of the Society for Advancement of Chicano and Native Americans in Science (SACNAS) in Dallas, Texas, October 15-18, 2009. The two a priori disparate fields of low dimensional topology and quantum geometry have recently been related through the involvement of categorical methods. Exciting progress has been recently made in each of these subjects independently, including the Weinstein conjecture in dimension three and Donaldson-Thomas/Gromov-Witten duality for all toric threefolds. Important open questions motivate continued study; these include the general Weinstein conjecture, and the trilogy of equivalences between Gromov-Witten theory, Donaldson-Thomas theory and Pandharipande-Thomas theory. Central to our symposium, these fields have recently been related through their use of derived categories and categorification. This includes the categorification of Khovanov and its many generalizations, the categorification of Donaldson-Thomas theory by Behrend and Joyce-Song and the derived categories of Pandharipande- Thomas theory. In addition, the program of Cautis and Kamnitzer shows categorification results in topology may be recast in some instances as isomorphisms of derived categories in algebraic geometry. The speakers at this symposium will announce accomplishments and also discuss new directions of research. Specifically, Renzo Cavalieri will discuss derived categories in GW and DT theories, Emille Davie will discuss applications to low dimensional topology, and Juan Ortiz- Navaro will discuss categorification, Khovanov homology and its generalizations. This symposium will enable and encourage students and other scientists to pursue research in areas related to the interaction of quantum geometry and topology, provide the opportunity for scientists to interact and foster collaboration and new research, and disseminate knowledge to a wide and extraordinarily diverse audience. While the reasons for organizing a scientific symposium on topology and quantum geometry are many, there is additionally an acute need to do so for an audience of underrepresented minorities. There is at this time significant underrepresentation of minorities in the mathematical sciences; this underrepresentation is evidently severe in both topology and quantum geometry and certainly the intersection of these subjects. There are very important questions that need to be addressed in these subjects, and it is necessary to attract a broad and diverse audience to work on these problems. Gromov-Witten theory and related fields have been extremely successful in solving outstanding problems, some over 100 years old, in several branches of mathematics and physics. Low dimensional topology is not only of basic importance in geometry and topology, but in several areas of applied mathematics as well, as highlighted in this symposium. It is predicted that underrepresented minorities will become the majority of United States Citizens in the not so distant future; as such, it is in the long term interest of topology and quantum geometry to have increased participation from members of these groups. Moreover, given the importance of these subjects to mathematics and science in general, it is in our national interest to work against the underrepresentation of minorities conducting research in these fields.