Mapping Class Groups and Categorification (13w5108)



(Australian National University)

(Columbia University)


The Banff International Research Station will host the "Mapping Class Groups and Categorification" workshop from April 7th to April 12th, 2013.

Take a towel and wrap it around an inner-tube. Did you wrap the towel so the long edge went around the outside of the tube? Or so that the short edge of the towel went around the outside? Or another way -- there are others, if you slant the towel. It turns out that the ways of wrapping a (very stretchy) towel around an inner-tube corresponds to the two by two matrices with integer entries and determinant one. These form a group, denoted SL_2(Z). The group SL_2(Z) shows up in many contexts in mathematics, from topology -- which is where we just saw it -- to number theory. In particular, it is what's known as an "arithmetic group."

Now, suppose your inner-tube had many holes instead of one. In this case, the analogue is another group, called the mapping class group of the surface. These new groups are not arithmetic, but share many properties with arithmetic groups. One explanation for some of these properties would be if mapping class groups could also be written with matrices. This question, called linearity, is currently open. At a higher categorical level, however, ideas from physics have led to an answer to the analogous question. This conference seeks to explore applications of those results.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).