Research in Geometry and Topology

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The proposal centers around 3 topics. The first is about questions ingroup theory motivated by the first order predicate calculus inlogic. The questions involve e.g. understanding the sets ofhomomorphisms of a fixed group into a free group that extend toanother fixed group. The second topic concerns the topology of modulispace of curves and of the Torelli subgroup of a mapping classgroup. The idea is to see if homology can be computed using a suitablychosen Morse function on the moduli space, and on the quotient ofTeichmuller space by the Torelli group, respectively. The Morsefunction is geometrically defined -- it is the systole, i.e. thelength of the shortest curve (or the homological version of this inthe case of Torelli groups). The last topic is on the quasi-isometricrigidity of right-angled Artin groups. The subject has seen somegreat advances in the last 15 years or so, but this particular classof groups raises many issues not addressed before.The subject of the proposal is the study of topological and geometricproperties of several objects that naturally appear inmathematics. For example, consider the collection of all possible"shapes" of a surface with a fixed number of holes. This collectionforms a space, called moduli space of curves. The calculation of the"number of holes" for this space has been recently completed by atopological tour de force. A different method for this calculation isproposed here, one that would also give additionalinformation. To describe the method, consider the followinganalogy. When water is slowly poured into anormal glass, the surface of water in the glass will always be adisk. By contrast, if water is poured into a glass with a hollowhandle, the surface of water will consist of two disks once waterreaches the handle. Thus, by examining the shape of the surface ofwater we can tell if a glass has a handle or not. Topological spaces,such as moduli space of curves, can be analyzed similarly. Aparticular way of "pouring water" is proposed here.