Principal Investigator: Sumio Yamada
The two main subjects of the proposed research are Teichmueller spaces of compact Riemann surfaces of genus greater than one and the universal Teichmueller space, which can be identified with the diffeomorphism group of the circle. There is a natural Riemannian metric on both classical and universal Teichmueller spaces, called the Weil-Petersson metric. Although those spaces are not symmetric spaces, and there are no classical Lie groups acting on them isometrically (unlike the hyperbolic spaces with the special linear group), one notes that on the classical Teichmueller spaces the mapping class group of the Riemann surfaces acts isometrically, and that on the universal Teichmueller space the diffeomorphism group of the circle acts isometrically. These group actions on Teichmueller spaces can be utilized to construct flat bundles over various manifolds with its fibers being copies of Teichmueller spaces, which leads to the questions of strong/super rigidity. It is in this context that the investigator intends to use the Weil-Petersson geometry of Teichmueller spaces.
The subject of Riemann surfaces has been one of the central themes of modern mathematics over the last two centuries. In recent years, it has received renewed attention due to the interest created by the so-called super-string theory, with which physicists hope to construct the grand unifying theory (GMT) of graviation and quantum physics. The Teichmueller space, which is the subject of this investigation, plays an important role since it is known to parametrizes the "shapes of the string." Interactions between mathematics and theoretical physics have proved to be fruitful for both sides, and it is the investigator's belief that the proposed research may offer some new way of understanding the outstanding issues in the relevant scientific fields.