In 1985 Jones assigned to a knot a polynomial with integer coefficients, and in 1989 Witten came up with a path-integral interpretation of this polynomial. Upon taking parallels, one can consider a sequence of polynomials. The volume conjecture relates an asymptotic behavior of this sequence (evaluated at complex roots of unity) with the Riemannian (mostly hyperbolic) geometry of the knot complement. The PIs propose to study this and related conjectures that relate quantum topology and geometrization.
A knot in 3-space is a flexible rope that is allowed to deform without crossing itself. The width and length of the rope are irrevelant, and its beginning and end coincide. For example, a rubber band is an example of a knot. In the middle of 19th century, Kelvin postulated that chemical elements (such as gold, aluminum) are made out of knots; thus different knot types explain the variation of observed chemical elements. More than a century later, string-theory offers a similar view of the world, where elementary particles are vibrating in the fabric of cosmos, along knotted circles. Whether string-theory explains the world is a key question. Meanwhile, it is known that string-theory has deeply influenced mathematics.
