The goal of this project is construct a rigorous framework for timelike Liouville theory. Timelike Liouville theory is a theory of quantum gravity with a "wrong sign" in the exponent, which makes it impossible to put it in the setting of ordinary probability theory. This project will aim to develop an extension of probability theory that accommodates Gaussian distributions with negative variance, and apply this extension to construct rigorous timelike Liouville theory. This project involves graduate students. While spacelike Liouville theory has received intense attention in the probability community over the last two decades, the timelike theory - which is closer to a true theory of quantum gravity - has remained beyond the reach of rigorous mathematics. The main reason, as in quantization efforts for other theories of gravity, is the appearance of Gaussian integrals with the "wrong sign" in front of the quadratic term. If the quadratic term comes with a positive sign, ordinary measure theory does not give the tools to make sense of it in a way that is consistent with physical calculations. This project will provide a rigorous framework for doing that, which has been outlined in near-complete detail in the project description. The long-term goal of this effort would be to derive the timelike DOZZ formula in the new rigorous setting, which has eluded mathematical understanding until now. This award reflects NSF's statutory mission and has been deemed worthy of support