Singularities and their partial resolutions

Abstract: The twentieth century saw the resolution of singularities for varieties in characteristic 0, and there has been progress toward generalizing the result and improving the methods in characteristic 0. On the other hand, significant progress had been made toward a resolution without making assumptions on the characteristic, at the cost of having a partial resolution. This means that the resulting birational model may have mild singularities. In this talk, I explain the situation and the methods used for obtaining Cohen-Macaulay partial resolutions. Then, I discuss further directions in partial resolutions.