ABHYANKAR 82nd BIRTHDAY CONFERENCE SPEAKER: Shreeram Abhyankar (Purdue) TITLE: Antiquadratic Transformations ABSTRACT: This talk exemplifies my fondness for polynomial manipulations which may be regarded as my primary field of interest. So we may think of this as high-school algebra. But my speciality is that I see great geometry in the interplay of terms caused by a change of variables. A QDT is a Quadratic Transformation of the local plane which enables us to resolve the singularities of a curve in it. It amounts to making the simplest type of substitution in a bivariate polynomial. This was initiated by Cremona in Italy, furthered by Max Noether in Germany, and localized by Zariski in America. Thus, ring theoretically, we are passing from a two dimentional regular local ring to a bigger one birationally dominating it. Seeing that Kuhlmann has written a long abstract, I decided to out-do him by following H. F. Baker and completely avoid mathematical symbols to see the geometry inside algebra. But now, following Jacobi, I want to invert by walking backwards, i.e., from the dominating regular local ring to the original one. This is an Antiqdt = Antiquadratic Transformation. It leads to the existence of plane curves qith preassigned singularities. It also helps with the Jacobian Problem discussed by Luengo, Sathaye, and Zhao. Nedless to say that the Polygon of Newton gets the center stage.