Abstract: We will discuss the problem of finding the maximum number of 4-point circles given 9 points, where a 4-point circle is a circle on which lie exactly 4 of the 9 given points. This problem is solved for points in $\mathbb{R}^2$, we present a generalization to $\mathbb{C}^2$. Our approach uses techniques from polyhedral geometry and group theory. In particular, we will construct a polytope whose lattice points represent candidate point-circle incidences of 9-point configurations. This method works until the number of circles is greater than 4, at which point solving for the lattice points becomes computationally infeasible. We present then another method which instead recursively constructs these configurations. We reduce computations by leveraging symmetries of $S_9$. Finally, we discuss the original motivation for this work, coming from an interpolation problem in mechanical engineering involving four-bar linkages.

This is a final presentation for the course AM 4999Z.

Abstract: A basic and naive problem in homotopy theory is to compute the sets $[S^m, S^n]$ of homotopy classes of maps between spheres of different dimensions. I will describe the preliminary results of a machine-based approach to these computations. Historically, there are two separate paradigms for such computations: the EHP sequence, and the unstable Adams spectral sequence. Our approach exploits both, and the interaction between them.

I will not assume any familiarity with either the EHP sequence or the unstable Adams spectral sequence.