Speaker: Dylan Airey

Title: Arithmetic progressions in Cantor-like sets

Abstract: One interesting property of the Cantor set is that it does not contain an arithmetic progression of length greater than 4. I will introduce a conjecture of V. Bergelson, L. Fishman, and D. Simmons about the behavior of the maximum length of arithmetic progressions in a class of Cantor-like sets. I will also prove a related theorem about constructing sets that contain arbitrarily long arithmetic progressions and have a specific Hausdorff dimension.

Speaker: Mike Trenfield

Title: Exploring the fractal landscape

Abstract: Fractals first captured the imagination of the general public in the 1960s with the advent of modern computer graphics and the work of the late Benoit Mandelbrot. It is well known that fractals that are subsets of the plane cannot be said to have finite length, so how can we best describe these? Traditional descriptions of fractals such as the Hausdorff dimension can often require very involved calculations. Are there certain instances in which we can ease our calculations and represent the Hausdorff dimension more compactly? Under which circumstances can we do this?