Anthony is an Undergraduate at the University of Wollongong. He graduates from a Bachelor of Advanced Mathematics in December 2014 and was awarded an Undergraduate Scholarship to complete this degree. During his degree, Anthony took on additional mathematics work on Sobelev’s Embedding Theorem and Interpolation Spaces. Currently, Anthony is committed to learning more about Lorentz Spaces and their properties. His current mathematics research looks at Wente’s Inequality.
Anthony has always had a passion for mathematics and science from a young age and, in particular, problem solving, physics and cosmology. Anthony starts a graduate position with PriceWaterhouse Coopers in 2015. In his spare time he is interested in philosophy, IT. art theory and classical music. He is eager to pursue further research in the future.
Wente’s Inequality in Higher Dimensions
CMC surfaces are the most natural generalisation of minimal surfaces. A minimal surface has vanishing mean curvature, while a CMC surface has constant mean curvature. In the real world CMC surfaces are realised in a variety of contexts, including hanging droplets and block copolymers. They have also enjoyed applications in architecture, including bridge-building, where the shape of an optimal load-bearing arc is typically the profile of one of Delaunay’s unduloids, CMC surfaces of revolution. Perhaps one of the most well-known examples of this is the Sydney Harbour Bridge.Although minimal surfaces and CMC surfaces have several similarities, there are also striking differences. One of these is in the study of their regularity and existence. In the last twenty years there has been remarkable progress in this direction, with F. Helein making fundamental contributions. The Gauss map of a conformally parametrised CMC surface is harmonic. Exploiting this fact, Helein used analysis on exotic function spaces to tie together work on CMC surfaces and harmonic maps. Along the way he made new deep observations about fundamental objects such as the Laplace operator and Riesz transform. The focus of this project is on understanding exactly how much of Helein’s fundamental work can be extended to the case of CMC hypersurfaces, with a view to tackling the four-dimensional case first.