The human brain is a complex organ with many folds and fissures, with considerable variability in the shape, size and depth of folds across individuals. This variability makes it difficult to relate brain function to anatomy and to compare results between different subjects. It is known that the majority of the functional processing of the brain occurs on the highly convoluted surface of the brain in the thin layer called the gray matter. Many areas of research involving the human brain are concerned with identifying regions and fissures that are responsible for particular functional tasks, such as vision or motion, or regions which are affected by diseases, such as Alzheimer's or schizophrenia.
Due to the complex shape and individual variability in the folding patterns and the surface-based functional processing of the brain, there is great interest in unfolding and flattening the cortical surface. Creating "flat" maps of the brain can lead to improved analysis and visualization techniques for functional and anatomical data. Flat maps will enable coordinate systems to be developed on the brain, which will make it easier to compare data from different subjects. Although, it is impossible to create length and area preserving maps, the 150-year-old Riemann Mapping Theorem says that conformal (angle preserving) maps exist. In this presentation I will discuss a mathematical method called "circle packing" which I am using to generate quasi-conformal maps of the human brain. I will present examples of some of the brain maps I have created and discuss how 150-year-old and modern mathematics is contributing to research that may enable neuroscientists to better understand the functioning of the human brain and elucidate new information.