I am currently investigating Abel’s historical proof of the insolubility of quintic polynomials. I hope to update my blog with either a page or a link to a separate blog when I have finished studying the proof. My idea is to explain the proof in plain language, including several elementary examples, so that it can be easily digested by a relative novice in the field.
Magnetic Resonance Guided High Intensity Focused Ultrasound (MRgHIFU)
The main body of my research is from a project that I worked on for over five years while I was a graduate student at the University of Utah. The goal of this work was to investigate the use of Magnetic Resonance guided High Intensity Focused Ultrasound (MRgHIFU) as a method for treating breast cancer. Unlike chemotherapy, surgery, and radiation, MrgHIFU offers the potential for safer and relatively non-invasive cancer treatments that facilitate quicker patient recovery.
My part in the research project was to run computer simulations of treatments using a variety of treatment parameters so that the treatments could be optimized. One of the obstacles that HIFU breast cancer treatments are facing that impede widespread clinical use are long treatment times. For even small tumors, treatments can take several hours.
The results of my research found that clinicians can potentially speed up treatments by an order of magnitude by using a tightly focused beam and stacking treatment locations along the axis of the ultrasound transducer. This approach works because it takes advantage of the physical properties of the system, including the idea of heat superposition.
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I also participated in two research projects as an undergraduate. Both projects were funded through a VIGRE grant given to the University of Utah and spanned two semesters.
The first problem was research investigating new methods of packing objects of different objects moving at different speeds through a small aperture. Such research has applications in the video game industry, particularly in games that have large army movements involving multiple sprites as part of the game.
The second research project involved investigating the connection between number systems and mathematical objects called fractals. Even though it seems like the topics of writing numbers in a particular number base (say ternary) and generating a fractal object seem unrelated at first, there are some deep connections between these problems. The Cantor gasket set can be formed by drawing the real number line between zero and one and systematically eliminating numbers with 1 in their ternary decimal expansion.