Products of Reflections in Smooth Bruhat Intervals

Overview

My research deepens connections between the two seemingly very different mathematical fields of algebraic combinatorics and algebraic geometry. In particular, I focus on the Bruhat order, a well-studied partial order structure in algebraic combinatorics, and deepen its relation to the algebraic geometric notion of smoothness. My main results show how chains in this Bruhat order emerge from smooth elements. I used this discovered structure to provide a new criterion for smoothness. Finding new criterions for smoothness has been a subject of many recent papers, and my results directly contribute to this effort, increasing possibilities for further study at the intersection of combinatorics and algebraic geometry. Outside pure mathematics, my research also has potential for a broader societal impact through the lens of physics. I build on methods used in cutting-edge research to analyze the amplituhedron, a geometric structure which is a tool in quantum field theory (QFT) and particle scattering. My research shows that the chains of the edges satisfy previously unknown relations, and could combine with other mathematical physics results in the near future to exponentially speed up computations of edges in the amplituhedron. Faster computations on this structure would lead further advances in QFT, thus deepening our knowledge of the subatomic nature of the universe. This research, by potentially advancing QFT, has proven potential for benefitting technologies in medical imaging, superconductivity, and more. SarcoSeg: A Convolutional Neural Network-based Sarcopenia Analysis System via Automatic Segmentation of Skeletal Muscle and Adipose Tissue in Cervical Computed Tomography Towards Precision Medicine

Competition history

• JSHS 2022 Category not listed

Resources

• National JSHS abstract book ↗

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