SOT STEP Awardee Explores Mathematical Frameworks for Paradigmatic Inspiration


By Alexandra Munoz posted 11-13-2014 15:27


With the generous support of the Society of Toxicology (SOT) Supplemental Training for Education Program (STEP) Award administered by the STEP (2).jpgGraduate Education Subcommittee, I was able to attend a mathematics masterclass hosted by the University of Copenhagen’s Center for Symmetry and Deformation in Copenhagen, Denmark. The topic of the masterclass was topological quantum field theories, quantum groups, and 2- and 3- manifold invariants and was taught by Christopher Douglas, André Henriques, and Ryszard Nest. In attendance were graduate students in the very abstract areas of mathematics that deal with categorizing topological spaces and characterizing their mathematical properties.

The format of the course was framed in a way that may seem unusual in the sciences but is more common in pure mathematics, and that is of telling a “story”—the story being how to derive three-dimensional topological quantum field theories (3d TFT).The process traced the mathematical steps needed to derive the 3d TFT. The story spanned five days, each filled with lectures, coffee breaks to process the information, and boards of abstract and intriguing drawings to detail the constructions needed to map the relations (see photo courtesy of Eivind Dahl).

The masterclass was truly an act of oral story telling—the kind that is rarely found anymore, but that persists in the math community where this type of story is often the only way some information is communicated. To that end, the objective was to bear witness to this story unfolding—the maddening brilliance of the storytellers and the very great task of communicating this information across the languages of algebras, categories, groups, and topology. I would describe this experience as transformative, paradigm-shifting, and enlightening. I expected to learn about quantum groups to use in modeling, and what I received was far more valuable and far more ineffable, an expanded sense of where models can go, the spaces which they can occupy, and the wildness of the terrain they can imagine.

This expansion has strengthened my ability to conceptualize, to think freely, to devise new structures, and to venture grandly into the intellectual unknown. In turn, it will significantly impact my research at the New York Medical Center and my path forward. In the next year and years to come, I will utilize this sense of freedom to articulate paradigmatic forms that require great complexity and great axiomatic clarity. Specifically, I will work to develop models that can be used to evaluate toxins in computational and theoretical settings. Such models require complex constructions of intracellular spaces that will be well-served by mathematics in this area. In addition, this experience has widened my professional network and has already led to new mentors who will support my mathematical and theoretical explorations now and in the future.