Hey! I'm Steve Trettel, a Minnesota transplant in southern California slowly coming to terms with the fact that winter is never just around the corner anymore. I am currently a 5th year graduate student at University of California, Santa Barbara trying to find balance in my life between hiking, cooking, and thinking about geometric manifolds. Wóiyukčaŋ, (or "thoughts" in the Dakhóta language) is my attempt to share some of the things I work on with you!
Research Wówiyawa kiŋ de theȟíka iwábdukčaŋ
My mathematical passions all stem from the desire to visualize and concretely understand various geometric phenomena. I work with Darren Long at UCSB, and am currently thinking about subgeometries of projective geometry, limits of geometries, and deformation spaces of geometric structures. My mathematical hobbies include accurately rendering homogeneous spaces, geometric group theory and quantum topology. I'm the kind of mathematician who enjoys "getting their hands dirty" with specific examples, and attempting to produce animations / computer models of my work when it helps convey meaning.
Low dimensional topology and geometry are full of beautiful mathematical objects and when I can I try to bring these to life in a way that they can be enjoyed by mathematicians and non-mathematicians alike. My primary focus is short video animations, but rendering these often leads to some beautiful stills as well. I have recently begun collaboration with the Media Arts and Technology department working on fully immersive virtual reality mathematical experiences as well.
Expository Writing Wíyawa wóuŋspe iwówawa
Occasionally I'll sit down and spend a week or so trying to really understand a particular phenomenon in low dimensional geometry or topology. When I succeed, I am trying to get better at writing up what I have learned so that it may benefit others. Here you'll find a mixture of posts on a variety of topics, with the common theme that they are all written in an accessible style and filled with pictures.
I've had many great teachers in my life that have helped me appreciate what a gift it is to be able to guide students to their own "ah-ha!" moments. Through a combination of learning the material myself and tutoring, I have written up some geometric viewpoints on math topics spanning early high school through college, which I will post here.
I don't have enough self-control to let myself read fiction all the time (else I end up staying up through the night telling myself "just one more chapter!"), but I do read a good amount of nonfiction. A selection of some memorable books I've come across appears under the "reading" tab.
The Dakota Language Dakhóta iápi kiŋ
I think my love for languages and math stem from the same general awe of the human mind's ability to abstract. Languages are themselves the abstract framework through which we can share our inner experiences with one another - and are so deeply a part of our being that even when we are thinking to ourselves, we do so in Language. I am a strong proponent of linguistic diversity and language revitalization, and have been studying the language of my home area, Dakhóta, for the past six years. The "Dakhóta" tab contains resources related to Dakhóta and language revitalization more generally.