Projects (Bachelor/Undergraduate theses)
Marte Lovise worked on the theory of 2-braids, and on constructing an invariant of 2-braids akin to the Kauffman bracket invariant of ordinary braids. Marte Lovise categorified the construction of the latter via a map from a braid group to a Temperley-Lieb algebra, followed by taking the Markov trace.
Therese constructed an invariant of 2-knots akin to the Jones polynomial of an ordinary knot. Therese's approach was geometric and cubical, akin to the construction of the Jones polynomial via the Kauffman bracket polynomial.
Tor explored Martin-Löf type theory, with a view towards linguistics. He placed Martin-Löf type theory in the context of other programming languages and other approaches to the foundations of mathematics, and presented its syntactics, along with its logical (the propositions-as-types correspondence) and set theoretic semantics.
Tina explored the theory of virtual knots, motivating the passage from ordinary knot theory to virtual knot theory, and discussing various ways of viewing a virtual knot. Tina worked out a classification of all virtual knots with up to four crossings, and explored the self-linking number invariant and the extension of the Jones polynomial to virtual knots.
Reidun gave an exposition of the eversion of the 2-sphere via Hamma-Nagase-Roseman moves due to Carter, up to roughly the half way point. After first presenting a knot-theoretic argument for why the circle cannot be inverted, Reidun worked on visualising the steps in Carter's eversion. She drew pictures of the knotted sphere that is obtained after each step in the eversion, and explaining the steps in a way which could be understood by somebody not trained in mathematics.
Håvard explored the possibility of calculating homotopy groups of spheres using higher categorical models of spheres, via the homotopy hypothesis. Håvard looked at a variety of algebraic models of higher categories, including Penon's higher categories and Gray 3-categories, and explored and made use of several important category theoretic tools.
Ulrik explored the theorem, due to McCord, that every homotopy type has a model that is an Alexandroff topological space, with a view towards calculating homotopy groups of spheres by means of their finite models. Along the way, Ulrik looked into adjunctions in category theory involving a presheaf category, as well as into aspects of the theory of cubical sets.
Therese gave a cubical reworking of the foundations of knot theory and 2-knot theory. Along the way, Therese looked at the theory of cubical sets, and reworked Reidemeister's original proof that isotopies of knots are detected by the Reidemeister moves. She outlined a proof that isotopies of 2-knots are detected by cubical Roseman-Homma-Nagase moves.
Last updated: 20:53 (GMT+2), 29/08/2015.