Title: Geometrically bounded embeddings
Speaker: Fedor Manin (Ohio State University)
Abstract: "Geometric" knot theory -- the study of knots with thickness -- has seen some study, motivated by applications such as protein folding. Less work has been done on the geometric complexity of embeddings of manifolds and simplicial complexes in higher dimensions. Indeed, the right notion of complexity depends on the dimensions and types of objects studied. Questions include: what is the least complexity of an embedding of a given M in a given N? What is the number of isotopy classes with representatives below a certain complexity? Is there a nice isotopy between two isotopic embeddings?
I am most interested in the case where the codimension of the embedding is at least 3. In this case, what little is known about the first question (through work of Gromov--Guth and Freedman--Krushkal) is tied up in questions of logic and computability. On the other hand, for certain notions of complexity, the second question has an answer: the number is polynomial in the complexity. This is joint work with Shmuel Weinberger.
Seminar URL: https://research.math.osu.edu/tgda/tgda-seminar.html