We hope you will join us as Eve Shaw presents her research for the Department of Mathematical Sciences Colloquium Series.
"Tangents, Pseudotangents, and Tangent Field"
November 19, 1:00 PM, RB 121
Abstract: Informally, given a set K in Rd and a point p in K, we say that T is a tangent to K at p if blowing up the set K centered at (or near) the point p converges to a picture that looks like T (or is contained in T). This idea admits lots of flexibility, we could consider blow-ups centered exactly at p or blow-ups centered at a sequence (pj) of points of K converging to p, the convergence of the blow-up pictures could be with respect to the Gromov-Hausdorff topology or the Attouch-Wets topology or something else, and the type of set T is allowed to be could be restricted to linear subspaces, for example. Classically, Rademacher's theorem says that if f:[0,1]-> Rd is Lipschitz, then f’(t) exists at almost every t in [0,1], which is in part a statement about tangents to the curve f([0,1]), which is to say, if a curve is rectifiable, then it has tangent lines at almost every point. We can then ask the reverse question; given a set K in Rd, if we know that its tangents are lines or are subsets of lines, then is K a rectifiable curve or is it contained in a rectifiable curve? Jones and Okikiolu answered this question in the affirmative in 1990 and 1992 with the celebrated analyst's traveling salesperson theorem, and since then questions of when a set can be captured by a rectifiable curve have commonly been investigated via versions of tangents. In this talk, we discuss the weak tangent fields of Alberti-Csörnyei-Preiss and the tangents and pseudotangents of Badger-Lewis, relating these notions to each other and to other notions of tangent and deducing regularity theorems for sets with sufficiently regular tangents and tangent fields.
This is based on joint works with Krandel (UC Davis), as well as with Garg (UIUC), Tyson (UIUC), and Yang (Zhejiang University).