Monday May 6th, 2024 | |
| Time: | 10:00 AM - 11:55 AM |
| Title: | Topics in Quantitative Rectifiability: Traveling Salesmen, Lipschitz Decompositions, Densities, and Big Pieces |
| Speaker: | Jared Krandel, Stony Brook University |
| Location: | Math 5-127 |
| Abstract: | |
| We present and prove assorted results in quantitative rectifiability. First, we study the quantitative rectifiability of Jordan arcs in Hilbert spaces, proving a version of the traveling salesman beta number estimate for length minus chord length analogous to an estimate recently attained by Bishop in Euclidean spaces. Second, we prove the existence of Lipschitz decompositions for domains with quantitatively flat boundaries. That is, we show any such domain has an "almost" decomposition into nice Lipschitz domains with control on the total surface area of the decomposition domains in terms of the original domain boundary area. Third, we study the regularity of Hausdorff measure on uniformly rectifiable metric spaces. We show that any such space satisfies the weak constant density condition of David and Semmes. Fourth, we study the iteration of the big pieces operator in Ahlfors regular metric spaces. We prove that iteration stabilizes after two iterations as a result of a more general extension theorem. | |