The Analyst's Traveling Salesman Theorem for large dimensional objects
Recent Developments in Harmonic Analysis May 15, 2017  May 19, 2017
Location: SLMath: Eisenbud Auditorium
geometric incidence problem
complex analysis
harmonic measure
harmonic analysis
Traveling Salesman Theorem
30D30  Meromorphic functions of one complex variable (general theory)
5211  Research data for problems pertaining to convex and discrete geometry
30D15  Special classes of entire functions of one complex variable and growth estimates
Azzam
The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a square sum of quantities called betanumbers that measure how nonflat the set is at each scale and location. Conversely, given such a curve, the square sum of its betanumbers is controlled by the total length of the curve, giving us quantitative information about how nonflat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional surfaces. This is joint work with Raanan Schul.
Azzam Notes

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