The APDE seminar on Monday, 10/23, will be given by Kevin Ren (Princeton) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (email@example.com) or Mengxuan Yang (firstname.lastname@example.org).
Title: Pinned Distances in R^d
Abstract: Given a set E in R^d with Hausdorff dimension > d/2, Falconer conjectured that the set of distances between any two points in E has positive Lebesgue measure. This conjecture remains open in all dimensions, despite significant progress in the last 30 years. Building upon this progress, we show that if d >= 3 and dim_H (E) > d/2 + 1/4 – 1/(8d+4), then the distance set of E has positive Lebesgue measure. The proof uses a new radial projection theorem in R^d applied to a variant of a decoupling framework of Guth-Iosevich-Ou-Wang. Joint work with Xiumin Du, Yumeng Ou, and Ruixiang Zhang.
The APDE seminar on Monday, 10/30, will be given by Yuzhou (Joey) Zou (Northwestern) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (email@example.com) or Mengxuan Yang (firstname.lastname@example.org).
Title: Weighted X-ray mapping properties on the Euclidean and Hyperbolic Disks
Abstract: We discuss recent works studying the sharp mapping properties of weighted X-ray transforms and weighted normal operators. These include a C^\infty isomorphism result for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We also discuss further work studying additional self-adjoint realizations of this operator, using the machinery of boundary triplets. In addition, we discuss ongoing work which applies these results to the X-ray transform on the hyperbolic disk by using a projective equivalence between the Euclidean and hyperbolic disks. Joint works with N. Eptaminitakis, R. K. Mishra, and F. Monard.