Alex Ortiz (Rice University) - Tangency counting for well-spaced circles скачать в хорошем качестве

Alex Ortiz (Rice University) - Tangency counting for well-spaced circles

Alex Ortiz (Rice University) - Tangency counting for well-spaced circles Recorded talk on Feb 3, 2026 at the Online Analysis Research Seminar (OARS) Abstract: In the late 1990s, Wolff introduced the circle tangency problem: among N circles, with no three tangent at a point, how many pairs can be internally tangent? He proved an N^{3/2+ε} upper bound, which was sharpened to N^{3/2} by Ellenberg, Solymosi, and Zahl in 2016 using the polynomial method. The conjectured sharp bound N^{4/3+ε} is achieved by a grid of circles. In recent joint work with Dominique Maldague, we show that for approximately gridlike families of circles, the N^{3/2} barrier can be broken: we obtain an upper bound of N^{25/18+ε}, coming close to the conjectured 4/3 exponent. I will outline the background and the connection of the circle tangency problem with restriction theory, and then describe our stopping-time framework, which uses a new square-function interpretation of refined decoupling for the truncated cone in R^3. https://sites.google.com/view/o-a-r-s