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Date:
Tuesday, 9 Sep. 2025,
09:00-10:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr. Sung-Jin Oh
(University of California, Berkeley )
Title: Integral formulas for under/overdetermined differential
operators via recovery on curves and the finite-dimensional
cokernel condition Abstract: Underdetermined differential operators arise naturally in
diverse areas of physics and geometry, including the
divergence-free condition for incompressible fluids, the
linearized scalar curvature operator in Riemannian geometry,
and the constraint equations in general relativity. The duals
of underdetermined operators, which are overdetermined, also
play a significant role. In this talk, I will present recent
joint work with Philip Isett (Caltech), Yuchen Mao (UC
Berkeley), and Zhongkai Tao (IHÉS) that introduces a novel
approach - called recovery on curves - to constructing integral
solution/representation formulas (i.e., right-/left-inverses)
for a broad class of under/overdetermined operators via solving
ODEs on curves. They are optimally regularizing and have
prescribed support properties (e.g., produce compactly
supported solutions for compactly supported forcing terms). A
key feature of our approach is a simple algebraic condition on
the principal symbol - called the finite-dimensional cokernel
(FC) condition - that implies the applicability of our method.
This condition simplifies and unifies various treatments of
related problems in the literature. If time permits, I will
discuss applications to studying the flexibility of initial
data sets in general relativity.