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SUMMARY:Ruixiang Zhang (IAS)
DTSTART:20210419T210000Z
DTEND:20210419T220000Z
DTSTAMP:20260423T024554Z
UID:OARS/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OARS/22/">St
 ationary set method for estimating oscillatory integrals</a>\nby Ruixiang 
 Zhang (IAS) as part of OARS Online Analysis Research Seminar\n\n\nAbstract
 \nGiven a polynomial $P$ of constant degree in $d$ variables and consider 
 the oscillatory integral $$I_P = \\int_{[0\,1]^d} e(P(\\xi)) \\mathrm{d}\\
 xi.$$ Assuming the number $d$ of variables is also fixed\, what is a good 
 upper bound of $|I_P|$? In this talk\, I will introduce a ``stationary set
 '' method that gives an upper bound with simple geometric meaning. The pro
 of of this bound mainly relies on the theory of o-minimal structures. As a
 n application of our bound\, we obtain the sharp convergence exponent in t
 he two dimensional Tarry's problem for every degree via additional analysi
 s on stationary sets. Consequently\, we also prove the sharp $L^{\\infty} 
 \\to L^p$ Fourier extension estimates for every two dimensional Parsell-Vi
 nogradov surface whenever the endpoint of the exponent $p$ is even. This i
 s joint work with Saugata Basu\, Shaoming Guo and Pavel Zorin-Kranich.\n
LOCATION:https://researchseminars.org/talk/OARS/22/
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