MATH Seminar

Title: Can computational math help settle down Morrey's and Iwaniec's conjectures?
Seminar: Analysis and Differential Geometry
Speaker: Wilfrid Gangbo, PhD of UCLA
Contact: Dr. Levon Nurbekyan, lnurbek@emory.edu
Date: 2025-02-14 at 11:00AM
Venue: MSC W303
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Abstract:
In 1987, D. L. Burkholder proposed a very simple looking and explicit energy functionals $I_p$ defined on $\mathbb{S}$, the set of smooth functions on the complex plane. A question of great interest is to know whether or not $\sup_{\mathbb{S}} I_p \geq 0$. Since the function $I_p$ is homogeneous of degree $p$, it is very surprising that it remains a challenge to prove or disprove that $\sup_{\mathcal{S}} I_p \geq 0$. Would $\sup_{\mathbb{S}} I_p \geq 0$, the so-called Iwaniec's conjecture on the Beurling--Ahlfors Transform in harmonic analysis would hold. Would $\sup_{\mathcal{S}} I_p > 0$, the so-called Morrey's conjecture in elasticity theory would hold. Therefore, proving or disproving that $\sup_{\mathbb{S}} I_p \geq 0$ is equally important. Since the computational capacity of computers has increased exponentially over the past decades, it is natural to hope that computational mathematics could help settle these two conjectures at once.

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