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APDE Seminar

Abstracts

Lei Zhang
(Professor @ University of Florida, United States)
Professor Lei Zhang got his PhD from Rutgers University in 2001 under the supervision of Yanyan Li. After that he was employed at Texas A&M, University of Alabama-Birmingham and University of Florida.
Blowup solutions and vanishing estimates for singular Liouville equations

The singular Liouville equation is a class of second order elliptic partial differential equations defined in two dimensional spaces: \[\Delta u+ H(x)e^{u}=4\pi \gamma \delta_0\] where \(H\) is a positive function, \(\gamma>-1\) is a constant and \(\delta_0\) stands for a singular source placed at the origin. This deceptively simply looking equation has a rich background in geometry, topology and Physics. In particular it interprets the Nirenberg problem in conformal geometry and is the reduction of Toda systems in Lie Algebra, Algebraic Geometry and Gauge Theory. Even if we only focus on the analytical aspects of this equation, it has wonderful and surprising features that attract generations of top mathematicians. The structure of solutions is particular intriguing when \(\gamma\) is a positive integer. In this talk I will report recent joint works with Juncheng Wei that give a satisfactory answer to important issues to this equation. I will report the most recent results, new insights, and the consequences of these results.

In this talk I will report recent joint works with Juncheng Wei that give a satisfactory answer to important issues to this equation. I will report the most recent results, new insights, and the consequences of these results.

Ryotaro Tanaka
(Associate Professor @ Tokyo University of Science, Japan)
Ryotaro Tanaka obtained his PhD from Niigata University in 2015 under the supervision of Professor Kichi-Suke Saito. AZer spending one year at Niigata University as a research fellow, and two years at Kyushu University as a JSPS postdoctoral fellow, he became a Junior Associate Professor at Tokyo University of Science, where he got promoted to Associate Professor in 2022.
Nonlinear Classification of Banach spaces based on Birkhoff-James orthogonality

Birkhoff-James orthogonality is a generalized orthogonality relation in Banach spaces which is strongly related to the geometric structure of Banach spaces. In this talk, I will present a recent work on the nonlinear classification of Banach spaces based on Birkhoff-James orthogonality. I will show that reflexive smooth Banach spaces are identified isomorphically by their Birkhoff-James orthogonality structure, that three or more dimensional Hilbert spaces are determined isometrically in terms of Birkhoff-James orthogonality, and that classical sequence spaces are classified under the equivalence based on Birkhoff-James orthogonality preservers.

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