New resummation techniques of divergent series: the Painlevé equation PII

I worked on this research project at The Ohio State University under Professor Ovidiu Costin from February 2018 to March 2019 and from August 2019 to May 2020. The research was on advanced methods for resummation of divergent series to convergent solutions for differential equations. When resumming a divergent series numerically, one generically has to deal with limited information due to having computed only a finite number of terms. To deal with this, we used novel approximation methods that are more precise than the standard Padé approximation. Additionally, Professor Costin developed a new resummation method which leads to rapidly convergent uniform rational expansions for solutions to differential equations. This method relies on resurgence theory, a new area in analysis with many applications to different areas of physics, such as quantum field theory. I applied this method to the tritronquée solutions of Painlevé equation PII. Our results are now being written up for publication, and I presented the ongoing results at the 2018 Ohio State Autumn Undergraduate Research Festival, the 2019 Ohio State Denman Undergraduate Research Forum (Denman poster), and the 2019 Young Mathematicians Conference at Ohio State (YMC abstract and YMC poster). Finally, I successfully completed and defended my undergraduate thesis on this research in April 2020 (thesis and defense slides).