:This course aims to show that many PDEs studied in differential geometry can be considered as moment map equations for some infinite-dimensional group action. This point of view gives a unifying perspective for the study of various problems and can provide important insights by establishing a parallel with known phenomena in finite dimensions. We will be most interested in explaining a formal connection between the existence of solutions to PDEs of geometric interest and algebraic stability conditions, along the lines of the Kempf-Ness Theorem. Time permitting, we will examine in some detail a famous example of this phenomenon: the Hermite-Einstein equation for holomorphic vector bundles over a complex curve.
You can find below some of the themes that will be discussed. Depending on the interests of the audience, topics can be added or deleted from this list. Introduction to symplectic and Kähler geometry Hamiltonian actions and symplectic quotients Elements of GIT, Kempf-Ness Theorem Introduction to infinite-dimensional manifolds Connections on holomorphic bundles Moment maps and the Hermite-Einstein equation The Hitchin-Kobayashi correspondence Moment maps and the constant scalar curvature equation references:
[1] Da Silva, Ana Cannas. "Lectures on symplectic geometry". Berlin: Springer, 2008.
[2] Donaldson, Simon K. "Moment maps in differential geometry." Surveys in differential geometry 8, no. 1 (2003): 171-189.
[3] Huybrechts, Daniel. "Complex geometry: an introduction". Springer Science & Business Media, 2005.
[4] McDuff, Dusa, and Dietmar Salamon. "Introduction to symplectic topology". Vol. 27. Oxford University Press, 2017.
[5] Trautwein, Samuel. "Infinite dimensional GIT and moment maps in differential geometry." PhD diss., ETH Zurich, 2018.