I am a Masters of Philosophy (research) student, studying at the University of Queensland, Brisbane, Queensland, Australia. I am a geometric analyst and differential geometer at heart, but I am keen on being fluent in algebraic geometry and topology. The purpose of this blog is to relate the language and ideas of differential geometry to those in algebraic geometry.

During my honours, I used geometric flows to solve the prescribed Ricci curvature problem, which asks "given some curvature T, can I find a metric whose (Ricci) curvature is T".
I posed this question on compact homogeneous spaces, whose rich algebraic structure reduced the problem to solving a system of ODEs. 

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During my Masters, I decided to broaden my horizons and invite more algebraic geometry into my differential/analytic life. Currently, I am learning the theory of holomorphic structures, connections and the Narasimhan-Seshadri theorem. So far, I have found this story (pioneered by Donaldson, Hitchens, Corlette, Uhlenbeck and others) to be an awesome simmering crockpot of differential/algebraic geometry, topology, elliptic PDEs and gauge theory. From then on, I hope to dive deeper in the story of  Higgs bundles and the non-abelian Hodge correspondence.

 

More generally, I am interested in moduli spaces. Since moduli spaces parametrises equivalence classes of objects, describing the geometry of a moduli space will provide a deep insight into the geometry of the objects themselves. This, to me, is a very powerful endeavour and so, I am keen on learning the differential/algebraic geometry and topology of moduli spaces. This includes the dynamics on moduli spaces, such as Teichmuller spaces, and Weil-Peterson geometry. 

I am definitely at the beginning of my journey and am still learning about myself in terms of mathematical interests. If there are some mistakes in my explanations, complaints or compliments, please let me know. My webpage is here.