Maxim is a third-year Mathematics student at the Australian National University, interested in areas of modern geometry influenced by Mathematical Physics. He has undertaken advanced studies in index theory, symplectic geometry and geometric gauge theory, as well as other topics including quantum field theory, Lie representation theory and the theory of schemes. In 2013 he was awarded the Boyapati Prize for first-year Mathematics and Computer Science and in 2014, the H. Neumann Prize for second-year Mathematics and the Australian Institute of Physics Prize. In 2016 he intends to complete an Honours thesis on the Donaldson-Floer-Fukaya category in Yang-Mills Theory, under the supervision of Dr. Bryan Wang.
The Donaldson-Floer-Fukaya Category In Yang-Mills Theory
This summer, I plan to investigate the symplectic geometry of $M_\Sigma$, the moduli space of flat connections, and apply the Fukaya category of $M_\Sigma$ to study the extended TQFT arising from Yang-Mills theory in dimension 2, 3 and 4; in dimension 2, for a closed surface $\Sigma$, we will study the Fukaya category whose objects are Lagrangian submanifolds, and whose morphisms between two objects $L_1$ and $L_2$ gives the Lagrangian Floer homology $HF(L_1, L_2)$. Some analytic breakthroughs in this direction have been achieved recently by Fukaya; it would be an exciting research project to uncover the underlying extended TQFT structures.