Towards a mathematical definition of Coulomb branches of 33-dimensional N=4N=4 gauge theories Hiraku Nakajima Kyoto University Consider the 33-dimensional N=4N=4 supersymmetric gauge theory associated with a compact Lie group GcGc and its quaternionic representation MM. Physicists study its Higgs and Coulomb branches, which are noncompact hyper-K\"ahler manifolds with SU(2)SU(2)-action, possibly with singularities. Higgs branches are just hyper-K\"ahler quotients of MM by GcGc, and the definition is clear. On the other hand, a physical definition of Coulomb branch involves a quantum correction, hence is hard to understand for mathematicians. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with C×C×-action when MM is of a form N⊕N∗N⊕N∗. The definition is motivated by hypothetical (2+1)D(2+1)D topological quantum field theories associated with monopole-like equations. The talk is based on my joint works with various others, e.g., A.Braverman and M.Finkelberg.