ABSTRACT:
I will start the talk by reviewing our recent work with M.Finkelberg
and H.Nakajima on the mathematical construction of Coulomb branches of
3-dimensional N=4 super-symmetric gauge theories as affine complex
(possibly singular) symplectic algebraic varieties admitting a
canonical quantization (no physics background will be assumed). I
will then proceed to the discussion of the generalization of this
construction to Coulomb branches of 4-dimensional gauge theories
compactified on a circle. I will present a set of conjectures about
the resulting varieties and give some examples.