Abstract:

Abstract:
A corollary of Hilbert's Nullstellensatz is that the quotient algebras of two
radical ideals in the ring of (commutative) polynomials in d complex variables
are isomorphic to each other if and only if the corresponding varieties are
isomorphic, in the sense that there exist polynomial maps between the ddimensional
complex space that restrict to mutually inverse bijections between
the corresponding varieties. In this talk, we consider the noncommutative (nc)
analogue of the above result and answer the following questions:
When are two nc varieties "isomorphic" to each other? What happens if we
replace the ring of complex polynomials with some other algebra of complex
nc functions?
We start with a soft introduction to nc function theory and discuss some of the
properties that general nc functions share. For the main result, we use a
remarkable theorem of Ball, Marx, and Vinnikov about extending nc functions
off of subvarieties to show that if the ambient space is "nice enough" then two
subvarieties are isomorphic (in the sense that there is a bijective nc map
between the subvarieties) if and only if this isomorphism is given by the
restriction of an isomorphism between the ambient spaces.