Algebraic Geometry and Commutative Algebra seminar
Tuesday, 31 January, 11:30 am
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Venue: Ramanujan Hall
Host: Sudarshan Gurjar
Speaker: Nitin Nitsure
Affiliation: TIFR Mumbai (retd)
Title: Multi-variable calculus via commutative algebra
Abstract: The nicest functions of all are the linear ones. But most of the important functions that we meet are non-linear. The derivative of a function gives a linear approximation to a nonlinear function. The Jacobian matrix is the version of this in several variables. Theorems such as the implicit function theorem of multivariable calculus say that under suitable hypothesis, a nonlinear function behaves in a small enough neighbourhood of a point quite like the linear function defined by its Jacobian matrix at that point. But what happens when we move from real numbers and Euclidean spaces (or smooth manifolds) to arbitrary fields, commutative rings, varieties and schemes? For example, what happens if functions are replaced by integers in a number field? What can differential calculus tell us about these? The amazing answer, discovered by Zariski, Grothendieck, Michael Artin and the modern algebraic geometers. is that we can essentially replicate the results of multivariable differential calculus via commutative algebra, in fact, it is possible to do much more, and then apply it to algebraic geometry. This led to a close study in the 1960s of the local behaviour of functions (or morphisms) on varieties and schemes, which occupies more than half the space in Grothendieck's famous `EGA'. This lecture is a semi-popular account of some of these ideas, and will not require any prior knowledge beyond standard undergraduate material. It is a special `opening lecture' of a more technical semester-long course on local structure of morphisms in algebraic geometry, which is in turn is the part II of the multi-semester course on Algebraic Stacks and Moduli which began in the last semester.