Dear Colleagues,
The 2012 Maxson Lectures will be given next week by Professor
Joe Harris, of Harvard University. We hope that you will be
able to come.
Time and place: Thursday, April 26 and Friday, April 27,
at 4 pm in Blocker 117, Texas A&M University,
College Station, Texas 77843-3368
Title: Interpolation Problems
Abstract: A fundamental theorem in algebra says that, given two
sets of complex numbers z_0,...,z_d and a_0,...,a_d, there exists
a unique polynomial f(z) in C[z] of degree at most d such that
f(z_i)=a_i for all i. We can extend this: given z_1,...,z_e and
integers m_1,...,m_e with sum d+1, there exists a unique
polynomial of degree at most d with arbitrary specified derivatives
up to order m_i at z_i.
This is a beautiful and highly useful result. But when we ask the
natural next question---what can we say about polynomials in
several variables---we enter a realm of mystery. The analogous
statement for polynomials in two or more variable is visibly false,
but no one knows exactly when, and by how much, it can fail. This
gives rise to a whole class of problems, collectively known as
interpolation problems.
The interpolation problem is like a number of problems in
algebraic geometry: it's completely elementary to state; a
general solution seems beyond us; and yet substantial progress
has been made and is currently being made. In these talks I'll
try to give an elementary introduction to the problem and what we
know about it. In particular, I'll try to describe a common thread
in the known and conjectured solutions of special cases, giving a
geometric characterization of when interpolation fails in general.
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