Introduction to Algebraic Geometry 
pcmi2008 
Course Description
Algebraic geometry is the study of solutions to systems of polynomial
equations. It is a central and very active area of modern mathematics with deep
connections to commutative algebra, complex analysis, number theory,
combinatorics, and topology. It has applications in physics, robotics, coding
theory, optimization theory, and more recently in biology and statistics. We
will begin by developing the standard "dictionary" for translating between
algebraic properties of polynomials and geometric properties of their solution
sets. Special topics include Hilbert functions and resolutions of ideals,
enumerative geometry, Grassmannians and the Schubert calculus, and
computational aspects involving Gröbner bases.
What can I do to prepare?
All students should visit this algebra
review page. I am assuming everyone has had a good course in linear
algebra. If you have not had a course in abstract algebra or if your algebra is
weak, please make sure to read the algebra review sheet on that page and do the
exercises. I will be assuming the ring theory presented there.
Schedule
There will be 15 weekday morning lectures and afternoon problem sessions.
The lecture titles are in the table below.
Lectures  
1  The Dictionary  2  Nullstellensatz, Noetherian  3  Mappings  4  Dimension, Singularities  5  Projective space I 

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6  Projective Space II  7  Hilbert Functions  8  The Syzygy Theorem  9  Grassmannians I  10  Grassmannians II 
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11  Schubert Calculus I  12  Schubert Calculus II  13  Gröbner Bases I  14  Gröbner Bases II  15  Sandpiles 
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