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
lecture     problem set Algebraic Curves by William Fulton. algebraic geometry algebraic varieties Zariski topology lecture1.nb		lecture    problem set David Hilbert the Nullstellensatz Emmy Noether Hilbert basis theorem		lecture    problem set spectrum of a ring lecture3.nb		lecture    problem set quotient field algebraic independence transcendence degree Krull dimension singularities Zariski tangent space Oscar Zariski lecture4.nb		lecture    problem set projective space conic sections perspective in graphic arts Bezout's theorem
6	Projective Space II	7	Hilbert Functions	8	The Syzygy Theorem	9	Grassmannians I	10	Grassmannians II
lecture    problem set Veronese surface Segre embedding Pappus' theorem and its dual conics lecture6.nb		lecture    problem set numerical polynomials lecture7.nb hw6.nb		lecture    problem set *CoCoA handout* module The ultimate free resolution An overview of the multiplicity conjecture, and a proof		lecture    problem set Grassmannian Plücker coordinates Hermann Grassmann Julius Plücker		lecture    problem set dual vector space exterior algebra
11	Schubert Calculus I	12	Schubert Calculus II	13	Gröbner Bases I	14	Gröbner Bases II	15	Sandpiles
lecture    problem set H. Schubert I H. Schubert II enumerative geometry Chow ring transversality meeting 4 lines conics problem projective plane		lecture    problem set An introduction to the Schubert calculus.		lecture    problem set Macaulay An introduction by Buchberger.		lecture    problem set Gröbner basis Buchberger's homepage		lecture     self-organized criticality Bak-Tang-Wiesenfeld model A nice sandpile applet

Links

• PCMI
• Some nice algebraic surfaces.
• Algebraic geometry and analytic geometry
• Algebraic Curves by William Fulton.
• Computational algebraic geometry.
  • CoCoA
    • reference card
    • manual
    • tutorial
  • Macaulay 2
  • Singular
• Natalie and Helen's picture of y-x^2 over the complex numbers.
• Jack's curve of conics (curve in P⁵).
• David White's solutions. Please notify us with any corrections/refinements/etc.
  • HW1, tex file
  • HW2, tex file
  • HW3, tex file
  • HW4, tex file
  • HW5, tex file
  • HW6, tex file
  • HW7, tex file
  • HW8, tex file
• Stefan Sabo's blowing-up of the PCMI t-shirt:
  • Paper.
  • Mathematica notebook.