Math 413/813 (2023)

Introduction to Algebraic Geometry (Projects)

enumerative geometry
Cubic plane curve
Kummer surface
Oscar Zariski
Task
Learn a new theorem related to the course material and communicate it as a written document and via a video presentation.
Minimum Requirements
  • Each student will focus on a different result.
  • The written document will introduce/motivate, correctly state, and prove a theorem. It will also include at least one interesting example, construction, or special case illustrating the theorem. The article will be as self-contained as possible. The new document must be typed, be at most eight pages in length (with one inch margins and a 12pt font), and be available in the PDF format.
  • The video presentation must introduce and state the theorem. It should also include at least one example, construction, or special case illustrating the theorem. This new video must be at most 20 minutes in length and available in a common format such as the MP4 file type.
Assessment
Project grades will be computed as follows:
Due Date Element Weight
2023.01.27 research 10%
2023.02.09 outline 10%
2023.03.06 draft 10%
2023.03.16 feedback 15%
2023.04.05 paper 30%
2023.04.13 video 25%
Advice
Paul R. Halmos and Steven L. Kleiman each provide some suggestions on how to write mathematics. Paul R. Halmos also offers some suggestions about how to talk mathematics.
Comments
By design, this project is very open-ended. Students are strongly encouraged to create their own examples. Consider what was the original motivation or historical context for your theorem. Does your theorem have any interesting specializations or important applications?
Potential Topics
The following are natural candidates:
  • Alexander duality — SANTIAGO GUTIERREZ
  • Automatic theorem proving — JOHN ALAJAJI
  • Bernstein theorem — JINGJING MAO
  • Budan-Fourier theorem — COLE GIGLIOTTI
  • Computations in local rings — DOMINIC AUSTRIA
  • Conditional independence models
  • Descartes rule of signs — SUNNIE ZHANG
  • Elliptic curves — GRAYSON PLUMPTON
  • Fröberg theorem
  • Grassmannians
  • Generic initial ideals
  • Going-up theorem — SOPHIA SHEN
  • Gröbner fan
  • Hilbert syzygy theorem — JULIA MCCLELLAN
  • Integer programming — TREVOR SHILLINGTON
  • Invariant theory — ANGUS MCGREGOR
  • Kushnirenko theorem
  • Lexsegment ideals
  • Linear partial differential equations — HOWIE HONG
  • Local-global principle — YUXIAN AN
  • Multivariate polynomial splines — BORIS ZUPANCIC
  • Multivariate resultants — JAMES CONNELLY
  • Newton polytopes — ALEXANDER KAPTY
  • Noether normalization
  • Puiseux series
  • Quillen–Suslin theorem
  • Real nullstellenstaz — XINYI LIANG
  • Sagbi basis
  • Secant varieties
  • Semidefinite programming — ALICE PETROV
  • Solving equations via eigenvectors — CLAIRE VANDESANDE
  • Sums of squares
  • Symmetric polynomials — ALYSSA GREEN
  • Triangulations and toric ideals
  • Universal Gröbner bases
References
  1. Ciro Ciliberto, An undergraduate primer in algebraic geometry, Unitext 129, Springer, 2021
  2. David A. Cox, John B. Little, and Donal O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Fourth Edition, Springer, 2015.
  3. David A. Cox, John B. Little, and Donal O’Shea, Using Algebraic Geometry, Second edition, Graduate Texts in Mathematics 185, Springer, 2005
  4. David Eisenbud, Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995
  5. Brendan Hassett, Introduction to algebraic geometry, Cambridge University Press, 2007
  6. Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2011
  7. Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer, 2005
  8. Frank Sottile, Real solutions to equations from geometry, University Lecture Series 57, American Mathematical Society, Providence, RI, 2011
  9. Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8. American Mathematical Society, Providence, RI, 1996
  10. Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics 97, American Mathematical Society, Providence, RI, 2002