Task:
Learn a new theorem related to the course material and communicate it in both verbal and written form.
Minimum requirements:

  • Each student is expected to learn a different theorem.
  • The written document must introduce, correctly state, and prove the theorem. It should also include at least one interesting example illustrating the theorem. The article should 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 PDF format.
  • The verbal presentation must introduce and state the theorem. It should also include at least one example illustrating the theorem.
Assessment and deadlines:
Project grades will be computed as follows:
due date element weight
2011-02-09 outline 10%
2011-03-23 rough draft 10%
2011-03-30 feedback on draft 10%
2011-04-09 presentation 30%
2011-04-09 feedback on presentations 10%
2011-04-09 final paper 30%
Comments:
By design, this assignment is very open-ended. Students are strongly encouraged to compute many examples. Students are also encouraged to formulate, test, and prove their own conjectures.
Advice:
Here are some suggestions on how to present mathmatics: Halmos on writing, Kleiman on writing, and Halmos on talking.
Potential topics:

  • algebraic statistics: [St2] §5 (Theorem 5.3); [St3] §8 (Theorem 8.14) — MARTIN HELMER;
  • Alexander duality: [MS] §5 (Theorem 5.24); [HH] §8.1 (Theorem 8.1.6) — ADRIAN MURESAN;
  • automatic theorem proving: [IVA] §6.4 (Proposition 6.5.8) — BENNETT KANUKA;
  • Bernstein's theorem: [UAG] §7.5 (Theorem 7.5.4); [St3] §3 (Theorem 3.2) — ALYSSA REED;
  • combinatorial Nullstellensatz: [TV] §9.1 (Theorem 9.2);
  • computation in local rings: [UAG] §4 (Theorem 4.2.2, Theorem 4.4.2); [GP] §6 (Theorem 6.2.6);
  • Fröberg's theorem: [HH] §9.2 (Theorem 9.2.3) — LARRY COGHLIN;
  • generic initial ideals: [E] §15.9 (Theorem 15.18); [HH] §4 (Theorem 4.1.2) — ADAM MCCABE;
  • Grassmannians: [H] §11 (Proposition 11.30) — BRITTANY HALVERSON-DUNCAN;
  • Hilbert syzygy theorem: [UAG] §6 (Theorem 6.2.1) [E] §15.5 (Theorem 15.10) — SUSANNAH MACDONALD;
  • integer programming: [UAG] §8.1-2 (Theorem 8.1.11); [St2] §5 (Theorem 5.5) — STEPHEN RO;
  • invariant theory of finite groups: [IVA] §7 (Theorem 7.3.5); [St1] §2 (Theorem 2.1.3) — DAVID MARQUIS;
  • Koszul complex: [E] §17.1 (Theorem 17.1); [GP] §7 (Theorem 7.6.14);
  • linear partial differential equations: [St3] §10 (Theorem 10.3) — JEREMY MACHATTIE;
  • Noether normalization: [E] §13.1 (Theorem 13.3) — JUNHO PETER WHANG;
  • Puiseux series: [E] §13.3 (Corollary 13.15); [St3] §1.4 (Theorem 1.7) — IVAN PENEV;
  • Quillen-Suslin theorem: [UAG] §5.1 (Theorem 5.1.8); [BG] §8 (Theorem 8.5) — NATALIE CORNEAU;
  • resolutions of monomial ideals: [MS] §3.5 (Theorem 3.17); [HH] §7 (Theorem 7.1.1);
  • resultants: [UAG] §3 (Theorem 3.2.3); [St3] §4 (Theorem 4.6) — CHIYO NISHIO;
  • SAGBI basis: [St2] §11 (Theorem 11.4); [MS] §14.3 (Theorem 14.11)
  • Stickelberger's theorem: [UAG] §2 (Theorem 2.4.5); [St3] §2.3 (Theorem 2.6);
  • straightening laws: [St1] §3 (Theorem 3.1.7, Theorem 3.2.1);
  • sums of squares: [St3] §7 (Theorem 7.3);
  • tropical hypersurfaces: [St3] § 9 (Theorem 9.17);
  • triangulations and toric ideals: [St2] §8 (Theorem 8.3); [BG] §7 (Theorem 7.18) — HARRY DE VALENCE;
  • universal Gröbner bases: [St2] §1 (Theorem 1.4), §7 (Theorem 7.1) — CONNOR BEHAN;
References:

[BG]  
Winfried Bruns and Joseph Gubeladze, Polytopes, rings, and K-theory, Monographs in Mathematics. Springer, 2009 ISBN 978-0-387-76355-2
[IVA]  
David A. Cox, John B. Little, and Don O'Shea, Ideals, Varieties, and Algorithms, third edition, Springer, 2007 ISBN 978-0-387-35650-1
[UAG]
David A. Cox, John B. Little, and Don O'Shea, Using Algebraic Geometry, GTM 185, Springer, 2005 ISBN 978-0-387-20733-9
[E]    
David Eisenbud, Commutative algebra with a view towards algebraic geometry, GTM 150. Springer, 1995 ISBN 0-387-94268-8
[GP]  
Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, 2nd edition, Springer, 2008 ISBN 978-3-540-73541-0
[H]    
Brendan Hassett, Introduction to Algebraic Geometry, Cambridge University Press, 2007 ISBN 978-0-521-69141-3
[HH]  
Jürgen Herzog and Takayuki Hibi, Monomial ideals, GTM 260. Springer, 2011 ISBN 978-0-85729-105-9
[MS]  
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, GTM 227. Springer, 2005 ISBN 0-387-22356-8
[St1]  
Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation. Springer, 1993 ISBN 3-211-82445-6
[St2]  
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8. American Mathematical Society, 1996 ISBN 0-8218-0487-1
[St3]  
Bernd Sturmfels, Solving systems of polynomial equations, CBMS 97, American Mathematical Society, 2002 ISBN 0-8218-3251-4
[TV]  
Terence Tao and Van H. Vu, Additive combinatorics, Cambridge University Press, 2010 ISBN 978-0-521-13656-3