1. A Structured Wikipedia for Mathematics
January 6, 2010
ICS poster talk
By henry Lin
OR “mathematics in a web 2.0 world”

2. Overview Background and motivation
A new system for organizing mathematics
Other interesting directions and open questions

3. The challenge of doing research
Finding related work and open questions
Related work is difficult to find
Naming problem
Lots of research being produced every year
Many different overlapping fields of research

4. A new project
Create a new website to organize existing results and open questions
A collaborative website like wikipedia, but results stored in a more structured manner
Structure provides better organization, and makes related work easier to find

5. Variations on Chernoff’s Bound
Initial Conditions:
Let x1, x2, …, xn be
Independent random variables
Binary random variables
With probability ½ of being 0 or 1
Let X = x1 + x2 + … + xn
Let μ = E[X]
Let 0 < δ < 1
Conclusion:
Pr[ X ≥ (1-δ)·μ ] ≤ exp(- μ·δ2)

6. Variations on Chernoff’s Bound
Initial Conditions:
Let x1, x2, …, xn be
Independent random variables
Binary random variables
With probability ½ of being 0 or 1
Let X = x1 + x2 + … + xn
Let μ = E[X]
Let δ > 0
Conclusion:
Pr[ X ≥ (1+δ)·μ ] ≤ exp(- μ·δ2)

7. Variations on Chernoff’s Bound
Initial Conditions:
Let x1, x2, …, xn be
Independent random variables
Binary random variables
With probability pi of being 1 and (1-pi) of being 0
Let X = x1 + x2 + … + xn
Let μ = E[X]
Let δ > 0
Conclusion:
Pr[ X ≥ (1+δ)·μ ] ≤ (eδ / (1 + δ)(1+δ) )μ

8. Hoeffding's Inequality
Initial Conditions:
Let x1, x2, …, xn be
Independent random variables
Such that xi є [ai , bi] almost surely
Let X = x1 + x2 + … + xn
Let μ = E[X]
Let δ > 0
Conclusion:
Pr[ X ≥ μ + δ] ≤ exp(- 2·δ2 / ∑ (ai - bi)2 )

9. Azuma–Hoeffding Inequality
Initial Conditions:
Let x1, x2, …, xn be
Such that Yi = x1 + … + xi forms a martingale
Such that |xi| < ci almost surely
Let X = x1 + x2 + … + xn
Let μ = E[X]
Let δ > 0
Conclusion:
Pr[ X ≥ δ ] ≤ exp(- δ2 / 2·∑ci2 )

10. Organizing Mathematical Concepts
Can be used to organize related:
Lemmas and Theorems
Mathematical Concepts
Algorithmic Problems
Complexity Classes

11. Related Work PlanetMath/Wikipedia: collaborative websites
by Nathan Egge, Aaron Krowne, and others
by Jimmy Wales, Larry Sanger, and others
Tricki: collaborative website to store techniques for solving math problems
by Tim Gowers and others
Open Problem Garden: collaborative website to store open problems
garden.irmacs.sfu.ca by Matt DeVos and Robert Šámal
Polymath: collaborative effort to solve open math problems
by Tim Gowers and others
Vdash: collaborative website to write proofs in a formal manner that can be checked
by Cameron Freer (MIT)

12. The Polymath Project
Collaborative work done over blogs, discussion forums, and Wiki pages
Derived a new simpler proof for a variation on the density Hales-Jewett theorem.
Initiated Feb 1, 2009
New result being written
Article about result published in Nature

13. Open Questions
Is the above system adequate for organizing and linking related results?
Should more rules be imposed on the system?
What other functionality should be implemented into the system?

14. Open Questions
Can a similar structured system be developed to link and organize proofs, or ideas for proofs of theorems?
Is there some structure that proofs can be written in to make them more understandable and checkable?
Are there other ways to make collaborative research more feasible?

15. Thanks!
Questions?