Preparation on Duality

You can go through this list to get prepared for this lecture on duality.

General Questions

1. What is duality?
2. What dualities in mathematics can you think of?
3. What do you think is the ultimate source of dualities?
4. Do you think that Cartesian products and disjoint union are similar? How to formulate this similarity?

Graph Theory

1. Are you familiar with planar graphs and its dual?
2. Can you try to give a definition of the dual without using geometric descriptions?
3. Can you try to list the most intrinsic defining properties of forests?

Linear Algebra

1. Do the graph theoretic problems above remind you about linear algebra?

Some Theorems and Topics We're About to Go Through

Take a look and gain basic familiarity about the following topics: graph theory (planar graphs, dual graphs), linear algebra (linear independence, orthogonal subspaces), and a little bit category theory. Only a little bit (what it is, and basic notions) is required.

Take a look and think a little bit on the proof of following theorems:

• Strong Duality Theorem, in linear programming.
• Minimax Theorem, in game theory.
• Kuratowski's Theorem, in graph theory.