The present talk is a tutorial one and gives some essence of submodular functions and discrete convexity.

The contents of my talk will be the following:

1. Submodular and Supermodular Functions
   1. Submodularity
   2. Distributive Lattices and Posets (Birkhoff-Iri)
2. Submodular Systems and Supermodular Systems
   1. Submodular polyhedron and Base Polyhedron
   2. Duality
   3. Polymatroids and Matroids
   4. Generalized Polymatroids
   5. Characterization by Edge-vectors
   6. Crossing- and Intersecting-submodular Functions
3. Intersection Theorem and Its Equivalents
   1. Intersection Theorem
   2. Discrete Separation Theorem
   3. Fenchel Duality Theorem
   4. Minkowski Sum Theorem
4. Minimum-Norm Base and Submodular Function Minimization
   1. Lexicographically Optimal Base
   2. Minimum-Norm Base
   3. Submodular Function Minimization
5. Maximum Weight Base Problem
   1. Greedy Algorithm
   2. Lovasz extension
   3. Subgradients and Subdifferentials of Submodular Functions
6. Discrete Convex Analysis
   1. L♮-convex Functions
   2. M♮-convex Functions
   3. Discrete Separation Theorem
   4. Fenchel Duality Theorem