1. Does tropical geometry look like…

2. No, it looks like …

3. “Life Math is pain, anyone who says differently is selling something” Goal: explain and prove some basic notions. Key words: complex numbers, algebraic curves, genus Basic example: exactly one conic passes through 5 points

4. Digression: polynomials Example: – roots Example: – roots

5. Digression: complex numbers

6. Theorem: any polynomial of degree n has exactly n complex roots

7. Algebraic curves Polynomials in two variables Degree

8. Algebraic curves Definition: real algebraic curve of degree n is the set of points Complex algebraic curve: the same but in x y y-x 2 =0

9. Examples: lines Line Through any two different points only one line passes – Fix one point – Write equation on a,b,c – The same for the second point Find the unique solution

10. Examples: conics Conic There is only one conic passing through 5 points. – Because is a linear equation.

11. Real algebraic curves 3 x=3 F(x,y)=0 X Y

12. Real algebraic curves

13. Complex algebraic curves

14. Complex algebraic curves = surfaces

15. The Earth = torus ?

16. Counting of algebraic curves How many lines pass through 1 point ? How many lines pass through 3 points? Theorem: Through 3d-1+g generic points the finite number of curves degree d and genus g pass.

17. How many curves? The simplest way to count – tropical geometry. In the tropical world things are simpler than in the complex world.

18. Hope you find your tropical world. Thank you!