The following beautiful result in the area of combinatorial geometry was established at the end of the 19th century: Given $$n$$ non-colinear points in the plane. A line that contains exactly two of the set of points is called an "ordinary line". The Sylvester-Gallai's Theorem claims that there exists at least one ordinary line. We will give several proofs to this theorem and investigate some more advanced cases: Can we develop a fast algorithm (deterministic or randomized) to find an "ordinary line"? Can we give a better lower bound for the number of ordinary lines? (At least $$3$$? At least $$\sqrt{n}$$? At least $$c n$$ for some constant $$c$$?) What if we replace the "lines" by "formal lines"? Namely, we may only use the axioms such as "there exists a unique formal line passing through two points" and "there exists at most one point passing through two formal lines"? What if we replace the "lines" by "pseudolines"? Namely now we may use those two axioms and that "pseudolines" are "actual" smooth curves and cross each other "properly"? In this case we may give an "order structure" for the points on a "pseudoline". Will things be different?

This class will cover some basic and advanced technology in algebraic inequalities. In this course, we will learn many cool inequalities and some new (in recent 10 years) elementary methods, then use them to prove and generalize interesting problems from Math Olympiads and beyond. Math Olympiad problems are expected to be solved in two hours without a calculator or a computer, but this is not the case for some other inequality problems. In this course, we will also learn (briefly) how to use MAPLE -- a symbolic computation software like MATHEMATICA (and better than that, just personally speaking). Some complicated calculation details will be verified by MAPLE. Some inequalities come from discretization of theorems from higher math. But don't get scared --we can solve them (or understand their solutions). Level of difficulty may vary based on student feedback.