First talk: 16-17
Noam Zimhoni: Some trees and some forests of some triangles.
Abstract: In 1934 B. Berggren first discovered the surprising result that every Pythagorean triangle is the result of the product of the vector $(3, 4, 5)$ given as a column vector by word in three matrices, and that every triangle is obtained in this manner exactly once and in primitive form. We will discuss a proof of this theorem given by Katayama S. and some generalization of it including original work.

Second talk: 17-18 
Michael Chapman: Almost Solutions
Abstract: When does an approximate solution to a problem close to an actual solution to the original problem? Which problems, if they have an instance that is almost solvable, then there is a complete solution for this instance? In this talk I will introduce a rigorous study of these kinds of questions, both from a theoretical computer science viewpoint (Property Testing) as well as from a group theoretical perspective (Group Stability). 
This talk will be great for three reasons:
1) Everything will be elementary. A 2nd year math student would be able to follow all the details easily.
2) The techniques used in these fields are varied and cool.
3) These elementary concepts are used to prove extremely deep results in pure mathematics, theoretical computer science and the mathematical modeling of quantum mechanics.