I only attended the major talks; the first one was by Richard Kenyon. The material, while interesting, flew by a little quickly (though it wouldn’t have for someone who researches full time). The main idea: piecewise approximation to smooth objects is extremely useful, not only topologically but also geometrically (example)
Something especially interesting to me: when trying to approximate certain smooth surfaces, the starting approximation doesn’t matter that much; there are many different piecewise linear sequences that converge to the same surface (not a surprise). There is much more there; this is a lecture I’d like to see again (if it gets posted).
The next one was the third Bernd Sturmfels; this was a continuation of his “algebraic geometry’s usefulness in optimization” series. One big idea: we know how to optimize a linear function on a polygon (e. g., simplex method). It turns out that we can sometimes speed up the process by the “central curve” method; the idea is to use algebraic geometry to do an optimization problem on the constraint plus a term involving logs: form where is the cost function. There is much more there.
The last talk was by an Ivy League professor; it was called “putting topology to work”. On one hand, it was great in the sense that there were many interesting applications. He then asked a sensible question: “how do we teach the essentials of this topology to engineers”?
His solution: revise the undergraduate curriculum so that…well…undergraduates had algebraic topology (or at least homological algebra) in their…linear algebra course. 🙂 It must be nice to teach Ivy league caliber undergraduates. 🙂
The elephant in the room: NO ONE seemed to ask the question: “do the students in our classrooms have the ability to learn this stuff to begin with?”
Do you really think that a class full of students with ACTs in the 22-26 range will be able to EVER handle the advanced stuff, no matter how well it is taught?