Mathematician Paul Erdös talked a lot about "The Book," where God supposedly kept perfect proofs to all mathematical theorems. Us mathematicians can't claim to have seen this book, but we do generally agree on some proofs that would definitely be there. A popular choice is Steve Fisk's solution to "The Art Gallery Problem." Basically, given a polygonal art gallery, we want to put guards at the vertices so they can see the entire interior. How many guards do we need? The answer and reasoning behind it are just about the coolest things I've seen, and as an MIT math major, that's saying something. If there's time, I'll also talk about generalizations, like art galleries in space.

Mathematician Paul Erdös talked a lot about "The Book," where God supposedly kept perfect proofs to all mathematical theorems. Us mathematicians can't claim to have seen this book, but we do generally agree on some proofs that would definitely be there. A popular choice is Steve Fisk's solution to "The Art Gallery Problem." Basically, given a (not necessarily convex) polygonal art gallery, we want to place guards at the vertices so they can see the entire interior. How many guards do we need? If there's time, I'll also discuss the Fortress Problem, in which we want to see the outside of a polygon, and the Prison Yard Problem, where we want to see both, as well as talking about some trickier cases like art galleries in space.

How big does a set in the plane need to be to allow a needle of length $$1$$ to be rotated completely around inside it? (You can slide the needle at any time.) As surprising math results go, this one ranks pretty high up there. I mean, a circle of diameter $$1$$ will obviously give an area of $$\frac{\pi}{4}$$, and if you're very clever you might even find a relatively simple shape that works with area $$\frac{\pi}{8}$$. That was a good as Kakeya himself could do. So what if I told you that the answer was $$0$$?! There is no minimum size for such a set! Come to this class to see the amazing proof of this fact. If there's time at the end we'll talk about related problems that are still unsolved.

Last year, I taught a class called "Formulas you were Never Meant to Know," in which we showed the messy formulas for things that most people thought didn't have formulas. At the end, I stated that the formula for the number of ways to tile an $$m$$ by $$n$$ rectangle with dominoes was $$\prod_{i=1}^m\prod_{j=1}^n\sqrt[4]{4\cos^2\left(\frac{i\pi}{m+1}\right)+4\cos^2\left(\frac{j\pi}{n+1}\right)}$$. I didn't prove it, however, claiming that even I wasn't that crazy. Well this Splash, I'm going that crazy. Come learn the insane proof of this insane statement, and discover how the &#$% it is that cosines and fourth roots found their way into a counting formula.

Geometry is sort of in a weird place these days. Look at a geometry paper today, and it will probably involve scary-sounding stuff like algebraic varieties, smooth manifolds, and Ricci flow. But what about lines, planes, and circles? What about all the stuff we learned in high school? Ask most contemporary mathematicians, and they will tell you that Euclidean "high school" geometry is a dead field, and all its problems can be easily solved by some fancy computer programs. Well I'm here to tell you that those mathematicians are all silly! Discrete (or Combinatorial) Geometry is the study of how geometric objects can exist and interact with each other given various constraints. For example, how many smaller copies of a given three-dimensional shape are needed to cover the larger version? What is the largest-area shape that can be continuously moved though an L-shaped bend with unit width? Ask these questions to any of those Euclidean geometry bashers I mentioned above, and they (and their fancy computer programs) will struggle, because in fact BOTH OF THESE PROBLEMS ARE UNSOLVED! Needless to say, we won't solve them either, but we will learn some other pretty cool stuff. In particular, we will see: -Whether guarding a prison is harder than guarding a fortress -How big of a net you need to catch fish in hyperspace -How best to pack infinitely many people into an infinitely big party -How a world-famous mathematician got shown up by one of his students -How a two-time Nobel laureate got shown up by a little-known Israeli materials scientist on sabbatical (who would then go on to win a Nobel prize himself) -That higher dimensions generally screw everything up, except in the rare cases when they don't If any of that sounds fun, come join me for a whirlwind tour of one of math's most exciting fields!

People tend to have a love-hate relationship with snow. On one hand, we love to throw it, build fortresses with it, and have school cancelled by it. On the other hand, we can definitely have too much; this past winter was a nightmare for anyone having to shovel! But us scientists have a love-love relationship with snow! Scientists love to ask "Why?" and the subject of snow immediately gives us plenty of why's. –"Why is snow white?" –"Why do snowflakes sometimes sting your face when they hit you?" –"Why does the artificial snow on ski slopes not quite feel the same?" –"Why so snowflakes always form such perfectly symmetrical hexagonal crystals?" –"For that matter, do snowflakes always form such perfectly symmetrical hexagonal crystals?" In addition, we'll see tons of beautiful photographs of six sided snowflakes, three sided snowflakes, and twelve sided snowflakes (but NO four sided ones!), as well as oddities with names like "arrowhead crystals," "chandelier crystals," "capped columns," and "stellar dendrites," and will learn learn precisely how each of them forms. Snow is really cool! (Pun intended)

Mathematician Paul Erdös talked a lot about "The Book," where God supposedly kept perfect proofs to all mathematical theorems. Us mathematicians can't claim to have seen this book, but we do generally agree on some proofs that would definitely be there. A popular choice is Steve Fisk's solution to "The Art Gallery Problem." Basically, given a (not necessarily convex) polygonal art gallery, we want to place guards at the vertices so they can see the entire interior. How many guards do we need? If there's time, I'll also discuss the Fortress Problem, in which we want to see the outside of a polygon, and the Prison Yard Problem, where we want to see both, as well as talking about some trickier cases like art galleries in space.

Ever wondered what the Taylor series for tangent was? The general formula for $$\sin(nx)$$? The sum of the first $$n$$ $$k$$th powers, for arbitrary $$k$$? If you ask your teacher any of these, he or she will probably say they're really, really ugly. So will I. And then I'll show them to you anyway. Come to this class to learn all the fantastically nasty general formulas that have been kept from your tender eyes.