The Winter 2016 SASMS will be held on Tuesday, March 22 at 5:00 in MC 5479. If you would like to give a talk, you may do so by clicking "Sign Up" above.
We've all learned how to use a compass and straightedge to draw stuff for those awful geometry problems we had to do for Euclid, COMC, etc in high school. You might also have wondered why the Ancient Greeks (and others) spent hundreds of years failing to solve easy problems like doubling the cube. Now that we're in undergrad and have some Galois theory at our disposal, we can finally prove some of the limitations of this "purist" compass and straightedge construction method.
First-order logic, in some sense, the "strongest" logic system for which the Compactness Theorem and Downwards Löwenheim-Skolem Theorem both hold. We will formalize this notion of "strength" and give a sketch of Lindström's proof of this result.
We constructively prove the existence of pizza and pop. Time permitting, we will provide examples.
As promised last term, I will prove that, when immersed in water, the general quintic equation does not dissolve into a water-radical solution. Time permitting, we'll also do some Galois theory.
Not a disproof of Zorn's lemma this time, sorry :P The Amitsur-Levitski theorem states that the ring of $n \times n$ matrices over any commutative ring $R$ satisfies a polynomial identity of degree $2n$. The theorem has received several proofs since the original one. However, they are either too bashy, or involving scary phrases like "non-trivial results in Lie algebras cohomology and group representations". This time I will present a proof by Rosset that is short and elementary.
Given a prophet that knows everything, inequality arises when the prophet takes everything and you get nothing. Disclaimer: No prophets are harmed in this talk. Recommended background for keeners: Be a prophet.
If you create two graphs by each time taking countably vertices and connecting two vertices with a probability of 1/2, what is the likelihood of those two graphs being isomorphic? The answer will (hopefully) surprise you!
What does the existence of first player winning strategies in Hex have to do with the Federal Election? As it turns out, quite a lot. We'll be looking at a fairly recent result that relates Arrow's Impossibility Theorem to the Hex Game Theorem.
For more than a year, ominous rumors had been privately circulating among high-level Western leaders that the Soviet Union had been at work on what was darkly hinted to be the ultimate weapon: Frobenius' Theorem...
We'll start at the beginning with classical mechanics, and sample the mathematical and physical ideas that led from there to quantum mechanics, quantum field theory, string theory, and beyond. Where does the uncertainty principle come from? What are Feynman diagrams? Why is gravity so hard to unify with other forces? We'll answer these questions, and possibly more! No background in physics assumed; mathematical requisites start at M136/138 but may slowly grow in an unbounded fashion.