February 5: Hayden Jananthan
Title: What’s Yellow and Equivalent to the Axiom of Choice?
Abstract: In the early and mid 20th century, the foundations of mathematics were a hot topic that plagued many of the finest mathematicians of the time. One particular issue was with the use of an axiom known as the “Axiom of Choice”. Today, we accept the Axiom of Choice without too much fuss, but we still often acknowledge its use with extra care than is taken with other axioms. Why was this axiom so contentious? What eventually won mathematics over to its favor? What’s Yellow and Equivalent to the Axiom of Choice? In our talk, we will explore the answers to these questions and related topics.
February 12: Professor Jesse Peterson
Title: Banach-Tarski Banach-Tarski
Abstract: Nearly a century ago, Stefan Banach and Alfred Tarski discovered a result which had been sought by alchemists for centuries prior. It is possible to take the unit ball, break it into finitely many pieces, and then translate and rotate the pieces so that their union give two disjoint unit balls. This paradox, seeming to create a ball ex nihilo, has had a lasting impact on mathematics, and the techniques involved continues to be expanded in modern research. In this talk I will discuss the proof, and also some of the implications, of the Banach-Tarski paradox.
February 19: Srivatsav Kunnawalkam Elayavalli
Title: Sharks in the Shallows
Abstract: On a fine sunny day, little Sri decided to go fishing because he really enjoyed it and was passionate about it and wanted to become a great fisher. Since he wasn’t an expert nor an experienced fisher, he decided to go to the local shallow waters thinking that he’ll find only small fish, and it’ll be a tractable job for him to catch them. Everything was going well until he discovered that several unconquerable beasts and sharks have been hiding there for centuries tricking several young and amateur fishers into catching them. Sometimes, even great expert fishermen go back to these shallows to try and catch them, and indeed a very few of them have succeeded and received immense glory and fame… Do you want to know more about the mystery of these legendary beasts??? Come to my talk.
February 26: Jon Ashbrock
Title: Computation v. Simulation or: How I learned to Stop Worrying and Love the RNG
Abstract: Computation of probabilities and averages is one of the most common tasks asked of applied mathematicians in industry. While most problems of this type are tractable via pen-and-paper calculation, I claim this is not the best approach. With the use of your favorite RNG and a little bit of coding knowledge, you can estimate the probability in a much faster method. Tonight, we’ll demonstrate this technique in two ways: showing you how to have the most fun playing roulette and by providing an introduction to Monte-Carlo integration.
March 12: Zack Tripp
Title: Fibonacci Numbers, Combinatorics, and Partitions
Abstract: Many people are aware of the Fibonacci numbers and may have heard about their omnipresence in nature and different areas of mathematics. However, not many people know that there is a simple (non-recursive) formula for them! In this talk, we will introduce one of the most essential ideas from combinatorics (that may look familiar to ideas from MATH1301 for those who have taken/are taking it) in order to find this formula. Using these ideas, we will also discuss how to compute something called the partition function, along with some of the most famous results by the great mathematician Ramanujan that will shed light on the nature of this function.
March 19: Professor Adam Prenosil
Title: Go forth and multiply in O(n^1.465)
Abstract: Multiplying two integers is surely one of the simplest operations in all of mathematics. It is literally so simple a child can do it: we all learn our multiplication tables in grade school, and soon thereafter we learn how to multiply large numbers using long multiplication. The idea that this obvious way of multiplying integers is the best possible way barely seems worth stating explicitly because, after all, how else could one possibly multiply two integers? It therefore came as a surprise even to eminent mathematicians when it was discovered in the 1960’s that we can in fact do substantially better than long multiplication. In this talk, I will present some of the fast algorithms for multiplying large integers that have been devised since then.
March 26: Levi Sledd
Title: Hyperbole and a Half-Plane
Abstract: For thousands of years, Euclid’s unwieldy fifth axiom, known as the parallel postulate, annoyed the mathematicians of the world. Countless tried, and failed, to prove the fifth axiom from the other four. Then in the nineteenth century, geometers János Bolyai and Nikolai Lobachevsky showed that, in some settings, it’s not true. This undergraduate seminar, we’ll talk about geometries in which the parallel postulate fails, with a special emphasis on the exceedingly strange and beautiful world of hyperbolic geometry. We’ll meet the hyperbolic disk and upper half-plane, and see how they—and your old friend from calculus class, the hyperboloid—are somehow one and the same. Come this Tuesday to witness the best undergraduate seminar talk ever! Well, maybe that’s just a bit of hyperbole.
April 2: Michael Montgomery
Title: Two Sides of the Same Coin
Abstract: Starting with two wildly different ideas only to learn that they describe the same thing is a common occurrence in mathematics. We can see a simple example from calculus where the anti-derivative and area under a curve are two representations of integration. This talk will explore another deep example of this phenomenon tying together a special type of algebra and topological spaces.