MathGEMs Archive

Abstract: We shall introduce the basic concepts of Einstein's theory of general relativity through discussions of Riemannian and Lorentzian geometry. Topics will include Riemannian manifolds, tensor fields, geodesics, curvature, and the Einstein field equations. Time permitting, we shall discuss the Schwarzschild metric and Schwarzschild black holes.

I aim for the talk to be accessible to anyone with a background in multivariable calculus (MATH 233), though knowledge of linear algebra and some familiarity with ordinary differential equations will be helpful.

Abstract: The Goldbach Conjecture is one of the oldest unsolved problems in all of mathematics. Its fame comes from its simplicity: all even integers greater than 2 can be written as the sum of two primes. This has been shown for even integers up to 4 QUINTILLION, yet no one has proven it.

We will go over the history of the Goldbach Conjecture, including its origin, attempts at solving it, and the general theory behind its potential proof. In addition, we will go over some computational intricacies, such as how to actually verify the conjecture for large even integers.

Abstract: While the real numbers are great, they can't do it all. Where the real numbers fail, the complex numbers save us! By doubling the dimension, we gain more structure and numbers to work with. In classic mathematician fashion, we must ask what happens if we do it again? Do we get 3-dimensional complex numbers? Or 4? And why stop there! We will discuss how to "climb the ladder" of the sets of complex numbers, constructing higher dimensional algebras. We will see the limits of this process and some properties of these new algebras.

Abstract: Factoring numbers is known to be a hard problem. By merely playing a literal game with tennis balls, randomized in much the same was as any board game, we will be able to collectively factor a certain family of numbers. No prerequisite knowledge of number theory or algebra will be necessary, but we will cover the basics of modular arithmetic and divisibility. Hand-Eye coordination is also not necessary, but will help.

Associated paper: https://arxiv.org/abs/2110.13313