Notes and Presentations

Talk Title: The Pre-Image Theorem and Consequences

Abstract: The Pre-Image theorem is a result from differential topology that allows one to construct manifolds easily. It also has some cool partial converses. Come check it out!

Talk Title: The Fundamental Group of the Circle and Applications

Abstract: The fundamental group of a space describes the possible loops you can make in it. The fundamental group of the circle is a very interesting group that has many applications towards proving nontrivial theorems, such as the Fundamental Theorem of Algebra.

Talk Title: Lattices, Fundamental Domains, and Minkowski Theory 

Abstract: Lattices, fundamental domains, and Minkowski Theory for the basis towards understanding results in number theory using geometric methods. Here they are used to prove the Four-Square and Two-Square Theorems.

Talk Title: Lebesgue Spaces and Important Inequalities

Abstract: Lebesgue Spaces form a very important class of function spaces that are also metric spaces. We will introduce Lebesgue spaces, then describe some important inequalities used in analysis.

Talk Title: Perceptrons: the Precursor to Artificial Neural Networks

Abstract: Modern neural networks have many parts. However, they all contain a fundamental unit called a perceptron. The perceptron by itself simulates a human neuron, and is key towards simulating learning.

Talk Title: Transversality is a Stable and Generic Property

Abstract: The intersection of two manifolds in Euclidean space can be very desirable when they intersect transversally. In this presentation we will define and describe transversality, as well as show why any intersection between manifolds can be made transveral easily.