We are excited to announce this semester’s math club reading groups! Reading groups are like weekly math book -clubs where undergrads can learn topics which may not be covered in their studies. Come check them out on the Reading Groups Discord server — this is where all reading groups communication will be sent from and where you will choose which groups to join.

 

Tensor Analysis and Differential Geometry

Michael Updike

Tensor analysis/differential geometry is a ubiquitous and powerful subject. Differential geometry shows up everywhere in both pure math and applied settings like physics, engineering, and computer science. For example, differential geometry can be used to classify manifolds, process data, and calculate the motion of black holes. In this course, I hope to appease both the applied and pure math audiences. We will focus on gaining an intuitive feel for geometry through many examples and derivations. That said, everything we do will be rigorous, with proofs provided for most statements. I hope to cover the basics of tensor analysis/differential geometry. We will start by going over tensors algebraically focusing on index calculus and getting a feel for tensor manipulation. We will then introduce the idea of tensors and “living” on manifolds and derive many key transformation laws. We will then use our new knowledge of tensor analysis to study differential geometry principally the idea of curvature. Depending on interest and time, we may also touch on subjects such as differential forms, lie theory, or general relativity. For this group, we won’t be using any textbook. Rather, I was planning to write my own notes loosely based on some combination of Jeevanjee’s An Introduction to Tensors and Group Theory, Nakahara’s Geometry, Topology, and Physics, and Lee’s “Smooth Manifolds.” Meetings are likely to consist of a short lecture (>30 min), followed by a “town-hall” where I’ll try to answer questions and guide people through the solutions to the short weekly problem set.

 

Hyperbolic Surfaces

John Teague

Non-Euclidean geometry is something of a lost art in mathematics. In the 19th and early 20th centuries, every serious student of mathematics and physics studied these geometries, but this has not been true of the modern generation of students. They have profound applications to the study of complex variables, to the topology of two- and three-dimensional manifolds, to the study of finitely presented infinite groups, to physics, and to other disparate fields of mathematics. In this course, I'm interested in developing the basic theory of the negatively curved geometry called hyperbolic geometry, as well as exploring some of its more modern applications in the study of low-dimensional topology (namely n = 2). The first half to two-thirds of the course will be studying the hyperbolic plane in its various incarnations and models, and the latter half will be dedicated to the study of 2-dimensional manifolds (surfaces) whose geometry reflect that of H^2. Ideally, after this reading group, members will have an excellent background for many of the applications in mathematics and physics of hyperbolic geometry, as well as a potential avenue for future study. I plan on using the Cannon et al. paper on arXiv titled "Hyperbolic Geometry" and the later chapters of Stillwell's book Geometry of Surfaces. I will be writing some problem sets to accompany the readings, and I hope to spend the majority of our meeting time discussing these problems. This group assumes you have a background in calculus and that you've had at least some basic interaction with linear algebra. Members would benefit from having worked with topology at the undergraduate level, but nothing is assumed past linear algebra.

 

Programming for Mathematics

Various Speakers

Guide people through using various languages for mathematics and setting up a working coding environment. 

 

Topics include, but are not limited to:

- Git

- Mathematica

- Sage

- Jupyter

- R

- Latex

 

Probability and Statistics

Riya Kattumenu & Marilyn Lionts

Probability and statistics have a variety of applications that range from measures to poker. In this reading group, we will introduce some of these topics, such as: measure theory, gambling probability, game probability, stochastic processes, and machine learning. Meetings will be one hour per week, with lectures and group activities to understand new concepts.