Titles and Abstracts (Spring 2019)

Just a reminder that we meet in RLM 5.122 every Tuesday 5-6 pm and Friday 4-5 pm.

 

April

Apr. 26 – Paul Tee

Title: One-Line Proof of the Fundamental Group of the Circle (MUST)

Abstract: It is a well-known fact that the fundamental group of the circle is isomorphic to \mathbb{Z}, the infinite cyclic group consisting of the integers. I will present a one-line proof of this result using loop-suspension adjunction. My talk will roughly follow this blog post: math3ma fundamental group of circle.

 

Apr. 23

Title: Math Club Heart-to-Heart

Abstract: For those of you wondering about the title, this is NOT intended to be a normal meeting. Our focus is going to be on addressing desires, issues, and potential solutions regarding Math Club and the surrounding community. If you have not been going to/never have gone to Math Club for reasons other than being busy then please, please, please consider coming and sharing your thoughts so that together we can build a better Math Club for the future. To everyone who cares, please bring your comments and concerns. I know it is a busy time of year and time might be tight, but please try to make it since this is important.

Update: To everyone who came today, thank you so much for sharing your feedback. For those of you who couldn’t make it but still want to share your thoughts, here is a link to an anonymous feedback form: https://docs.google.com/forms/d/e/1FAIpQLSf5B7j2V3ZbzY7nGG1S2uCX7pDjt1gNexeXt2MQad7_nds9kA/viewform?usp=sf_link. Math Club should be a welcoming and accessible place for undergrads to share their passion for math. Achieving that goal requires help from the whole community, so your feedback is very much appreciated.

 

Apr. 19 – Sam Mokhtar

Title: Representation Theory of \mathfrak{sl}_3(\mathbb{C}) and Beyond (MUST)

Abstract: This is the fourth and final in a series of lectures aimed at introducing the finite dimensional representation theory of Lie groups and Lie algebras. This talk will be focused on the representation theory of \mathfrak{sl}_3(\mathbb{C}) and further topics if there is time.

Complete Lecture Notes: Mokhtar and Green – A Mini Course in Representation Theory

 

Apr. 16

Title: Fall 2019 Course Schedule Exploration

Come join us to explore the fall 2019 course schedule and unpack the math course offerings. Even if you are graduating we would appreciate it if you show up and help field questions from younger students.

 

Apr. 12 – David Green

Title: Representation Theory of \mathfrak{sl}_2(\mathbb{C}) (MUST)

Abstract: This is the third in a series of four lectures aimed at introducing the finite dimensional representation theory of Lie groups and Lie algebras. This talk will be focused on the representation theory of \mathfrak{sl}_2(\mathbb{C}), the Lie algebra of the Lie group \mathrm{SL}_2(\mathbb{C}).

 

Apr. 5 – Sam Mokhtar

Title: Representations and the Baker-Campbell-Hausdorff Formula (MUST)

Abstract: This is the second in a series of four lectures aimed at introducing the finite dimensional representation theory of Lie groups and Lie algebras. This talk will be focused on representations and applications of the exponential map.

Sam – Representation Theory Lecture 2

 

Apr. 2 – Zachary Gardner

Title: What is reciprocity?

Abstract: Reciprocity is a phenomenon that lies at the heart of algebraic number theory and has motivated many interesting developments both inside and outside of number theory. We will start with an informal discussion of quadratic reciprocity, move onto generalizations, and motivate class field theory if there is time.

March

Mar. 29 – Sam Mokhtar

Title: Lie Groups and Lie Algebras (MUST)

Abstract: This is the first in a series of four lectures aimed at introducing the finite dimensional representation theory of Lie groups and Lie algebras. This talk will be focused on definitions and examples.

Sam – Representation Theory Lecture 1

 

Mar. 26 – Tom Gannon

Title: Representation Theory and Whittaker Vectors

Abstract: In this talk, we’ll discuss some of the basics of representation theory, a subject which studies abstract groups by embedding them into matrix groups. We’ll talk about how this naturally leads to representation theory of Lie algebras, and talk about the basics of this subject. We’ll talk about a generalization of finite dimensional representations of Lie algebras known as Whittaker vectors and discuss how these Whittaker vectors play into the notion of groups acting on categories.

 

Mar. 15 – Sebastian Lozano-Munoz

Title: Stats: Part II (MUST)

Abstract: We will discuss a more rigorous treatment of the mathematical foundations underlying the previous talk.

 

Mar. 12 – Tam Cheetham-West

Title: Profinite groups and how to “find” them

Abstract: Profinite groups naturally arise as colimits of systems of finite groups (each endowed with the discrete topology). Profinite completion, a process that produces a profinite group from a given group, is a useful tool in 3-manifold topology and more generally for the computation of coefficients in group cohomology. This talk will be a measured introduction to concepts and results. Knowledge of group theory and some point set topology is helpful but not required.

 

Mar. 8 – Sebastian Lozano-Munoz

Title: Torturing the data: Stats, what is it good for? (MUST)

Abstract: The majority of our interactions with statistics come through its use in the public sphere, but mathematically stats is a rich field of study. In this talk I’ll give an overview of what statistics is, the kinds of stats we might engage in, and then delve into some examples. I’ll be presenting three examples first from an applied perspective and a then from a theoretical perspective. I will present examples of time series, neural networks, and generalized linear regression. Our discussion will be centered around the probabilistic underpinnings of stats and will only require calculus and linear algebra.

 

Mar. 5

Title: Mathematica 11 in Education and Research

Abstract: This technical talk will show live calculations in Mathematica 11 and other Wolfram technologies relevant to courses and research. Specific topics include:
* Enter calculations in everyday English, or using the flexible Wolfram Language
* Visualize data, functions, surfaces, and more in 2D or 3D
* Store and share documents locally or in the Wolfram Cloud
* Use the Predictive Interface to get suggestions for the next useful calculation or function options
* Access trillions of bits of on-demand data
* Use semantic import to enrich your data using Wolfram curated data
* Easily turn static examples into mouse-driven, dynamic applications
* Access 10,000 free course-ready applications
* Utilize the Wolfram Language’s wide scope of built-in functions, or create your own
* Get deep support for specialized areas including machine learning, time series, image processing, parallelization, and control systems, with no add-ons required
Current users will benefit from seeing the many improvements and new features of Mathematica 11 (https://www.wolfram.com/mathematica/new-in-11/), but prior knowledge of Mathematica is not required.

 

Mar. 1 – Hunter Stufflebeam

Title: Tensor Field Soup for the Soul (MUST)

Abstract: In this talk we’ll take a brisk informal tour of Riemannian geometry, with a special emphasis on the geometry of submanifolds. We’ll first meet the basic tools which allow us to view differentiable manifolds as objects endowed with notions of distance, angle, and curvature, and we’ll explore some of the neat ways to study such quantities. Then, we’ll touch on some core results, such as Gauss’ famous Theorem Egregium. If time permits, at the end I’ll throw all the strong regularity conditions away and talk about a generalized notion of surfaces, through the language of rectifiable varifolds.

February

Feb. 26 – Leon Liu

Title: Introduction to Morse Theory via Pictures

Abstract: Morse theory (unrelated to Morse codes!) is a way to cut a geometric object (manifolds) into pieces. Using the torus as an example, I am going to describe how to glue the torus together using only a real-valued function and four special points on the torus called critical points. We are also going to talk about the concept of flow lines and gradient descent. This talk is going to focus heavily on geometry and motivation. We will draw lots of pictures!

 

Feb. 22 – Jeffrey Jiang

Title: Differential Geometry Minicourse Lecture 4 (MUST)

This is the fourth in a series of four talks on the basics of differential geometry.

Jeffrey – Diff Geo Lecture 4 Notes (Handwritten)

 

Feb. 19 – Zachary Gardner

Title: Math Fundamentals: Compactness

Abstract: In order to acclimate newer students to the world of pure math, we will from time to time give talks on math fundamentals. This week, the focus is on compactness – a notion that arises in algebra, topology, geometry, analysis, and elsewhere.

Math Fundamentals – Compactness

A Pedagogical History of Compactness

 

Feb. 15 – Jeffrey Jiang

Title: Differential Geometry Minicourse Lecture 3 (MUST)

This is the third in a series of four talks on the basics of differential geometry. This talk covers roughly chapters 11-15 of Lee.

Jeffrey – Diff Geo Lecture 3 Notes (TeX)

Jeffrey – Diff Geo Lecture 3 Notes (Handwritten)

 

Feb. 12 – Max Stolarski

Title: Math Interview Puzzles

Abstract: I’ll go over some math puzzles that I’ve encountered myself or that my friends have related to me from various job interviews over the years. The techniques involved won’t require anything other than some elementary calculus, probability, linear algebra, and topology. Instead, the emphasis will be on thinking creatively and approaching the problems in a clever way.

Max Stolarski – Math Interview Puzzles

 

Feb. 8 – Jeffrey Jiang

Title: Differential Geometry Minicourse Lecture 2 (MUST)

This is the second in a series of four talks on the basics of differential geometry. This talk covers roughly chapters 8-9 of Lee.

Jeffrey – Diff Geo Lecture 2 Notes (TeX)

Jeffrey – Diff Geo Lecture 2 Notes (Handwritten)

 

Feb. 5 – Leon Liu

Title: The bar construction and its applications

Abstract: The bar construction provides a canonical way to extract a simplicial object from a module object of an algebra. We will go through a couple of simple examples before delving into the full generality of the construction. We will also examine the particular example of BG (and EG) for G a (discrete) topological group.

 

Feb. 1 – Jeffrey Jiang

Title: Differential Geometry Minicourse Lecture 1 (MUST)

This is the first in a series of four talks on the basics of differential geometry. This course will be following John Lee’s Introduction to Smooth Manifolds. This talk covers roughly chapters 1-3 of Lee. See the notes below for more information.

Jeffrey – Diff Geo Preliminaries

Jeffrey – Diff Geo Lecture Notes (TeX)

Jeffrey – Diff Geo Lecture Notes (Handwritten)