Department of Mathematics

Special Mathematics Classes: Capstones

MTH 396 Prerequisites:

MTH 396 Prerequisites: Completion of Tier I Writing Requirement, MTH 309, MTH 310, and MTH 320 (or the honors equivalents, or approval of department) and approval of the department. Typically the department expects a cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Note: Email notification will be given once your override has been issued.

MTH 496 Prerequisites:

Completion of Tier I Writing Requirement and approval of the department. Typically the department expects students to have completed MTH 309, MTH 310, and MTH 320 (or the honors equivalents) with cumulative GPA of at least 2.0 and an average of at least 2.0 across MTH 309, MTH 310 and MTH 320. Additional prerequisite courses may be required and can be found in the descriptions below. Note: Email notification will be given once your override has been issued.

FS22 MTH 496 Section 1: Machine Learning

Instructor: Guowei Wei

Machine learning (ML), including deep learning (DL), has had tremendous success in science, technology, industry, social media, etc. in the past decade. It is regarded as one of the most transformative technologies in human history. ML is the core technology in data science and is of fundamental importance for data-driven discovery. In this course, we will not only discuss the theoretical framework of ML algorithms and architectures but also put an emphasis on programming skills so that each student is able to implement ML algorithms for real-world problems. The course will start with linear regression (LR), logistic regression (LR), gradient descent, regularization, k-nearest neighbors (KNN), k-means, dimensionality reduction (i.e., principal component analysis (PCA) and uniform manifold approximation and reduction (UMAP)), support vector machine (SVM), kernel learning (KL), and decision trees (DT), including random forest (RF) and gradient boosting trees (GBT). If time permits, we will discuss some basic DL materials, such as artificial neural networks (ANN) and convolutional neural networks (CNN).

Prerequisites: (MTH 309 or MTH 314) & (CSE 231 or CMSE 202)

Text: There is no required textbook for this course A full set of lecture notes and tutorial materials will be provided.

FS22 MTH 496 Section 2: Intro. to Functional Analysis

Instructor: Brent Nelson

We will begin by studying the so-called lp spaces, which consist of sequences whose pth powers yield absolutely convergent series. For p>=1, it turns out that these are examples of Banach spaces, which are a fundamental objects in functional analysis. We will conduct a thorough study of Banach spaces and the continuous functions on them (called bounded operators), including proving the open mapping theorem, the inverse mapping theorem, the closed graph theorem, the principle of uniform boundedness, and the Hahn–Banach theorem. As a special case, we will also consider Hilbert spaces (which corresponds to p=2 for the lp spaces) and applications to Fourier analysis.

Prerequisites: (MTH 309 or MTH 317H) and (MTH 320 or MTH 327H). Some exposure to metric spaces in the form of MTH 421 or MTH 461 is recommended, but not necessary.

SS23 MTH 496 Section 1: Fourier and Time Frequency Analysis

Instructor: Mark Iwen

This course will discuss Fourier series, Orthogonal Functions, Inner Product Spaces, Hilbert Spaces Convergence issues, The Fourier Transform, Distributions, and introductory time-frequency analysis. Applications in computer science and engineering of the powerful ideas presented in this class will also be covered to motivate the mathematical material.

Prerequisites: MTH 320 or MTH 327H.

SS23 MTH 496 Section 2: Intro to Representation Theory of Groups

Instructor: Igor Rapinchuk

This course will provide an introduction to the representation theory of groups, a beautiful and extremely useful mathematical subject that demonstrates remarkable interactions between linear algebra and abstract algebra. In addition to being one of the core subjects of modern algebra, representation theory has numerous deep connections to various other areas, including algebraic geometry and topology, combinatorics, number theory, and some branches of analysis. Moreover, it is used extensively in such sciences as physics, quantum chemistry, and statistics, to name a few. So, some exposure to representation theory is extremely beneficial, particularly for those students who intend to pursue graduate studies either in mathematics or in sciences that rely heavily on mathematical techniques. Some specific topics that we will cover include various aspects of the theory of bilinear forms and inner products spaces, a quick review of the basic definitions and constructions of group theory, and some fundamental results in the representation theory of finite groups (including the complete reducibility of complex representations and character theory).

Prerequisites: MTH 309 & MTH 310 & MTH 411(or Concurrently)