# Capstone Courses

## 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.

## FS23 – MTH 496 section 1 - Harmonic analysis on hyper cube and learning

**Instructor: Alexander Volberg**

Hypercube (also called Hamming cube) is the collection of long strings of +1 and −1. The functions on hypercube are called Boolean functions if they also assume only values ±1. In “big data” one can think that each string is the collection of positive and negative attributes of your “clients”, and the value of your function means whether the client “drops” the service or joins it. Knowing part of your function you wish to predict what will happen with the next client, if you are given client’s attributes. Can you recognize a function if you know it only on a part of its domain of definition? Of course not, but you have more chances if you know something about its spectral concentration. This is what we will try to study for functions on Hamming cube. We describe the basics of analysis of Boolean functions, we also describe some rudiments of the mathematics of learning theory and another theory called social choice, a topic studied by economists, political scientists, mathematicians, and computer scientists. Harmonic analysis on Hamming cube has many unusual and interesting features, and, recently started to play a role as one of mathematical foundation of big data.

Recommended Background – STT 351 or STT 441

## FS23 – MTH 496 section 2 - Nonlinear Dynamics and Chaos

**Instructor: Keith Promislow**

This course introduces the exciting field of dynamical systems with an emphasis on bifurcation and chaos in applied models. Nonlinear differential equations and iterative maps model a beating heart, tumor growth, animal conflict, ecological systems, as well as mechanical, electrical, and economic oscillations. Through linear and nonlinear techniques we will analyze the behavior of models with emphasis on prediction of the onset of qualitative change.

Required Background: MTH 320 & (MTH 235 or MTH 340)

## SS24 – MTH 496 section 1 - Introduction to Differential Geometry (Curves and Surfaces)

**Instructor: Xiaodong Wang**

This course aims to introduce students to the beautiful ideas and results of classical differential geometry. More specifically, we use the tools of multivariable calculus and linear algebra to study curves and surfaces in space. A central theme is the study of different kinds of curvature which are defined locally and their global implications. We start with plane curves and curves in space. The core of the course is about smooth surfaces in space. We will discuss 1st and 2ndfundamental forms, connection, geodesics, curvature, Gauss-Codazzi eqautions, Gauss-Bonnet formula, etc. If time allows, we will talk about minimal surfaces and more global results. There will be lots of hands-on examples as well as proofs.

Required Background: MTH 421 (or concurrently)

## SS24 – MTH 496 section 2 - Introduction to Machine Learning

**Instructor: Ekaterina Rapinchuk**

This course provides a broad introduction to machine learning. Topics include supervised learning ( support vector machines, neural networks, etc.) and unsupervised learning (clustering, dimensionality reduction, etc.). We will also discuss the different tasks of machine learning: classification, regression, clustering, density estimation and dimensionality reduction. During the last part of the course, we will discuss deep learning.

Required Background: CSE 231 or CMSE 201