# Courses

# Course Listing

## Introduction to Applied Mathematics

**APMTH 50**

2024 Spring

**Cengiz Pehlevan**

Monday, Wednesday, Friday

9:00am to 10:15am

This course provides an introduction to the problems and issues of applied mathematics, focusing on areas where mathematical ideas have had a major impact on diverse fields of human inquiry. The course is organized around two-week topics drawn from a variety of fields, and involves reading classic mathematical papers in each topic. The course also provides an introduction to mathematical modeling and programming.

## Supervised Reading and Research

**APMTH 91R**

2024 Spring

**Margo Levine, Sarah Iams**

Supervised reading or research on topics not covered by regular courses. For AM concentrators, work may be supervised by faculty in other departments. For non-concentrators, work must be supervised by an AM faculty member. Students must receive the approval of an (Associate) Director of Undergraduate Studies and obtain their signature before submitting AM91r forms.

## Thesis Research

**APMTH 99R**

2024 Spring

**Margo Levine, Sarah Iams**

Provides an opportunity for students to engage in preparatory research and the writing of a senior thesis. Graded on a SAT/UNS basis as recommended by the thesis supervisor. The thesis is evaluated by the supervisor and by one additional reader.

## Statistical Inference for Scientists and Engineers

**APMTH 101**

2024 Spring

**Robert D. Howe**

Tuesday, Thursday

11:15am to 12:30pm

Introductory statistical methods for students in the applied sciences and engineering. Random variables and probability distributions; the concept of random sampling, including random samples, statistics, and sampling distributions; the Central Limit Theorem; parameter estimation; confidence intervals; hypothesis testing; simple linear regression; and multiple linear regression. Introduction to more advanced techniques as time permits.

## Ordinary and Partial Differential Equations

**APMTH 105**

2024 Spring

**Margo Levine**

Monday, Wednesday, Friday

9:00am to 10:15am

Ordinary differential equations: power series solutions; special functions; eigenfunction expansions. Elementary partial differential equations: separation of variables and series solutions; diffusion, wave and Laplace equations. Brief introduction to nonlinear dynamical systems and to numerical methods.

## Graph Theory and Combinatorics

**APMTH 107**

2024 Spring

**Adam Hesterberg**

Monday, Wednesday

11:15am to 12:30pm

Topics in combinatorial mathematics that find frequent application in computer science, engineering, and general applied mathematics. Course focuses on graph theory on one hand, and enumeration on the other. Specific topics include graph matching and graph coloring, generating functions and recurrence relations, combinatorial algorithms, and discrete probability. Emphasis on problem solving and proofs.

## Nonlinear Dynamical Systems

**APMTH 108**

2024 Spring

**Sarah Iams**

Monday, Wednesday, Friday

1:30pm to 2:45pm

An introduction to nonlinear dynamical phenomena, focused on identifying the long term behavior of systems described by ordinary differential equations. The emphasis is on stability and parameter dependence (bifurcations). Other topics include: chaos; routes to chaos and universality; maps; strange attractors; fractals. Techniques for analyzing nonlinear systems are introduced with applications to physical, chemical, and biological systems such as forced oscillators, chaotic reactions, and population dynamics.

## Introduction to PDEs and their Applications

**APMTH 109**

2024 Spring

**Nick Trefethen**

Tuesday, Thursday

12:00pm to 1:15pm

This course serves as an introduction to partial differential equations (PDE) and their applications across the sciences. The course will familiarize students with the process of starting with a model, deriving the appropriate PDE, and solving it. Examples include wave equations, diffusion equations, the Laplace equation, and several nonlinear equations such as the Burgers and KdV equations. To build intuition for the analytical solutions, simple numerical simulations will be utilized.

## Mathematical Modeling

**APMTH 115**

2024 Spring

**Michael P. Brenner**

Tuesday, Thursday

10:30am to 11:45am

Abstracting the essential components and mechanisms from a natural system to produce a mathematical model, which can be analyzed with a variety of formal mathematical methods, is perhaps the most important, but least understood, task in applied mathematics. This course approaches a number of problems without the prejudice of trying to apply a particular method of solution. Topics drawn from biology, economics, engineering, physical and social sciences.

## Applied Linear Algebra and Big Data

**APMTH 120**

2024 Spring

**Eli Tziperman**

Tuesday, Thursday

1:30pm to 2:45pm

Topics in linear algebra that frequently arise in applications, especially in the analysis of large data sets: linear equations, eigenvalue problems, linear differential equations, principal component analysis, singular value decomposition; data mining and machine learning methods: clustering (unsupervised learning) and classification (supervised) using neural networks and random forests. Examples from physical sciences, biology, climate, commerce, the internet, image processing, and more will be given. The approach is application-motivated, focusing on an intuitive understanding of the algorithms behind these methods obtained by analyzing small data sets. Programming assignments can be done using Python or Matlab.

## Geometric Methods for Machine Learning

**APMTH 220**

2024 Spring

**Melanie Weber**

Monday, Wednesday

3:00pm to 4:15pm

Recently, there has been a surge of interest in exploiting geometric structure in data and models in machine learning. This course will give an overview of this emerging research area and its mathematical foundation, with a focus on recent literature and open problems. We will cover a range of topics at the intersection of geometry and machine learning including basic differential geometry, graph representation learning, manifold learning, graph neural networks, machine learning on manifolds, and geometric deep learning. Lectures will be complemented by student-led discussions of relevant papers.

## Decision Theory

**APMTH 231**

2024 Spring

**Demba Ba**

Tuesday, Thursday

11:15am to 12:30pm

ES 201/AM 231 is a course in statistical inference and estimation from a signal processing perspective. The course will emphasize the entire pipeline from writing a model, estimating its parameters and performing inference utilizing real data. The first part of the course will focus on linear and nonlinear probabilistic generative/regression models (e.g. linear, logistic, Poisson regression), and algorithms for optimization (ML/MAP estimation) and Bayesian inference in these models. We will play particular attention to sparsity-induced regression models, because of their relation to artificial neural networks, the topic of the second part of the course. The second part of the course will introduce students to the nascent and exciting research area of model-based deep learning. At present, we lack a principled way to design artificial neural networks, the workhorses of modern AI systems. Moreover, modern AI systems lack the ability to explain how they reach their decisions. In other words, we cannot yet call AI explainable or interpretable which, as a society, poses important questions as to the responsible use of such technology. Model-based deep learning provides a framework to develop and constrain neural-network architectures in a principled fashion. We will see, for instance, how neural-networks with ReLU nonlinearites arise from sparse probabilistic generative models introduced in the first part of the course. This will form the basis for a rigorous recipe we will teach you to build interpretable deep neural networks, from the ground up. We will invite an exciting line up of speakers. Time permitting, we will provide a model-based pespective of the building blocks of modern language and image generative models.

## Learning, Estimation, and Control of Dynamical Systems

**APMTH 232**

2024 Spring

**Na Li**

Monday, Wednesday

9:45am to 11:00am

This graduate level course studies dynamic systems in time domain with inputs and outputs. Students will learn how to design estimator and controller for a system to ensure desirable properties (e.g., stability, performance, robustness) of the dynamical system. In particular, the course will focus on systems that can be modeled by linear ordinary differential equations (ODEs) and that satisfy time-invariance conditions. The course will introduces the fundamental mathematics of linear spaces, linear operator theory, and then proceeds with the analysis of the response of linear time-variant systems. Advanced topics such as robust control, model predictive control, linear quadratic games and distributed control will be presented based on allowable time and interest from the class. The material learned in this course will form a valuable foundation for further work in systems, control, estimation, identification, detection, signal processing, and communications.

## Special Topics in Applied Mathematics

**APMTH 299R**

2024 Spring

**Madhu Sudan**

Supervision of experimental or theoretical research on acceptable problems in applied mathematics and supervision of reading on topics not covered by regular courses of instruction.