# Courses

# Course Listing

## Computing for Science and Engineering

**APMTH 10**

2020 Fall

**Efthimios Kaxiras**

Tuesday, Thursday

12:00pm to 1:15pm

This course is a systematic introduction to a computing environment (python with jupyter notebooks) suited for applications to science and engineering. It consists of three Modules: 1. Basics: essential elements of computing: types of variables (integer, floating, logical), lists, arrays, basic operations (for, while loops, if statement), definition of functions, file handling and plotting. 2. Elementary: numerical differentiation, root finding, series expansions, numerical integration, fitting of curves and error analysis, plotting and simulating in higher dimensions (contours). 3. Advanced: solving simple first and second-order ordinary differential equations, solving partial differential equations, use of random numbers for sampling and simulations, such as Monte Carlo integration and modeling realistic problems, like the spread of the COVID-19 pandemic. The course work consists of attending lectures and labs, weekly homework assignments, a mid-term project and a final project; all work is developed in small groups, but assignments must be written by students individually.

## Solving and Optimizing

**APMTH 22A**

2020 Fall

**Steven Gortler**

Monday, Wednesday, Friday

12:00pm to 1:15pm

This course covers a combination of linear algebra and multivariate calculus with an eye towards solving systems of equations and optimization problems. Students will learn how to prove some key results, and will also implement these ideas with code.Linear algebra: matrices, vector spaces, bases and dimension, inner products, least squares problems, eigenvalues, eigenvectors, singular values, singular vectors.Multivariate calculus: partial differentiation, gradient and Hessian, critical points, Lagrange Multipliers.

## Integrating and Approximating

**APMTH 22B**

2021 Spring

**Sarah Iams**

Monday, Wednesday, Friday

10:30am to 11:45am

Multivariable and vector calculus, supplemented with numerical methods. Multivariate calculus: multiple integration, functions of two or three variables, approximating functions. Parameterized curves, line and surface integrals. Vector calculus: gradient, divergence and curl, Green’s, divergence theorems. Complex numbers. Select differential equations topics.

## Introduction to Applied Mathematics

**APMTH 50**

2021 Spring

**Cengiz Pehlevan**

Monday, Wednesday

1:30pm to 2:45pm

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**

2020 Fall

**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.

## Supervised Reading and Research

**APMTH 91R**

2021 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**

2020 Fall

**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.

## Thesis Research

**APMTH 99R**

2021 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**

2020 Fall

**Robert D. Howe, Jeffrey Paten**

Monday, Wednesday

6:00pm to 7:15pm

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.

## Statistical Inference for Scientists and Engineers

**APMTH 101**

2020 Fall

**Robert D. Howe, Jeffrey Paten**

Monday, Wednesday

10:30am to 11:45am

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.

## Complex and Fourier Analysis with Applications to Art, Science and Engineering

**APMTH 104**

2020 Fall

**L Mahadevan**

Complex analysis: complex numbers, functions, mappings, Laurent series, differentiation, integration, contour integration and residue theory, conformal mappings and circle packings. Applications to visualization, art (especially M.C. Escher) and photography. Fourier Analysis: orthogonality, Fourier Series, Fourier transforms. Signal processing: sampling theorems (Nyquist, Shannon), fast Fourier and other discrete transforms, wavelets and filtering. Applications to image, video, audio and morphological analysis: filtering and cleaning images, musical analysis, fraud and authentication, filter banks for engineering.

## Complex and Fourier Analysis with Applications to Art, Science and Engineering

**APMTH 104**

2020 Fall

**L Mahadevan**

Tuesday, Thursday

12:00pm to 1:15pm

Complex analysis: complex numbers, functions, mappings, Laurent series, differentiation, integration, contour integration and residue theory, conformal mappings and circle packings. Applications to visualization, art (especially M.C. Escher) and photography. Fourier Analysis: orthogonality, Fourier Series, Fourier transforms. Signal processing: sampling theorems (Nyquist, Shannon), fast Fourier and other discrete transforms, wavelets and filtering. Applications to image, video, audio and morphological analysis: filtering and cleaning images, musical analysis, fraud and authentication, filter banks for engineering.

## Ordinary and Partial Differential Equations

**APMTH 105**

2021 Spring

**Zhigang Suo, Ethan Levien**

Tuesday, Thursday

7:30pm to 8:45pm

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.

## Ordinary and Partial Differential Equations

**APMTH 105**

2021 Spring

**Zhigang Suo, Ethan Levien**

Tuesday, Thursday

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**

2021 Spring

**Leslie Valiant**

Tuesday, Thursday

10:30am to 11:45am

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**

2020 Fall

**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.

## Nonlinear Dynamical Systems

**APMTH 108**

2020 Fall

**Sarah Iams**

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.

## Mathematical Modeling

**APMTH 115**

2021 Spring

**Zhiming Kuang**

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**

2021 Spring

**Eli Tziperman**

Tuesday, Thursday

1:30pm to 2:45pm

Topics in linear algebra which arise frequently 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 methods including frequent pattern analysis, clustering, outlier detection, classification, and machine learning, including neural networks and random forests. Examples will be given from physical sciences, biology, climate, commerce, internet, image processing and more.

## Introduction to Optimization: Models and Methods

**APMTH 121**

2020 Fall

**Yiling Chen, Margo Levine**

Monday, Wednesday

10:30am to 11:45am

Introduction to basic mathematical ideas and computational methods for solving deterministic optimization problems. Topics covered: linear programming, integer programming, branch-and-bound, branch-and-cut. Emphasis on modeling. Examples from business, society, engineering, sports, e-commerce. Exercises in AMPL, complemented by Mathematica or Matlab.

## Introduction to Optimization: Models and Methods

**APMTH 121**

2020 Fall

**Yiling Chen, Margo Levine**

Monday, Wednesday

9:00pm to 10:15pm

Introduction to basic mathematical ideas and computational methods for solving deterministic optimization problems. Topics covered: linear programming, integer programming, branch-and-bound, branch-and-cut. Emphasis on modeling. Examples from business, society, engineering, sports, e-commerce. Exercises in AMPL, complemented by Mathematica or Matlab.

## Physical Mathematics I

**APMTH 201**

2020 Fall

**Michael P. Brenner**

Introduction to methods for developing accurate approximate solutions for problems in the sciences that cannot be solved exactly, and integration with numerical methods and solutions. Topics include: dimensional analysis, algebraic equations, complex analysis, perturbation theory, matched asymptotic expansions, approximate solution of integrals.

## Advanced Scientific Computing: Numerical Methods

**APMTH 205**

2020 Fall

**Christopher Rycroft, Zhiming Kuang**

Tuesday, Thursday

10:30am to 11:45am

An examination of the mathematical foundations of a range of well-established numerical algorithms, exploring their use through practical examples drawn from a range of scientific and engineering disciplines. Emphasizes theory and numerical analysis to elucidate the concepts that underpin each algorithm. There will be a significant programming component. Students will be expected to implement a range of numerical methods through individual and group-based project work to get hands-on experience with modern scientific computing.

## Advanced Scientific Computing: Stochastic Methods for Data Analysis, Inference and Optimization

**APMTH 207**

2020 Fall

**Weiwei Pan**

Tuesday, Thursday

9:00am to 10:15am

Develops skills for computational research with focus on stochastic approaches, emphasizing implementation and examples. Stochastic methods make it feasible to tackle very diverse problems when the solution space is too large to explore systematically, or when microscopic rules are known, but not the macroscopic behavior of a complex system. Methods will be illustrated with examples from a wide variety of fields, like biology, finance, and physics.

## Advanced Scientific Computing: Stochastic Methods for Data Analysis, Inference and Optimization

**APMTH 207**

2020 Fall

**Weiwei Pan**

Tuesday, Thursday

1:30pm to 2:45pm

Develops skills for computational research with focus on stochastic approaches, emphasizing implementation and examples. Stochastic methods make it feasible to tackle very diverse problems when the solution space is too large to explore systematically, or when microscopic rules are known, but not the macroscopic behavior of a complex system. Methods will be illustrated with examples from a wide variety of fields, like biology, finance, and physics.

## Inverse Problems in Science and Engineering

**APMTH 216**

2021 Spring

**Michael P. Brenner**

Many problems in science and engineering are inverse problems. Any experiment that requires an explanation can be couched thus - given the data, what is the theory/model that provides it - this is an inverse problem. In engineering, a given function (in a product/software …. ) requires a design - again an inverse problem. This course will introduce a wide array of features of inverse problems from science and engineering - from oil prospecting and seismology to cognitive science, from particle physics to engineering design. We will then introduce deterministic and probabilistic algorithms for solving these problems. Much of the class will be spent studying how the recent revolution in deep neural networks can (and cannot) be used to solve such inverse problems. The class will have a substantial computational component -- part of every class session will contain instruction and computer implementation of the algorithms in question. Students will carry out final projects in their own area of interest. Programming will be taught and carried out in Python and Tensorflow.

## Advanced Scientific Computing: Numerical Methods for Partial Differential Equations

**APMTH 225**

2021 Spring

**Christopher Rycroft**

Monday, Wednesday

10:30am to 11:45am

This course examines a variety of advanced numerical methods, with a focus on those relevant to solving partial differential equations that arise in physical problems. Topics include the finite volume method, finite element method, and interface tracking methods. Associated problems in numerical linear algebra and optimization will be discussed. The course will examine the mathematical underpinnings of each method, as well as look at their practical usage, paying particular attention to efficient implementations on modern multithreaded and parallel computer architectures.

## Neural Computation

**APMTH 226**

2020 Fall

**Cengiz Pehlevan**

This course introduces advanced mathematical methods and models used in theoretical neuroscience and theory of neural networks. We will explore computations and functions performed by the brain, and how they are implemented by neurons and their networks. We will cover selected topics from deep learning theory; spiking neuron models; population codes; normative theories of sensory representations; models of synaptic plasticity; computing with dynamics in recurrent neural networks; attractor network models of memory and spatial maps; neural models of probabilistic inference in the brain and drift-diffusion models of decision making. Concrete examples of applications of these ideas to the brain will be discussed. Topics at the research frontier will be emphasized.

## Decision Theory

**APMTH 231**

2021 Spring

**Demba Ba**

Tuesday, Thursday

10:30am to 11:45am

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) in these models. We will play particular attention to sparsity-induced regression models, that arise for instance in compressed sensing, 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 generative models of deep networks called 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 nonlinearities 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. Speakers will suggest papers that a group of students will present at the beginning of lecture, which will build up to a final project/paper that utilizes/on model-based deep learning applied to problems of interest to students.

## Special Topics in Applied Mathematics

**APMTH 299R**

2020 Fall

**Yiling Chen**

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.

## Special Topics in Applied Mathematics

**APMTH 299R**

2021 Spring

**Cengiz Pehlevan**

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.