**EGR 192 / MAT 192 / PHY 192 / APC 192**

## An Integrated Introduction to Engineering, Mathematics, Physics

**Professor/Instructor**

**Christine Jiayou Taylor**

Taken concurrently with EGR/MAT/PHY 191. An integrated course that covers the material of PHY 103 and MAT 201 with the emphasis on applications to engineering. Math topics include: vector calculus; partial derivatives and matrices; line integrals; simple differential equations; surface and volume integrals; and Green's, Stokes's, and divergence theorems. One lecture, two preceptorials.

**APC 199 / MAT 199**

## Math Alive

**Professor/Instructor**

An exploration of some of the mathematical ideas behind important modern applications, from banking and computing to listening to music. Intended for students who have not had college-level mathematics and are not planning to major in a mathematically based field. The course is organized in independent two-week modules focusing on particular applications, such as bar codes, CD-players, population models, and space flight. The emphasis is on ideas and mathematical reasoning, not on sophisticated mathematical techniques. Two 90-minute classes, one computer laboratory.

**MAT 323 / APC 323**

## Topics in Mathematical Modeling

**Professor/Instructor**

**Zahra Aminzare**

Draws problems from the sciences & engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.

**MAT 377 / APC 377**

## Combinatorial Mathematics

**Professor/Instructor**

**Noga Mordechai Alon**

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.

**GEO 441 / APC 441**

## Computational Geophysics

**Professor/Instructor**

**Jeroen Tromp**

An introduction to weak numerical methods used in computational geophysics. Finite- and spectral-elements, representation of fields, quadrature, assembly, local versus global meshes, domain decomposition, time marching and stability, parallel implementation and message-passing, and load-balancing. Parameter estimation and "imaging" using data assimilation techniques and related "adjoint" methods. Labs provide experience in meshing complicated surfaces and volumes as well as solving partial differential equations relevant to geophysics. Prerequisites: MAT 201; partial differential equations and basic programming skills. Two 90-minute lectures.

**ELE 486 / APC 486**

## Transmission and Compression of Information

**Professor/Instructor**

An introduction to lossless data compression algorithms, modulation/demodulation of digital data, error correcting codes, channel capacity, lossy compression of analog and digital sources. Three hours of lectures. Prerequisites: 301, ORF 309.

**MAE 501 / APC 501**

## Mathematical Methods of Engineering Analysis I

**Professor/Instructor**

**Clarence Worth Rowley III**

Methods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to functional analysis, Sturm-Liouville theory, Green's functions for the solution of ordinary differential equations and Poisson's equation, and the calculus of variations.

**CBE 502 / APC 502**

## Mathematical Methods of Engineering Analysis II

**Professor/Instructor**

**Luc Deike**

Linear ordinary differential equations (systems of first-order equations, method of Frobenius, two-point boundary-value problems); spectrum and Green's function; matched asymptotic expansions; partial differential equations (classification, characteristics, uniqueness, separation of variables, transform methods, similarity); and Green's function for the Poisson, heat, and wave equations, with applications to selected problems in chemical, civil, and mechanical engineering.

**APC 503 / AST 557**

## Analytical Techniques in Differential Equations

**Professor/Instructor**

Local analysis of solutions to linear and nonlinear differential and difference equations. Asymptotic methods, asymptotic analysis of integrals, perturbation theory, summation methods, boundary layer theory, WKB theory, and multiple scale theory. Prerequisite: MAE 306 or equivalent.

**MAE 502 / APC 506**

## Mathematical Methods of Engineering Analysis II

**Professor/Instructor**

**Clarence Worth Rowley III**

A complementary presentation of theory, analytical methods, and numerical methods. The objective is to impart a set of capabilities commonly used in the research areas represented in the Department. Standard computational packages will be made available in the courses, and assignments will be designed to use them. An extension of MAE 501.

**APC 509**

## Methods and Concepts in Electronic Structure Theory

**Professor/Instructor**

**John A. Keith**

This course derives how and why chemical bonds between atoms form, leading to the creation of molecules and condensed matter. State-of-the-art electronic structure theory methods are discussed and compared in terms of strengths, weaknesses, and numerical implementations. Students will learn how to predict molecular structure, qualitative character of the electron distribution (hybridization), and relative chemical bond strengths directly from a simple multi-electron wavefunction. Condensed matter electronic structure will be introduced via band theory, followed by an analysis of the pros and cons of modern density functional theory methods.

**MAT 586 / APC 511 / MOL 511 / QCB 513**

## Computational Methods in Cryo-Electron Microscopy

**Professor/Instructor**

**Amit Singer**

This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

**MAT 585 / APC 520**

## Mathematical Analysis of Massive Data Sets

**Professor/Instructor**

**Amit Singer**

This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.

**MAT 522 / APC 522**

## Introduction to PDE

**Professor/Instructor**

**Alexandru D. Ionescu**

The course is a basic introductory graduate course in partial differential equations. Topics include: Laplacian, properties of harmonic functions, boundary value problems, wave equation, heat equation, Schrodinger equation, hyperbolic conservation laws, Hamilton-Jacobi equations, Fokker-Planck equations, basic function spaces and inequalities, regularity theory for second order elliptic linear PDE, De Giorgi method, basic harmonic analysis methods, linear evolution equations, existence, uniqueness and regularity results for classes of nonlinear PDE with applications to equations of nonlinear and statistical physics.

**APC 523 / AST 523 / MAE 507**

## Numerical Algorithms for Scientific Computing

**Professor/Instructor**

**Gabriel Perez-Giz**

A broad introduction to scientific computation using examples drawn from astrophysics. From computer science, practical topics including processor architecture, parallel systems, structured programming, and scientific visualization will be presented in tutorial style. Basic principles of numerical analysis, including sources of error, stability, and convergence of algorithms. The theory and implementation of techniques for linear and nonlinear systems of equations, ordinary and partial differential equations will be demonstrated with problems in stellar structure and evolution, stellar and galactic dynamics, and cosmology.

**APC 524 / MAE 506 / AST 506**

## Software Engineering for Scientific Computing

**Professor/Instructor**

The goal of this course is to teach basic tools and principles of writing good code, in the context of scientific computing. Specific topics include an overview of relevant compiled and interpreted languages, build tools and source managers, design patterns, design of interfaces, debugging and testing, profiling and improving performance, portability, and an introduction to parallel computing in both shared memory and distributed memory environments. The focus is on writing code that is easy to maintain and share with others. Students will develop these skills through a series of programming assignments and a group project.

**ORF 550 / APC 550**

## Topics in Probability

**Professor/Instructor**

**Ramon van Handel**

An introduction to nonasymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics. Emphasis is on developing a common set of tools that has proved to be useful in different areas. Topics may include: concentration of measure; functional, transportation cost, martingale inequalities; isoperimetry; Markov semigroups, mixing times, random fields; hypercontractivity; thresholds and influences; Stein's method; suprema of random processes; Gaussian and Rademacher inequalities; generic chaining; entropy and combinatorial dimensions; selected applications.

**MAE 541 / APC 571**

## Applied Dynamical Systems

**Professor/Instructor**

**Clarence Worth Rowley III**

Phase-plane methods and single-degree-of-freedom nonlinear oscillators; invariant manifolds, local and global analysis, structural stability and bifurcation, center manifolds, and normal forms; averaging and perturbation methods, forced oscillations, homoclinic orbits, and chaos; and Melnikov's method, the Smale horseshoe, symbolic dynamics, and strange attractors. Offered in alternate years.

**MAT 572 / APC 572**

## Topics in Combinatorial Optimization

**Professor/Instructor**

**Paul Douglas Seymour**

This course covers current topics in combinatorial optimization. More specific topic details are provided when the course is offered.

**AOS 576 / APC 576**

## Current Topics in Dynamic Meteorology

**Professor/Instructor**

**Isaac M. Held**

An introduction to topics of current interest in the dynamics of large-scale atmospheric flow. Possible topics include wave-mean flow interaction and nonacceleration theorems, critical levels, quasigeostrophic instabilities, topographically and thermally forced stationary waves, theories for stratospheric sudden warmings and the quasi-biennial oscillation of the equatorial stratosphere, and quasi-geostrophic turbulence.