## Appl and Computational Math

- APC 199/MAT 199: Math AliveMathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those without college mathematics who want to understand the mathematical concepts behind important modern applications. The course consists of individual modules, each focusing on a particular application (e.g. compression, animation and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set requirements. Students will learn by doing simple examples.
- APC 350/MAT 322: Introduction to Differential EquationsThis course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations.
- APC 523/AST 523/MAE 507: Numerical Algorithms for Scientific ComputingA broad introduction to numerical algorithms used in scientific computing. The course begins with a review of the basic principles of numerical analysis, including sources of error, stability, and convergence. The theory and implementation of techniques for linear and nonlinear systems of equations and ordinary and partial differential equations are covered in detail. Examples of the application of these methods to problems in engineering and the sciences permeate the course material. Issues related to the implementation of efficient algorithms on modern high-performance computing systems are discussed.
- ELE 486/APC 486: Transmission and Compression of InformationOur digital world relies heavily on the ability to extract, store, and transfer information. Over the years much effort has been devoted to the development of methodologies that perform these tasks efficiently. This course covers the fundamental algorithms and limits of data compression and transmission, detailing key components of information theory and coding theory such as entropy, source/channel codes, and information measures. We also draw connections between these theories and several techniques in supervised and unsupervised machine learning, including data clustering, principal component analysis and graphical models.
- MAE 502/APC 506: Mathematical Methods of Engineering Analysis IITopics in complex analysis and functional analysis, with emphasis on applications in physics and engineering. Topics include power series, singularities, contour integration, Cauchy's theorems, and Fourier series; an introduction to measure theory and the Lebesgue integral; Hilbert spaces, linear operators, and adjoints; the spectral theorem, and its application to Sturm-Liouville problems.
- MAT 490/APC 490: Mathematical Introduction to Machine LearningThis course gives a mathematical introduction to machine learning. There are three major components in this course. (1) Machine learning models (kernel methods, shallow and deep neural network models) for both supervised and unsupervised learning problems. (2) Optimization algorithms (gradient descent, stochastic gradient descent, EM). (3) Mathematical analysis of these models and algorithms.
- MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in Cryo-Electron MicroscopyThis 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.
- MSE 515/APC 515/CHM 559: Random Heterogeneous MaterialsComposites, porous media, foams, colloids, geological media, and biological media are all examples of heterogeneous materials. The relationship between the macroscopic (transport, mechanical, electromagnetic, and chemical) properties and material microstructure is formulated. Topics include statistical characterization of the microstructure; percolation theory; fractals; sphere packings; Monte Carlo techniques; image analysis; homogenization theory; cluster and perturbation expansions; variational bounding techniques; topology optimization methods; and cross-property relations. Biological and cosmological applications are discussed.