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 problemset requirements. Students will learn by doing simple examples.

APC 350/CEE 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.

GEO 441/APC 441: Computational GeophysicsAn introduction to weak numerical methods, in particular finiteelement and spectralelement methods, used in computational geophysics. Basic surface & volume elements, representation of fields, quadrature, assembly, local versus global meshes, domain decomposition, time marching & stability, parallel implementation & messagepassing, and loadbalancing. In the context of parameter estimation and 'imaging', will explore data assimilation techniques and related adjoint methods. The course offers handson lab experience in meshing complicated surfaces & volumes as well as numerically solving partial differential equations relevant to geophysics

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 SturmLiouville problems.

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 crossproperty relations. Biological and cosmological applications are discussed.

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 highperformance computing systems are discussed.

AST 559/APC 539: Turbulence and Nonlinear Processes in Fluids and PlasmasA comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including NavierStokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudospectral and particleincell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and directinteraction approximation; PDF methods and intermittency; variational techniques. Applications from neutral fluids, fusion plasmas, and astrophysics.

MAT 586/APC 511/MOL 511/QCB 513: Computational Methods in CryoElectron MicroscopyThis course focuses on computational methods in cryoEM, including threedimensional abinitio 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 nonlinear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

MAT 588/APC 588: Topics in Numerical Analysis: Optimization On Smooth ManifoldsThe key player is the problem: minimize f(x), where x lives on a smooth manifold M, and f is a smooth cost function on M. Applications abound in scientific computing, signal processing, computer vision, machine learning and statistics. Manifolds arise in optimization as a result of constraints (e.g., lowrank, orthogonality) and as a result of symmetry (quotient spaces). By endowing the manifold with a Riemannian structure, we obtain meaningful notions of gradient and Hessian on the manifold. This enables us to generalize standard algorithms such as gradient descent and trustregions. The course mixes mathematical analysis and coding.