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CME 307: Optimization (MS&E 311)

Applications, theories, and algorithms for finite-dimensional linear and nonlinear optimization problems with continuous variables. Elements of convex analysis, first- and second-order optimality conditions, sensitivity and duality. Algorithms for unconstrained optimization, and linearly and nonlinearly constrained problems. Modern applications in communication, game theory, auction, and economics. Prerequisites: MATH 113, 115, or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ye, Y. (PI)

CS 205L: Continuous Mathematical Methods with an Emphasis on Machine Learning

A survey of numerical approaches to the continuous mathematics with emphasis on machine and deep learning. Although motivated from the standpoint of machine learning, the course will focus on the underlying mathematical methods including computational linear algebra and optimization, as well as special topics related to training/using neural networks including automatic differentiation via backward propagation, steepest/gradient decent, momentum methods and adaptive time stepping for ordinary differential equations, etc. Students have the option of doing written homework and either a take-home or in class exams with no programming required, or may skip the exams and instead do a programming project. (Replaces CS205A, and satisfies all similar requirements.) Prerequisites: Math 51; Math 104 or 113 or equivalent or comfortable with the associated material.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Fedkiw, R. (PI)

CS 229T: Statistical Learning Theory (STATS 231)

How do we formalize what it means for an algorithm to learn from data? How do we use mathematical thinking to design better machine learning methods? This course focuses on developing mathematical tools for answering these questions. We will present various learning algorithms and prove theoretical guarantees about them. Topics include generalization bounds, implicit regularization, the theory of deep learning, spectral methods, and online learning and bandits problems. Prerequisites: A solid background in linear algebra (Math 104, Math 113 or CS205) and probability theory (CS109 or STAT 116), statistics and machine learning (STATS 315A, CS 229 or STATS 216).
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

CS 329M: Topics in Artificial Intelligence: Algorithms of Advanced Machine Learning

This advanced graduate course explores in depth several important classes of algorithms in modern machine learning. We will focus on understanding the mathematical properties of these algorithms in order to gain deeper insights on when and why they perform well. We will also study applications of each algorithm on interesting, real-world settings. Topics include: spectral clustering, tensor decomposition, Hamiltonian Monte Carlo, adversarial training, and variational approximation. Students will learn mathematical techniques for analyzing these algorithms and hands-on experience in using them. We will supplement the lectures with latest papers and there will be a significant research project component to the class. Prerequisites: Probability (CS 109), linear algebra (Math 113), machine learning (CS 229), and some coding experience.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

ENGR 205: Introduction to Control Design Techniques

Review of root-locus and frequency response techniques for control system analysis and synthesis. State-space techniques for modeling, full-state feedback regulator design, pole placement, and observer design. Combined observer and regulator design. Lab experiments on computers connected to mechanical systems. Prerequisites: 105, MATH 103, 113. Recommended: Matlab.
Terms: Aut | Units: 3 | Grading: Letter (ABCD/NP)
Instructors: ; Rock, S. (PI)

MATH 104: Applied Matrix Theory

Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and EE 103/CME 103 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of EE 103 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106.
Terms: Aut, Win, Spr, Sum | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

MATH 113: Linear Algebra and Matrix Theory

Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization. (Math 104 offers a more application-oriented treatment.) Prerequisites: Math 51
Terms: Aut, Win, Spr, Sum | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

MATH 121: Galois Theory

Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite: Math 120 and (also recommended) 113.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Tsai, C. (PI); Cotner, S. (TA)

MATH 122: Modules and Group Representations

Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite: Math 120. Also recommended: 113.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Bump, D. (PI)

MATH 146: Analysis on Manifolds

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Hershkovits, O. (PI)

MATH 215A: Algebraic Topology

Topics: fundamental group and covering spaces, basics of homotopy theory, homology and cohomology (simplicial, singular, cellular), products, introduction to topological manifolds, orientations, Poincare duality. Prerequisites: 113, 120, and 171.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit

MS&E 310: Linear Programming

Formulation of standard linear programming models. Theory of polyhedral convex sets, linear inequalities, alternative theorems, and duality. Variants of the simplex method and the state of art interior-point algorithms. Sensitivity analyses, economic interpretations, and primal-dual methods. Relaxations of harder optimization problems and recent convex conic linear programs. Applications include game equilibrium facility location. Prerequisite: MATH 113 or consent of instructor.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ye, Y. (PI); Li, X. (TA)

MS&E 311: Optimization (CME 307)

Applications, theories, and algorithms for finite-dimensional linear and nonlinear optimization problems with continuous variables. Elements of convex analysis, first- and second-order optimality conditions, sensitivity and duality. Algorithms for unconstrained optimization, and linearly and nonlinearly constrained problems. Modern applications in communication, game theory, auction, and economics. Prerequisites: MATH 113, 115, or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ye, Y. (PI)

MS&E 321: Stochastic Systems

Topics in stochastic processes, emphasizing applications. Markov chains in discrete and continuous time; Markov processes in general state space; Lyapunov functions; regenerative process theory; renewal theory; martingales, Brownian motion, and diffusion processes. Application to queueing theory, storage theory, reliability, and finance. Prerequisites: 221 or STATS 217; MATH 113, 115. (Glynn)
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit

MS&E 322: Stochastic Calculus and Control

Ito integral, existence and uniqueness of solutions of stochastic differential equations (SDEs), diffusion approximations, numerical solutions of SDEs, controlled diffusions and the Hamilton-Jacobi-Bellman equation, and statistical inference of SDEs. Applications to finance and queueing theory. Prerequisites: 221 or STATS 217: MATH 113, 115.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

MS&E 351: Dynamic Programming and Stochastic Control

Markov population decision chains in discrete and continuous time. Risk posture. Present value and Cesaro overtaking optimality. Optimal stopping. Successive approximation, policy improvement, and linear programming methods. Team decisions and stochastic programs; quadratic costs and certainty equivalents. Maximum principle. Controlled diffusions. Examples from inventory, overbooking, options, investment, queues, reliability, quality, capacity, transportation. MATLAB. Prerequisites: MATH 113, 115; Markov chains; linear programming.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

PHYSICS 113: Computational Physics

Numerical methods for solving problems in mechanics, astrophysics, electromagnetism, quantum mechanics, and statistical mechanics. Methods include numerical integration; solutions of ordinary and partial differential equations; solutions of the diffusion equation, Laplace's equation and Poisson's equation with various methods; statistical methods including Monte Carlo techniques; matrix methods and eigenvalue problems. Short introduction to Python, which is used for class examples and active learning notebooks; independent class projects make up more than half of the grade and may be programmed in any language such as C, Python or Matlab. No Prerequisites but some previous programming experience is advisable.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-AQR, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Cabrera, B. (PI)

STATS 231: Statistical Learning Theory (CS 229T)

How do we formalize what it means for an algorithm to learn from data? How do we use mathematical thinking to design better machine learning methods? This course focuses on developing mathematical tools for answering these questions. We will present various learning algorithms and prove theoretical guarantees about them. Topics include generalization bounds, implicit regularization, the theory of deep learning, spectral methods, and online learning and bandits problems. Prerequisites: A solid background in linear algebra (Math 104, Math 113 or CS205) and probability theory (CS109 or STAT 116), statistics and machine learning (STATS 315A, CS 229 or STATS 216).
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit
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