## 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

Instructors:
Kerckhoff, S. (PI)
;
Fauteux-Chapleau, F. (TA)

## 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

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

## 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

Instructors:
Blanchet Mancilla, J. (PI)

## 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.

Last offered: Spring 2019

## 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.

Last offered: Winter 2019

## 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

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).

Last offered: Autumn 2018

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