# Courses

**Courses offered**

**EL-GY 6233: System Optimization Methods **

Formulations of system optimization problems. Elements of functional analysis applied to system optimization. Local and global system optimization with and without constraints. Variational methods, calculus of variations, and linear, nonlinear and dynamic programming iterative methods. Examples and applications. Newton and Lagrange multiplier algorithms, convergence analysis.

**EL-GY 9213: Game Theory for Multi-Agent Systems **

The goal of this class is to provide a broad and rigorous introduction to the theory, methods and algorithms of multi-agent systems. The material spans disciplines as diverse as engineering (including control theory and signal processing), computer science (including artificial intelligence, algorithms and distributed systems), micro-economic theory, operations research, public policies, psychology and belief systems. A primary focus of the course is on the application of cooperative and non- cooperative game theory for both static and dynamic models, with deterministic as well as stochastic descriptions. The coverage will encompass both theoretical and algorithmic developments, with multi- disciplinary applications.

**EL-GY 9223: Reinforcement Learning for Complex Networks **

This course is a graduate level course focusing on the theory and practice of reinforcement learning. Reinforcement learning is a paradigm that focuses on the question: How to interact with an environment when the decision maker's current action affects future consequences. This course provides an accessible in-depth treatment of reinforcement learning and dynamic programming methods using function approximators. The course starts with a concise introduction to Markov Decision Processes and optimal control problems, in order to build the foundation. We present an extensive review of state-of-the-art approaches to dynamic programming and reinforcement learning with approximations.

**EL-GY 5213 Introduction to Systems Engineering**

This course introduces fundamentals of systems engineering process. Topics: Multidisciplinary systems methodology, design and analysis of complex systems. Brief history of systems engineering. Mathematical models. Objective functions and constraints. Optimization tools. Topics to be covered include identification, problem definition, synthesis, analysis and evaluation activities during conceptual and preliminary system design phases. Decision analysis and utility theory. Information flow analysis in organizations. Elements of systems management, including decision styles, human information processing, organizational decision processes and information system design for planning and decision support. Basic economic modeling and analysis. Requirements development, life-cycle costing, scheduling and risk analysis. Application of computer-aided systems engineering (CASE) tools.

**Recommended Courses**

### EL-GY 6253 Linear Systems

Basic system concepts. Equations describing continuous and discrete-time linear systems. Time domain analysis, state variables, transition matrix and impulsive response. Transform methods. Time-variable systems. Controllability, observability and stability. SISO pole placement, observer design. Sampled data systems.

### EL-GY 6063 Information Theory

Mathematical information measures: entropy, relative entropy and mutual information. Assymptotic equipartition property, entropy rates of stochastic processes. Lossless source encoding theorems and source coding techniques. Channel capacity, differential entropy and the Gaussian channel. Lossy source coding rate distortion theory. Brief overview of network information theory.

### EL-GY 7253 State Space Design for Linear Control Systems

Topics covered in this course include canonical forms; control system design objectives; feedback system design by MIMO pole placement; MIMO linear observers; the separation principle; linear quadratic optimum control; random processes; Kalman filters as optimum observers; the separation theorem; LQG; Sampled-data systems; microprocessor-based digital control; robust control and the servocompensator problem.

### MATH-GA 2430.001 REAL VARIABLES (one-term format)

Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.

### MATH-GA 2170.001 INTRODUCTION TO CRYPTOGRAPHY

The primary focus of this course will be on definitions and constructions of various cryptographic objects, such as pseudo-random generators, encryption schemes, digital signature schemes, message authentication codes, block ciphers, and others time permitting. We will try to understand what security properties are desirable in such objects, how to properly define these properties, and how to design objects that satisfy them. Once we establish a good definition for a particular object, the emphasis will be on constructing examples that provably satisfy the definition. Thus, a main prerequisite of this course is mathematical maturity and a certain comfort level with proofs. Secondary topics that we will cover only briefly will be current cryptographic practice and the history of cryptography and cryptanalisys.

### MATH-GA 2180.001 ADVANCED CRYPTOGRAPHY (Randomness in Cryptography)

We will cover a variety of topics revolving around randomization, entropy, information-theoretic crypto, extractors and (time permitting) leakage-resilient cryptography. Some of the topics include:(im)possibility of authentication with weak sources; impossibility on basing privacy on entropy alone; encryption => extraction; differential privacy with SV sources; using public randomness; extractors (LHL, etc.); "square-friendly" privacy applications; randomness condensers; robust extractors; fuzzy extractors; entropic security and privacy (incl. private fuzzy extractors); privacy amplification and non-malleable extractors; locally computable extractors, bounded storage/retrieval model; computational (HILL, unpredictability) entropy and computational extractors; dense model theorem; randomized MACs; and time permitting, leakage-resilient cryptography.

**MATH-GA 2550.001 FUNCTIONAL ANALYSIS**

The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting.

**MATH-GA 2911.001, 2912.001 PROBABILITY: LIMIT THEOREMS I, II**

Fall Term: Probability, independence, laws of large numbers, limit theorems including the central limit theorem. Markov chains (discrete time). Martingales, Doob inequality, and martingale convergence theorems. Ergodic theorem.

Spring Term: Independent increment processes, including Poisson processes and Brownian motion. Markov chains (continuous time). Stochastic differential equations and diffusions, Markov processes, semi-groups, generators and connection with partial differential equations.

**MATH-GA 2563.001 HARMONIC ANALYSIS**

Classical Fourier Analysis: Fourier series on the circle, Fourier transform on the Euclidean space, Introduction to Fourier transform on LCA groups. Stationary phase. Topics in real variable methods: maximal functions, Hilbert and Riesz transforms and singular integral operators. Time permitting: Introduction to Littlewood-Paley theory, time-frequency analysis, and wavelet theory.

### MATH-GA 2704.001 APPLIED STOCHASTIC ANALYSIS

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments.