Mathematics

Sparse and redundant representations constitute a fascinating area of research in signal and image processing. This is a relatively young field that has been taking form for the last 15 years or so, with contributions from harmonic analysis, numerical algorithms and machine learning, and has been vastly applied to myriad of problems in computer vision and other domains. This course will focus on sparsity as a model for general data, generalizing many different other constructions or priors. This idea - that signals can be represented with just a few coefficients - leads to a long series of beautiful (and surprisingly, solvable) theoretical and numerical problems, and many applications that can benefit directly from the new developed theory. In this course we will survey this field, starting with the theoretical foundations and systematically covering the knowledge that has been gathered in the past years. This course will touch on theory, numerical algorithms, and applications in image processing and machine learning.

Prerequisite(s): Mathematical Methods or equivalent

The course covers fundamental mathematical ideas for certain approximation and statistical learning problems in high dimensions. We start with basic approximation theory in low-dimensions, in particular linear and nonlinear approximation by Fourier and wavelets in classical smoothness spaces, and discuss applications in imaging, inverse problems and PDE’s. We then introduce notions of complexity of function spaces, which will be important in statistical learning. We then move to basic problems in statistical learning, such as regression and density estimation. The interplay between randomness and approximation theory is introduced, as well as fundamental tools such as concentration inequalities, basic random matrix theory, and various estimators are constructed in detail, in particular multi scale estimators. At all times we consider the geometric aspects and interpretations, and will discuss concentration of measure phenomena, embedding of metric spaces, optimal transportation distances, and their applications to problems in machine learning such as manifold learning and dictionary learning for signal processing.

This course covers a broad spectrum of mathematical techniques essential to the solution of advanced problems in physics and engineering. Topics include ordinary and partial differential equations, contour integration, tabulated integrals, saddlepoint methods, linear vector spaces, boundary-value problems, eigenvalue problems, Green’s functions, integral transforms, and special functions. Application of these topics to the solution of problems in physics and engineering is stressed. Prerequisite(s): Vector analysis and ordinary differential equations (linear algebra and complex variables recommended).

The course will introduce students to the basic concepts of nonlinear physics, dynamical system theory, and chaos. These concepts will be studied by examining the behavior of fundamental model systems that are modeled by ordinary differential equations and, sometimes, discrete maps. Examples will be drawn from physics, chemistry, and engineering. Some mathematical theory is necessary to develop the material. Practice through concrete examples will help to develop the geometric intuition necessary for work on nonlinear systems. Students conduct numerical experiments using provided software, which allows for interactive learning. Prerequisite(s): Mathematics through ordinary differential equations. Familiarity with MATLAB is helpful. Consult instructor for more information.

This course exposes the student to the physical principles underlying satellite observations of Earth by optical, infrared, and microwave sensors, as well as techniques for extracting geophysical information from remote sensor observations. Topics will include spacecraft orbit considerations, fundamental concepts of radiometry, electromagnetic wave interactions with land and ocean surfaces and Earth’s atmosphere, radiative transfer and atmospheric effects, and overviews of some important satellite sensors and observations. Examples from selected sensors will be used to illustrate the information extraction process and applications of the data for environmental monitoring, oceanography, meteorology, and climate studies.

To understand the forces that cause global climate variability, we must understand the natural forces that drive our weather and our oceans. This course covers the fundamental science underlying the nature of the Earth’s atmosphere and its ocean. This includes development of the basic equations for the atmosphere and ocean, the global radiation balance, description of oceanic and atmospheric processes, and their interactions and variability. Also included will be a description of observational systems used for climate studies and monitoring, fundamentals underlying global circulation, and climate prediction models. Prerequisite(s): Undergraduate degree in physics or engineering or equivalent, with strong background in mathematics through the calculus level.

This course introduces statistical methods that are widely used in modern applications. A balance is struck between the presentation of the mathematical foundations of concepts in probability and statistics and their appropriate use in a variety of practical contexts. Foundational topics of probability, such as probability rules, related inequalities, random variables, probability distributions, moments, and jointly distributed random variables, are followed by foundations of statistical inference, including estimation approaches and properties, hypothesis testing, and model building. Data analysis ranging from descriptive statistics to the implementation of common procedures for estimation, hypothesis testing, and model building is the focus after the foundational methodology has been covered. Software, for example R-Studio, will be leveraged to illustrate concepts through simulation and to serve as a platform for data analysis. Prerequisite(s): Multivariate calculus.

Optimization models play an increasingly important role in financial decisions. This course introduces the student to financial optimization models and methods. We will specifically discuss linear, integer, quadratic, and general nonlinear programming. If time permits, we will also cover dynamic and stochastic programming. The main theoretical features of these optimization methods will be studied as well as a variety of algorithms used in practice. Prerequisite(s): Multivariate calculus and linear algebra. Course Note(s): Due to overlap in subject matter in EN.625.615 and EN.625.616, students may not receive credit towards the MS or post-master’s certificate for both EN.625.615 and EN.625.616.

This course is an introduction to fundamental tools in designing, conducting, and interpreting Monte Carlo simulations. Emphasis is on generic principles that are widely applicable in simulation, as opposed to detailed discussion of specific applications and/or software packages. At the completion of this course, it is expected that students will have the insight and understanding to critically evaluate or use many state-of-the-art methods in simulation. Topics covered include random number generation, simulation of Brownian motion and stochastic differential equations, output analysis for Monte Carlo simulations, variance reduction, Markov chain Monte Carlo, simulation-based estimation for dynamical (state-space) models, and, time permitting, sensitivity analysis and simulation-based optimization. Course Note(s): This course serves as a complement to the 700-level course EN.625.744 Modeling, Simulation, and Monte Carlo. EN.625.633 Monte Carlo Methods and EN.625.744 emphasize different topics, and EN.625.744 is taught at a slightly more advanced level. EN.625.633 includes topics not covered in EN.625.744 such as simulation of Brownian motion and stochastic differential equations, general output analysis for Monte Carlo simulations, and general variance reduction. EN.625.744 includes greater emphasis on generic modeling issues (bias-variance tradeoff, etc.), simulation-based optimization of real-world processes, and optimal input selection.

Prerequisite(s): Linear algebra and a graduate-level statistics course such as EN.625.603 Statistical Methods and Data Analysis.

This course offers a rigorous treatment of the subject of investment as a scientific discipline. Mathematics is employed as the main tool to convey the principles of investment science and their use to make investment calculations for good decision making. Topics covered in the course include the basic theory of interest and its applications to fixed-income securities, cash flow analysis and capital budgeting, mean-variance portfolio theory and the associated capital asset pricing model, utility function theory and risk analysis, derivative securities and basic option theory, and portfolio evaluation.

Prerequisite(s): Multivariate calculus and a course in probability and statistics (such as EN.625.603 Statistical Methods and Data Analysis).

The concept of options stems from the inherent human desire and need to reduce risks. This course starts with a rigorous mathematical treatment of options pricing, and related areas by developing a powerful mathematical tool known as Ito calculus. We introduce and use the well-known field of stochastic differential equations to develop various techniques as needed, as well as discuss the theory of martingales. The mathematics will be applied to the arbitrage pricing of financial derivatives, which is the main topic of the course. We treat the Black-Scholes theory in detail and use it to understand how to price various options and other quantitative financial instruments. Topics covered in the course include options strategies, binomial pricing, Weiner processes and Ito’s lemma, the Black-Scholes-Merton Model, futures options and Black’s Model, option Greeks, numerical procedures for pricing options, the volatility smile, the value at risk, exotic options, martingales and risk measures. Course Note(s): This class is distinguished from EN.625.641 Mathematics of Finance: Investment Science (formerly 625.439) and EN.625.714 Introductory Stochastic Differential Equations with Applications, as follows: EN.625.641 Mathematics of Finance: Investment Science gives a broader and more general treatment of financial mathematics, and EN.625.714 Introductory Stochastic Differential Equations with Applications provides a deeper (more advanced) mathematical understanding of stochastic differential equations, with applications in both finance and non-finance areas.

Prerequisite(s): Multivariate calculus, linear algebra and matrix theory (e.g., EN.625.609 Matrix Theory), and a graduate-level course in probability and statistics (such as EN.625.603 Statistical Methods and Data Analysis).

This course will be a rigorous and extensive introduction to modern methods of time series analysis and dynamic modeling. Topics to be covered include elementary time series models, trend and seasonality, stationary processes, Hilbert space techniques, the spectral distribution function, autoregressive/ integrated/moving average (ARIMA) processes, fitting ARIMA models, forecasting, spectral analysis, the periodogram, spectral estimation techniques, multivariate time series, linear systems and optimal control, state-space models, and Kalman filtering and prediction. Additional topics may be covered if time permits. Some applications will be provided to illustrate the usefulness of the techniques. Course Note(s): This course is also offered in the Department of Applied Mathematics and Statistics (Homewood campus) as EN.553.639.

Prerequisite(s): Graduate course in probability and statistics (such as EN.625.603 Statistical Methods and Data Analysis) and familiarity with matrix theory and linear algebra.

The goal of this course is to give basic knowledge of stochastic differential equations useful for scientific and engineering modeling, guided by some problems in applications. The course treats basic theory of stochastic differential equations, including weak and strong approximation, efficient numerical methods and error estimates, the relation between stochastic differential equations and partial differential equations, Monte Carlo simulations with applications in financial mathematics, population growth models, parameter estimation, and filtering and optimal control problems. Prerequisite(s): Multivariate calculus and a graduate course in probability and statistics, as well as exposure to ordinary differential equations.

This course will focus on both the theoretical and the practical aspects of designing, training, and testing reinforcement learning systems. The course begins with an examination of Markov decision processes (MDPs), which provide a sound mathematical basis for modeling and solving complex sequential decision problems. The more traditional analytical method for solving MDPs, dynamic programming, will be reviewed. We will then examine the major reinforcement learning approaches, such as Monte Carlo methods, temporal difference methods, policy gradient methods, and deep learning methods, comparing them as appropriate to dynamic programming techniques. Fundamental issues and limitations on the performance of reinforcement learning algorithms (e.g., the credit assignment problem, the exploration / exploitation tradeoff, on-policy learning versus off-policy learning, partial observability, and algorithm convergence properties) will be examined for each approach. Weekly exercises and discussion topics will reinforce and expand on the classroom material. In addition, students will gain practical experience during a semester-long project by programming, training, and testing various reinforcement learning algorithms.

Prerequisite(s): EN.625.638/EN.605.647 - Neural Networks or experience programming artificial neural networks in a high-level language.

Modern Cryptography includes seemingly paradoxical notions such as communicating privately without a shared secret, proving things without leaking knowledge, and computing on encrypted data. In this challenging but rewarding course we will start from the basics of private and public key cryptography and go all the way up to advanced notions such as zero-knowledge proofs, functional encryption and program obfuscation. The class will focus on rigorous proofs and require mathematical maturity.

Prerequisite(s): Students may receive credit for only one of EN.600.442, EN.601.442, EN.601.642.;(EN.601.230 OR EN.601.231) AND (EN.553.310 OR EN.553.311 OR EN.553.420 OR EN.553.421)

Distribution Area: Engineering, Quantitative and Mathematical Sciences

AS Foundational Abilities: Science and Data (FA2)

Same material as 601.442, for graduate students. Modern Cryptography includes seemingly paradoxical notions such as communicating privately without a shared secret, proving things without leaking knowledge, and computing on encrypted data. In this challenging but rewarding course we will start from the basics of private and public key cryptography and go all the way up to advanced notions such as zero-knowledge proofs, functional encryption and program obfuscation. The class will focus on rigorous proofs and require mathematical maturity. Required course background: Probability & Automata/Computation Theory

Prerequisite(s): Students may receive credit for only one of EN.601.442 OR EN.601.642.

Distribution Area: Engineering, Quantitative and Mathematical Sciences

The components of a compiler appear in every software application that handles input from an external source. This course shows how the components of a compiler are built and how they fit together to extract meaning from the input and how the data flows through the compiler’s components to become useful to applications. Students will get practical experience in how to use the LLVM tools to build a complete compiler for a subset of the C++ programming language that can target almost any platform. Students will also get experience in developing a “Just In Time” component for an application that will accept code as input into the application while it is running, to be compiled and linked into the application so the application can execute it. Prerequisites: This course has no formal prerequisites, but experience with C++ is highly recommended because LLVM is written in C++, and therefore, all homework will be in C++, and this course is software homework intensive.

This course will teach the fundamentals needed to utilize the ever-increasing power of the GPUs housed in the video cards attached to our computers. For years, this capability was limited to the processing of graphics data for presentation to the user. With the CUDA and OpenCL frameworks, programmers can develop applications that harness this power directly to search, modify, and quickly analyze large amounts of various types of data. Students will be introduced to core concurrent programming principles, along with the specific hardware and software considerations of these frameworks. In addition, students will learn canonical algorithms used to perform high-precision mathematics and data transformations. Class time will be split between lectures and hands-on exercises. There will be two individual projects in both CUDA and OpenCL programming, which will allow students to independently choose demonstrable goals, develop software to achieve those goals, and present the results of their efforts.

This follow-on course to data structures (e.g., EN.605.202) provides a survey of computer algorithms, examines fundamental techniques in algorithm design and analysis, and develops problem-solving skills required in all programs of study involving computer science. Topics include advanced data structures (red-black and 2-3-4 trees, union-find), recursion and mathematical induction, algorithm analysis and computational complexity (recurrence relations, big-O notation, NP-completeness), sorting and searching, design paradigms (divide and conquer, greedy heuristic, dynamic programming, amortized analysis), and graph algorithms (depth-first and breadth-first search, connectivity, minimum spanning trees, network flow). Advanced topics are selected from among the following: randomized algorithms, information retrieval, string and pattern matching, and computational geometry. Prerequisite(s): EN.605.202 Data Structures or equivalent. EN.605.203 Discrete Mathematics or equivalent is recommended. Course Note(s): The required foundation courses may be taken in any order but must be taken before other courses in the degree. Students can only earn credit for one of EN.605.620, EN.605.621, EN.685.621 or EN.705.621

This course introduces algorithms and architectures for the analysis and processing of digital signals, taking the computer science perspective. It emphasizes computational complexity and efficiency and the design and implementation of computer algorithms for processing signals, designing digital filters, and effectively presenting and displaying information. Topics include signal analysis, discrete Fourier transform (definition, applications, and fast algorithms), convolution and correlation, spectral estimation and display, filter design, signal encoding/decoding, time-frequency analysis, Software Defined Radio (SDR), arithmetic computational complexity, and applications Background in signal processing and mathematics will be introduced as needed.

Prerequisite(s): EN.605.621 Foundations of Algorithms or equivalent background, some knowledge of complex numbers and linear algebra (vectors and matrices).

Fundamentals of image processing are covered, with an emphasis on digital techniques. Topics include digitization, enhancement, segmentation, the Fourier transform, filtering, restoration, reconstruction from projections, and image analysis including computer vision. Concepts are illustrated by laboratory sessions in which these techniques are applied to practical situations, including examples from biomedical image processing. Prerequisite(s): Familiarity with Fourier transforms.

Today an immense social media landscape is being fueled by new applications, growth of devices (e.g., Smartphones and devices), and human appetite for online engagement. With a myriad of applications and users, significant interest exists in the obvious question, “How does one better understand human behavior in these communities to improve the design and monitoring of these communities?” To address this question a multidisciplinary approach that combines social network analysis (SNA), natural language processing, and data analytics is required. This course combines all these topics to address contemporary topics such as marketing, population influence, etc. There will be several small projects. Prerequisite(s): Knowledge of Python or R; matrix algebra.

Prerequisite(s): Computer Science majors need to complete foundation requirement first.;Foundation Prerequisites for Cybersecurity Majors:EN.605.621 AND EN.695.601 AND EN.695.641

This is a foundational course in Artificial Intelligence. Although we hear a lot about machine learning, artificial intelligence is a much broader field with many different aspects. In this course, we focus on three of those aspects: reasoning, optimization, and pattern recognition. Traditionally, the first was covered under “Symbolic AI” or “Good Old Fashioned AI” and the latter two were covered under “Numeric AI” (or more specifically, “Connectionist AI” or “Machine Learning”). However, despite the many successes of machine learning algorithms, practitioners are increasingly realizing that complicated AI systems need algorithms from all three aspects. This approach falls under the ironic heading “Hybrid AI”. In this course, the foundational algorithms of AI are presented in an integrated fashion emphasizing Hybrid AI. The topics covered include state space search, local search, example based learning, model evaluation, adversarial search, constraint satisfaction problems, logic and reasoning, expert systems, rule based ML, Bayesian networks, planning, reinforcement learning, regression, logistic regression, and artificial neural networks (multi-layer perceptrons). The assignments weigh conceptual (assessments) and practical (implementations) understanding equally.

Prerequisite(s): EN.605.621 Foundations of Algorithms. A working knowledge of Python programming is assumed as all assignments are completed in Python.

This course surveys the principal difficulties of working with written language data, the fundamental techniques that are used in processing natural language, and the core applications of NLP technology. Topics covered in the course include language modeling, text classification, labeling sequential data (tagging), parsing, information extraction, question answering, machine translation, and semantics. The dominant paradigm in contemporary NLP uses supervised machine learning to train models based on either probability theory or deep neural networks. Both formalisms will be covered. A practical approach is emphasized in the course, and students will write programs and use open source toolkits to solve a variety of problems. Course prerequisite(s): There are no formal prerequisite courses, although having taken any of EN.605.649 Introduction to Machine Learning, EN.605.744 Information Retrieval, or EN.605.645 Artificial Intelligence is helpful. Course note(s): A working knowledge of Python is assumed. While some of the assigned exercises can be done in any programming language, we will sometimes provide example code in Python, and many of the labs are best solved in Python.Course note(s): A working knowledge of Python is assumed. While some of the assigned exercises can be done in any programming language, we will sometimes provide example code in Python, and many of the labs are best solved in Python.

This course provides an introduction to concepts in neural networks and connectionist models. Topics include parallel distributed processing, learning algorithms, and applications. Specific networks discussed include Hopfield networks, bidirectional associative memories, perceptrons, feedforward networks with back propagation, and competitive learning networks, including self-organizing and Grossberg networks. Software for some networks is provided. Prerequisite(s): Multivariate calculus and linear algebra.

Analyzing large data sets (“Big Data”), is an increasingly important skill set. One of the disciplines being relied upon for such analysis is machine learning. In this course, we will approach machine learning from a practitioner’s perspective. We will examine the issues that impact our ability to learn good models (e.g., inductive bias, the curse of dimensionality, the bias-variance dilemma, and no free lunch). We will then examine a variety of approaches to learning models, covering the spectrum from unsupervised to supervised learning, as well as parametric versus non-parametric methods. Students will explore and implement several learning algorithms, including logistic regression, nearest neighbor, decision trees, and feed-forward neural networks, and will incorporate strategies for addressing the issues impacting performance (e.g., regularization, clustering, and dimensionality reduction). In addition, students will engage in online discussions, focusing on the key questions in developing learning systems. At the end of this course, students will be able to implement and apply a variety of machine learning methods to real-world problems, as well as be able to assess the performance of these algorithms on different types of data sets. Prerequisite(s): EN.605.202 – Data Structures or equivalent.

Prerequisite(s): EN.605.202 – Data Structures or equivalent, EN.605.621 – Foundations of Algorithms or EN.685.621 – Algorithms for Data Science or 705.621 – Introduction to Algorithms

This course explores the underlying theory and practical concepts in creating visual representations of large amounts of data. It covers the core topics in data visualization: data representation, visualization toolkits, scientific visualization, medical visualization, information visualization, flow visualization, and volume rendering techniques. The related topics of applied human perception and advanced display devices are also introduced. Prerequisite(s): Experience with data collection/analysis in data-intensive fields or background in computer graphics (e.g., EN.605.667 Computer Graphics) is recommended.

This course examines the principles of computer graphics, with a focus on the mathematics and theory behind 2D and 3D graphics rendering. Topics include graphics display devices, graphics primitives, 2D and 3D transformations, viewing and projection, color theory, visible surface detection and hidden surface removal, lighting and shading, and object definition and storage methods. Practical application of these concepts is emphasized through laboratory exercises and code examples. Laboratory exercises use the C++ programming language and OpenGL on a PC. Prerequisite(s): Familiarity with linear algebra.

This course provides an introduction to the field of data communications and computer networks. It covers the principles of data communications, the fundamentals of signaling, basic transmission concepts, transmission media, circuit control, line sharing techniques, physical and data link layer protocols, error detection and correction, data compression, network security techniques and protocols, common carrier services and data networks, the mathematical techniques used for network design and performance analysis, Ethernet and Wi-Fi local area networks, and the TCP/IP-based Internet. Potential topics include analog and digital signaling; data encoding and modulation; Shannon channel capacity; synchronous and asynchronously transmission; RS232 physical layer interface standards; FDM, TDM, and STDM multiplexing techniques; inverse multiplexing; analog and digital transmission; V series modem standards; PCM encoding and T1 transmission circuits; LRC, VRC, and CRC error detection techniques; Hamming and Viterbi forward error correction techniques; character and bit-oriented protocols, information transparency, and BSC and HDLC data link layer protocols; Huffman, MNP5, and Lempel-Ziv-Welch data compression algorithms; circuit, message, packet, and cell switching techniques; public key and symmetric encryption algorithms, authentication, digital signature, and message digest techniques, secure e-mail, PGP, TLS/SSL, Kerberos, and IPsec security algorithms; reliability, availability, and queuing analysis performance techniques; Ethernet, Fast Ethernet, Gigabit and 10 Gigabit Ethernet LANs; Wi-Fi 4, Wi-Fi 5, Wi-Fi 6, and Wi-Fi 7 LANs; IPv4 and IPv6 network layer protocols, RIP, OSPF, and BGP4 routing protocols; and TCP and UDP transport layer protocols.

This multi-disciplinary course focuses on the application of modeling and simulation principles to complex systems. A complex system is a large-scale nonlinear system consisting of interconnected or interwoven parts (such as a biological organism, an ecological system, the economy, fluids or strongly-coupled solids). The subject is interdisciplinary with foundations in mathematics, nonlinear science, numerical simulations and statistical physics. The course begins with an overview of complex systems, followed by modeling techniques based on nonlinear differential equations, networks, and stochastic models. Simulations are conducted via numerical calculus, analog circuits, Monte Carlo methods, and cellular automata. In the course we will model, program, and analyze a wide variety of complex systems, including dynamical and chaotic systems, cellular automata, and iterated functions. By defining and iterating an individual course project throughout the term, students will gain hands-on experience and understanding of complex systems that arise from combinations of elementary rules. Students will be able to define, solve, and plot systems of linear and non-linear systems of differential equations and model various complex systems important in applications of population biology, epidemiology, circuit theory, fluid mechanics, and statistical physics.Course prerequisite(s): Knowledge of elementary probability and statistics and previous exposure to differential equations. Students applying this course to the MS in Bioinformatics should also have completed at least one Bioinformatics course prior to enrollment.Course note(s): This course may be counted toward a three-course concentration in Bioinformatics.

This course covers fundamental algorithms for efficiently solving geometric problems, especially ones involving 2D polygons and 3D polyhedrons. Topics include elementary geometric operations; polygon visibility, triangulation, and partitioning; computing convex hulls; proximity searching, Voronoi diagrams, and Delaunay triangulations with applications; special polygon and polyhedron algorithms such as point containment and extreme point determination; point location in a planar graph subdivision; dimension reduction in data; and robot motion planning around polygon obstacles. Applications to such areas as computer graphics, big data analytics and pattern recognition, geometric databases, numerical taxonomy, and robotics will be addressed. The course covers theory to the extent that it aids in understanding how the algorithms work. Emphasis is placed on algorithm design and implementation. Programming projects are an important part of the coursework. Prerequisite(s): Foundations of algorithms. Some familiarity with linear algebra.

Prerequisite(s): EN.605.621 Foundations of Algorithms. Some familiarity with linear algebra.

Quantum computing is no longer a purely theoretical notion. The NSA and NIST are preparing to transition to quantum resistant cryptography. We have now entered the intermediate-scale quantum era, with near-term applications such as quantum machine learning being explored. Scalable quantum computers aren't here yet, but ongoing developments suggest they are on their way. This course provides an introduction to quantum computation for computer scientists: the focus is on algorithms rather than physical devices, and familiarity with quantum mechanics (or any physics at all) is not a prerequisite. Instead, pertinent aspects of the quantum mechanics formalism are developed as needed in class. The course begins with an introduction to the QM formalism. It then develops the abstract model of a quantum computer, and discusses how quantum algorithms enable us to achieve, for some problems, a significant speedup (in some cases an exponential speedup) over any known classical algorithm. This discussion provides the basis for a detailed examination of quantum integer factoring, quantum search, and other quantum algorithms. The course also explores quantum error correction, quantum teleportation, and quantum cryptography. It concludes with a glimpse at what the cryptographic landscape will look like in a world with quantum computers. Required work includes problem sets and a research project. Prerequisites: Some familiarity with linear algebra and with the design and analysis of algorithms or instructor permission.

All science requires mathematics. Formal methods used in developing computer systems are mathematically based techniques for describing system properties. These formal methods then can provide frameworks within which developers can specify, develop, verify, and prove systems in a systematic, rather than ad hoc manner. According to some researchers, the application of formal specification and verification methods could avoid disasters such as Heartbleed bug, Ariane 5 rocket explosion, and Therac-25 radiation therapy machine harms. This course is an introduction to the vast world of formal methods. The course starts with review of propositional logic, predicate logic, and covers set theoretic specification methods via Z, temporal specification via PTL, grammars, and logic based methods via Caml and Coq proof assistant. Each student will carry out an investigation of an existing formal verification system, applying it to a suitable problem of the student's choice. Among possible projects will be the formal verification of problem solutions such as designing a semaphore, designing a machine learning algorithm, a web interface, a test suite, a sophisticated data structure, or a theorem.

This course provides a practical introduction to deep neural networks (DNN) with the goal to extend student’s understanding of the latest and cutting-edge technology and concepts in deep learning (DL) field. DNNs are simplified representation of neurons in the brain that are suited in complex applications, such as natural language processing (NLP), computer vision (CV), speech processing, and many other predictive models rising from non-linear and unstructured data, including text, images, video, audio. The course starts with a brief review of machine learning (ML) and neural networks (NN), including anatomy of neural networks, model evaluation techniques and feature engineering in Python with TensorFlow (TF) and Keras. It then defines and exemplifies the deep learning with convolutional neural networks (CNN), recurrent neural networks (RNN), long-short term memory (LSTM) networks with attention mechanism, generative adversarial networks (GAN) and deep reinforcement learning (DRL), and transfer learning among other key concepts. This is a hands-on course with significant Python coding requirements. Students will apply neural networks to the computer vision (CV) tasks, natural language processing (NLP) tasks, and domains with structured data. Since DL is a rapidly developing field, the course will also rely on recent seminal publications, which students may be asked to reproduce with small scale datasets as an exercise. Prerequisites: Multivariate calculus, linear algebra, probability/statistics; A neural network OR machine learning course: Examples: EN.605.647, EN.625.638, EN.525.670,EN.605.649, EN.705.601, EN.605.646. A working knowledge of Python is assumed. Prior coding experience data munging, numerical linear algebra, ML, and visualization libraries is highly recommended: Example: Python, Numpy, Pandas, ScikitLearn, Matplotlib.

Prerequisite(s): A course in Machine Learning

Systems biology is the study of complex biological systems using theoretical, mathematical, and computational tools and concepts. The advent of genomics, big data, and highpowered computing is allowing better understanding and elucidation of these systems. Central to systems biology is the development of computational models, based on sound statistics, which incorporate biological structures and networks, and can be informed and tested, with data on multiple scales. In this class, students will learn to develop and use different types of models of complex biological systems and how to test and perturb them. Students will learn basic biological system components and dynamics, as well as the data formats, sources, and modeling tools required to interrogate them. Tools will be used relating to functional genomics, evolution, biochemical systems, and cell biology. Students will utilize a model they have developed and available data from public repositories to investigate both a discovery-based project and a hypothesis based project. Prerequisite(s): Courses in molecular biology (EN.605.205 Molecular Biology for Computer Scientists or AS.410.602 Molecular Biology) and differential equations.Prerequisite(s): Courses in molecular biology (EN.605.205 Molecular Biology for Computer Scientists or AS.410.602 Molecular Biology) and differential equations.

Prerequisite(s): Courses in molecular biology (EN.605.205 Molecular Biology for Computer Scientists or AS.410.602 Molecular Biology) and differential equations.

The age of Cyber-Physical Systems (CPS) has officially begun. Not long ago, these systems were separated into distinct domains, cyber and physical. Today, the rigid dichotomy between domains no longer exists. Cars have programmable interfaces, Unmanned Aerial Vehicles (UAVs) roam the skies, and critical infrastructure and medical devices are now fully reliant on computer control. With the increased use of CPS and the parallel rise in cyber-attack capabilities, it is imperative that new methods for securing these systems be developed. This course will investigate key concepts behind CPS including: control systems, protocol analysis, behavioral modeling, and Intrusion Detection System (IDS) development. The course will be comprised of theory, computation, and projects to better enhance student learning and engagement . The course will begin with the mathematics of continuous and digital control systems and then shift the focus to the complex world of CPS, where both a general overview for the different domains (Industrial Control, Transportation, Medical Devices, etc.) and more detailed case studies will be provided . Students will complete a number of projects, both exploiting security vulnerabilities and developing security solutions for UAVs and industrial controllers. Several advanced topics will be introduced including behavioral analysis and resilient CPS.Course Notes: There are no prerequisite courses; however, students will encounter many concepts and technologies in a short period of time. Student should have a basic understanding of python programming, networking, matrices, and Windows and Linux operating systems.

This course provides an introduction to the principles and practice of contemporary cryptography. It begins with a brief survey of classical cryptographic techniques that influenced the modern development of the subject. The course then focuses on more contemporary work: symmetric block ciphers and the Advanced Encryption Standard, public key cryptosystems, digital signatures, authentication protocols, and cryptographic hash functions. The course also provides an overview of quantum resistant cryptography and, as time permits, other recent developments such as homomorphic encryption. Complexity theory and computational number theory provide the foundation for much of the contemporary work in cryptology; pertinent ideas from complexity and number theory are introduced, as needed, throughout the course.

This course offers an in-depth journey through the algorithmic concepts vital for mastering the intricacies of data science. It begins with an intensive examination of algorithm analysis, with a special focus on understanding the runtime complexities essential for addressing real-world data problems. The curriculum encompasses thorough training in data preprocessing, along with foundational knowledge in probability and statistics, equipping students to proficiently clean and interpret data. The course introduces key mathematical transformations such as Eigen decomposition, FFT, DCT, and Wavelets. These tools are crucial for unearthing underlying patterns in data by creating innovative feature spaces. Students will explore a seamless blend of diverse algorithm types, including intelligent algorithms, statistical algorithms, optimization algorithms, graph algorithms, and learning algorithms. This comprehensive approach, enriched with optimization techniques, forms a holistic toolkit for the contemporary Data Scientist. Moving beyond theoretical concepts, the course delves into practical aspects of analysis, visualization, and understanding of complexity classes. Occasional forays into algorithmic proofs enhance the theoretical grounding of students, bridging theory with practical application. The course culminates in modules focused on data modeling and visualization, enabling students to adeptly apply algorithmic techniques to produce insightful and meaningful data representations. Upon completing this course, students will be thoroughly equipped with both practical and theoretical algorithmic strategies, preparing them to confidently address a wide array of challenges in the data science field. Students can only earn credit for one of EN.605.620, EN.605.621, or EN.685.621.

This course will cover the core concepts and skills in the interdisciplinary field of data science. These include problem identification and communication, probability, statistical inference, visualization, extract/transform/load (ETL), exploratory data analysis (EDA), linear and logistic regression, model evaluation and various machine learning algorithms such as random forests, k-means clustering, and association rules. The course recognizes that although data science uses machine learning techniques, it is not synonymous with machine learning. The course emphasizes an understanding of both data (through the use of systems theory, probability, and simulation) and algorithms (through the use of synthetic and real data sets). The guiding principles throughout are communication and reproducibility. The course is geared towards giving students direct experience in solving the programming and analytical challenges associated with data science. The assignments weight conceptual (assessments) and practical (labs, problem sets) understanding equally. Prerequisite(s): A working knowledge of Python scripting and SQL is assumed as all assignments are completed in Python.

Prerequisite(s): EN.685.652 Data Engineering Principles and Practice or equivalent course.

This course provides a background in engineering electromagnetics required for more advanced courses in the field. Topics include vector calculus, Poisson’s and Laplace’s equations, Vector potentials, Green’s functions, magnetostatics, magnetic and dielectric materials, Maxwell’s equations, plane wave propagation and polarization, reflection and refraction at a plane boundary, frequency-dependent susceptibility functions, transmission lines, waveguides, and simple antennas. Practical examples are used throughout the course.

This course provides a foundation in the theory and applications of probability and stochastic processes and an understanding of the mathematical techniques relating to random processes in the areas of signal processing, detection, estimation, and communication. Topics include the axioms of probability, random variables, and distribution functions; functions and sequences of random variables; stochastic processes; and representations of random processes. Prerequisite(s): A working knowledge of multi-variable calculus, Fourier transforms, and linear systems theory.

In this course, students receive an introduction to the principles, performance and applications of communication systems. Students examine analog modulation/demodulation systems (amplitude - AM, DSB & SSB; and angle - PM & FM) and digital modulation/demodulation systems (binary and M-ary) in noise and interference. Sub-topics include filtering, sampling, quantization, encoding and the comparison of coherent & noncoherent detection techniques to improve signal-to-noise ratio (SNR) and bit error rate (BER) performance. Special topics and/or problems will be assigned that provide knowledge of how communication systems work from a system engineering viewpoint in real-world environments. Prerequisite(s): A working knowledge of Fourier transforms, linear systems, and probability theory. Basic working knowledge of MATLAB.

This course reviews electromagnetic theory and introduces the interaction of light and matter with an emphasis on laser theory. A fundamental background is established, necessary for advanced courses in optical engineering. Topics include Maxwell’s equations, total power law, introduction to spectroscopy, classical oscillator model, Kramers-Kroenig relations, line broadening mechanisms, rate equations, laser pumping and population inversion, laser amplification, laser resonator design, and Gaussian beam propagation.

Prerequisite(s): EN.525.605 Intermediate Electromagnetics or equivalent.

This course examines fundamental principles and applications of Digital Signal Processing. Introductory topics include linear, time-invariant systems, discrete-time convolution, and frequency-domain representations of discrete-time signals and systems. Sampling and quantization of continuous-time signals are covered. The Discrete Fourier Transform and efficient algorithms for its computation are studied in detail. The z-transform and its application to linear discrete-time systems analysis is studied. The design of digital filters using the windowing, equiripple, impulse invariance, and bilinear transformation methods is treated, along with the implementation of digital filter difference equations using canonical structures. MATLAB is utilized to demonstrate and implement Digital Signal Processing techniques.Prerequisite(s): A working knowledge of linear systems and Fourier analysis. Familiarity with MATLAB.

This course explores the use of satellite, terrestrial, celestial, radio, magnetic, and inertial systems for the real-time determination of position, velocity, acceleration, and attitude. Particular emphasis is on the historical importance of navigation systems; avionics navigation systems for high performance aircraft; the Global Positioning System; the relationships between navigation, cartography, surveying, and astronomy; and emerging trends for integrating various navigation techniques into single, tightly coupled systems.

This hardware-supplemented course covers the guidance, navigation- and control principles common to many small fixed-wing and multirotor unmanned aerial vehicles (UAVs). Building on classical control systems and modeling theory, students will learn how to mathematically model UAV flight characteristics and sensors, develop and tune feedback control autopilot algorithms to enable stable flight control, and fuse sensor measurements using extended Kalman filter techniques to estimate the UAV position and orientation. Students will realize these concepts through both simulation and interaction with actual UAV hardware. Throughout the course, students will build a full 6-degree-of-freedom simulation of controlled UAV flight using MATLAB and Simulink. Furthermore, students will reinforce their UAV flight control knowledge by experimenting with tuning and flying actual open-source quadrotor UAVs. Prerequisite(s): Background in control systems (e.g., EN.525.609 Continuous Control Systems) and matrix theory along with a working knowledge of MATLAB. Experience using Simulink is desired. Existing familiarity with C programming language, electronics, and microcontrollers will be helpful but is not required.

This course will cover machine perception with a focus on computer vision (i.e., feature detection, stereovision, structure from motion, deep learning object detection) as the primary use case. Additional sensor modalities will be addressed (i.e., radar, lidar) along with data fusion (i.e., Kalman filtering, target tracking) in order to provide a broad understanding of multi-modality machine perception.

This course presents error-control coding with a view toward applying it as part of the overall design of a data communication or storage and retrieval system. Block, trellis, and turbo codes and associated decoding techniques are covered. Topics include system models, generator and parity check matrix representation of block codes, general decoding principles, cyclic codes, an introduction to abstract algebra and Galois fields, BCH and Reed-Solomon codes, analytical and graphical representation of convolutional codes, performance bounds, examples of good codes, Viterbi decoding, BCJR algorithm, turbo codes, and turbo code decoding.

Prerequisite(s): Background in linear algebra, such as EN.625.609 Matrix Theory; in probability, such as EN.525.614 Probability and Stochastic Processes for Engineers; and in digital communications, such as EN.525.616 Communication Systems Engineering. Familiarity with MATLAB or similar programming capability.

The fundamentals of statistical signal processing are presented in this course. Topics include matrix factorizations and least squares filtering, optimal linear filter theory, classical and modern spectral estimation, adaptive filters, and optimal processing of spatial arrays.

Prerequisite(s): EN.525.614 Probability and Stochastic Processes for Engineers, EN.525.627 Digital Signal Processing, linear algebra, and familiarity with a scientific programming language such as MATLAB.

This course focuses on the underlying principles of pattern recognition and on the methods of machine intelligence used to develop and deploy pattern recognition applications in the real world. Emphasis is placed on the pattern recognition application development process, which includes problem identification, concept development, algorithm selection, system integration, and test and validation. Machine intelligence algorithms to be presented include feature extraction and selection, parametric and non-parametric pattern detection and classification, clustering, artificial neural networks, support vector machines, rule-based algorithms, fuzzy logic, genetic algorithms, and others. Case studies drawn from actual machine intelligence applications will be used to illustrate how methods such as pattern detection and classification, signal taxonomy, machine vision, anomaly detection, data mining, and data fusion are applied in realistic problem environments. Students will use the MATLAB programming language and the data from these case studies to build and test their own prototype solutions.

Prerequisite(s): EN.525.614 Probability and Stochastic Processes for Engineers or equivalent. A course in digital signal or imageprocessing is recommended, such as EN.525.627 Digital Signal Processing, EN.525.619 Introduction to Digital Image and Video Processing, 525.643 Real-Time Computer Vision, or 525.746 Image Engineering.

This is an introductory course on wavelet analysis, with an emphasis on the fundamental mathematical principles and basic algorithms. We cover the mathematics of signal (function) spaces, orthonormal bases, frames, time-frequency localization, the windowed Fourier transform, the continuous wavelet transform, discrete wavelets, orthogonal and biorthogonal wavelets of compact support, wavelet regularity, and wavelet packets. It is designed as a broad introduction to wavelets for engineers, mathematicians, and physicists.Prerequisite: Competence with multivariable calculus, linear algebra, and a scientific programming language is required, as well as familiarity with Fourier transforms and signal processing fundamentals such as the discrete Fourier transform, convolutions, and correlations.

Intelligent algorithms are, in many cases, practical alternative techniques for tackling and solving a variety of challenging engineering problems. For example, fuzzy control techniques can be used to construct nonlinear controllers via the use of heuristic information when information on the physical system is limited. Such heuristic information may come, for instance, from an operator who has acted as a "human-in-the-loop" controller for the process. This course investigates several concepts and techniques commonly referred to as intelligent algorithms; discusses the underlying theory of these methodologies when appropriate; and takes an engineering perspective and approach to the design, analysis, evaluation, and implementation of Intelligent Systems. Fuzzy systems, genetic algorithms, particle swarm and ant colony optimization techniques, and neural networks are the primary concepts discussed in this course, and several engineering applications are presented along the way. Expert (rule-based) systems are also discussed within the context of fuzzy systems. An intelligent algorithms research paper must be selected from the existing literature, implemented by the student, and presented as a final project. Prerequisite(s): Student familiarity of system-theoretic concepts is desirable.

Information theory concerns the fundamental limits for data compressibility and the rate at which data may be reliably communicated over a noisy channel. Course topics include measures of information, entropy, mutual information, Markov chains, source coding theorem, data compression, noisy channel coding theorem, error-correcting codes, and bounds on the performance of communication systems. Classroom discussion and homework assignments will emphasize fundamental concepts, and advanced topics and practical applications (e.g., industry standards, gambling/finance, machine learning) will be explored in group and individual research projects.

Prerequisite(s): EN.525.614 Probability and Stochastic Processes for Engineers or equivalent.

The fundamental concepts of multidimensional digital signal processing theory as well as several associated application areas are covered in this course. The course begins with an investigation of continuous-space signals and sampling theory in two or more dimensions. The multidimensional discrete Fourier transform is defined, and methods for its efficient calculation are discussed. The design and implementation of two-dimensional non-recursive linear filters are treated. The final part of the course examines the processing of signals carried by propagating waves. This section contains descriptions of computed tomography and related techniques and array signal processing. Several application oriented software projects are required.

Prerequisite(s): EN.525.614 Probability and Stochastic Processes for Engineers and EN.525.627 Digital Signal Processing or equivalents. Knowledge of linear algebra and MATLAB is helpful.

This course introduces statistical analyses and techniques of experimental design appropriate for use in environmental applications. The methods taught in this course allow the experimenter to discriminate between real effects and experimental error in systems that are inherently noisy. Statistically designed experimental programs typically test many variables simultaneously and are very efficient tools for developing empirical mathematical models that accurately describe physical and chemical processes. They are readily applied to production plant, pilot plant, and laboratory systems. Topics covered include fundamental statistics; the statistical basis for recognizing real effects in noisy data; statistical tests and reference distributions; analysis of variance; construction, application, and analysis of factorial and fractional-factorial designs; screening designs; response surface and optimization methods; and applications to pilot plant and waste treatment operations. Particular emphasis is placed on analysis of variance, prediction intervals, and control charting for determining statistical significance as currently required by federal regulations for environmental monitoring.Prerequisite: Undergraduate statistics is strongly recommended

This course is an introduction to decision support models used in environmental planning and management. We will develop and apply analytical methods and mathematical models that help decision makers solve complex environmental and socio-economic problems and formulate associated policies. There is a focus on real-world problems in the public sector, including urban facility location, transportation planning, water resources management, biological conservation, and landscape resources management. You will learn how to structure and analyze problems and formulate optimization models to make the most of limited resources and achieve specified objectives related to efficiency, cost-effectiveness, environmental protection, public health, and fairness to stakeholder groups and to the public. The types of models covered in this course are linear programming, integer programming, and multi-objective models. Algorithmic solution methods are reviewed, and computer-based solution methods are applied in the context of a course project. Prerequisite: pre-calculus mathematics including algebra with multiple variables.

A development of stochastic processes with substantial emphasis on the processes, concepts, and methods useful in mathematical finance. Relevant concepts from probability theory, particularly conditional probability and conditional expection, will be briefly reviewed. Important concepts in stochastic processes will be introduced in the simpler setting of discrete-time processes, including random walks, Markov chains, and discrete-time martingales, then used to motivate more advanced material. Most of the course will concentrate on continuous-time stochastic processes, particularly martingales, Brownian motion, diffusions, and basic tools of stochastic calculus. Examples will focus on applications in finance, economics, business, and actuarial science.

This is the key introductory course for the financial mathematics program and introduces the major topics of investment finance. The investment universe, its context of markets, and the flow of global capital are introduced. Details of equities, interest, bonds, commodities, forwards, futures, and derivatives are introduced to varying degree. The concepts of deterministic cash flow stream, valuation, term structure theories, risk, and single- and multi-period random cash flows are presented. Here the neoclassical theory of finance is introduced including the topics of efficient markets, the risk-return twins leading to the mean variance Capital Asset Pricing Model (CAPM), the efficient frontier, the intertemporal models, and Arbitrage Pricing Theory (APT). Some introductory models of asset dynamics (including the binomial model), basic options theory, and elements of hedging are also included in this course. Course Note(s): This course is the same as EN.553.642 offered by through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

This is the first of a two-course sequence devoted to the mathematical modeling of securities and the markets in which they are created and exchanged. The basic cash, hybrid, and derivative instruments are reviewed and set in a rigorous mathematical context. This includes equities, bonds, options, forwards, futures, and swaps, as well as their dealer, overthe-counter, and exchange environment. Models of the term structure of interest rates, spot rates and, the forward rate curve are treated; derived from cash instruments (e.g., bonds and interest rates like LIBOR) as well as from derivatives (such as Eurodollar futures and swaps). Principles of static, discrete, continuous and dynamic probabilistic models for derivative analysis (including the Weiner process, Ito’s Lemma, and an introduction to risk-neutral valuation) are applied to develop the binomial tree approach to option valuation, the Black-ScholesMerton differential equation, and the Black-Scholes formulas for option pricing. Course Note(s): This course is the same as EN.553.644 offered by through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

This is the second of a two-course sequence devoted to the mathematical modeling of securities and the markets in which they are created and exchanged. Focus turns to interest rate derivatives and the credit markets. The martingale approach to risk-neutral valuation is covered, followed by interest rate derivatives and models of the short rate process (including Heath, Jarrow & Morton and the Libor Market Model); analysis of bonds with embedded options and other interest rate derivatives (e.g., caps, floors, swaptions). Credit risk and credit derivatives, including copula models of time to default, credit default swaps, and a brief introduction to collateralized debt obligations will be covered. A major component of this course is computational methods. This includes data and time series analysis (e.g., estimation of volatilities), developing binomial and trinomial lattices and derivative analysis schemes, and numerical approaches to solving the partial differential equations of derivatives. Course Note(s): This course is the same as EN.553.645 offered through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

Prerequisite(s): EN.555.644 Introduction to Financial Derivatives

This course applies advanced mathematical techniques to the measurement, analysis, and management of risk. The focus is on financial risk. Sources of risk for financial instruments (e.g., market risk, interest rate risk, credit risk) are analyzed; models for these risk factors are studied and the limitation, shortcomings, and compensatory techniques are addressed. Throughout the course, the environment for risk is considered, be it regulatory or social (e.g., Basel capital accords). A major component of the course are the Value at Risk (VaR) and Conditional VaR measures for market risk in trading operations, including approaches for calculating and aggregating VaR, testing VaR, VaR-driven capital for market risk, and limitations of the VaR-based approach. Asset Liability Management (ALM), where liquidity risk as well as market risk can affect the balance sheet, is analyzed. Here, models for interest rate, spread, and volatility risks are applied to quantify this exposure. Another major component of the course is credit risk. Sources of credit risk, how measured risk is used to manage exposure, credit derivatives, techniques for measuring default exposure for a single facility (including discriminant analysis and Mertonbased simulation), portfolio risk aggregation approaches (including covariance, actuarial, Merton-based simulation, macro-economic default model, and the macro-economic cashflow model - for structured and project finance). Finally, there is a brief introduction to concepts and tools that remain valid for large and extreme price moves, including the theory of copulas and their empirical testing and calibration. Course Note(s): This course is the same as EN.553.646 offered through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

This course focuses on modern quantitative portfolio theory, models, and analysis. Topics include intertemporal approaches to modeling and optimizing asset selection and asset allocation; benchmarks (indexes), performance assessment (including Sharpe, Treynor, and Jenson ratios) and performance attribution; immunization theorems; alpha-beta separation in management, performance measurement, and attribution; Replicating Benchmark Index (RBI) strategies using cash securities/derivatives; Liability-Driven Investment (LDI); and the taxonomy and techniques of strategies for traditional management (Passive, Quasi-Passive [Indexing] Semi-Active [Immunization & Dedicated] Active [Scenario, Relative Value, Total Return and Optimization]). In addition, risk management and hedging techniques are also addressed. Course Note(s): This course is the same as EN.553.647 offered through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

This course focuses on structured securities and the structuring of aggregates of financial instruments into engineered solutions of problems in capital finance. Topics include the fundamentals of creating asset-backed and structured securities—including mortgage-backed securities (MBS), stripped securities, collateralized mortgage obligations (CMOs), and other asset-backed collateralized debt obligations (CDOs)— structuring and allocating cash-flows as well as enhancing credit; equity hybrids and convertible instruments; asset swaps, credit derivatives, and total return swaps; assessment of structure-risk interest rate-risk and credit-risk as well as strategies for hedging these exposures; managing portfolios of structured securities; and relative value analysis (including OAS and scenario analysis). Course Note(s): This course is the same as EN.553.648 offered through the full-time Applied Mathematics & Statistics department for the residence Master of Science in Engineering in Financial Mathematics.

Mathematics is so much more that simply the language of science, or a set of techniques for solving quantitative-based problems. In fact, it is not a science at all, but an art, a construct of the imagination that not only provides structure to the reality of the world, but also gives form to anything and everything we can possibly imagine. Many of its fundamental principles and methods of employment are shared by artists of all types, from musicians to painters, sculptors, and poets. In this First-Year Seminar, we will explore these principles and methods shared by mathematicians and artists, like the notions of abstraction, metaphor, and pattern, the aesthetic quality both mathematicians and artists give to their work, the geometry of representation and visualization, the imagination as a tool of discovery and structure, and the use of mathematics in art, as well as the use of art in mathematics. Along the way, we will talk to artists and mathematicians, and hopefully visit the studios and galleries of each.

Distribution Area: Quantitative and Mathematical Sciences

This First-Year Seminar is designed for students of all backgrounds to provide a mathematical introduction to social choice theory, weighted voting systems, apportionment methods, and gerrymandering. In the search for ideal ways to make certain kinds of political decisions, a lot of wasted effort could be averted if mathematics could determine that finding such an ideal were actually possible in the first place. The seminar will analyze data from recent US elections as well as provide historical context to modern discussions in politics, culminating in a mathematical analysis of the US Electoral College. Case studies, future implications, and comparisons to other governing bodies outside the US will be used to apply the theory of the course. Students will use Microsoft Excel to analyze data sets. There are no mathematical prerequisites for this course.

The course is an introduction to methods in harmonic analysis, in particular Fourier series, Fourier integrals, and wavelets. These methods will be introduced rigorously, together with their motivations and applications to the analysis of basic partial differential equations and integral kernels, signal processing, inverse problems, and statistical/machine learning.

Prerequisite(s): (AS.110.201 OR AS.110.212 OR EN.553.291) AND (AS.110.202 OR AS.110.211) AND (AS.110.405 OR AS.110.415)

Distribution Area: Quantitative and Mathematical Sciences

AS Foundational Abilities: Science and Data (FA2)

We will cover several topics in the mathematical and computational foundations of Data Science. The emphasis is on fundamental mathematical ideas (basic functional analysis, reproducing kernel Hilbert spaces, concentration inequalities, uniform central limit theorems), basic statistical modeling techniques (e.g. linear regression, parametric and non-parametric methods), basic machine learning techniques for unsupervised (e.g. clustering, manifold learning), supervised (classification, regression), and semi-supervised learning, and corresponding computational aspects (linear algebra, basic linear and nonlinear optimization to attack the problems above). Applications will include statistical signal processing, imaging, inverse problems, graph processing, and problems at the intersection of statistics/machine learning and physical/dynamical systems (e.g. model reduction for stochastic dynamical systems).

Distribution Area: Quantitative and Mathematical Sciences

AS Foundational Abilities: Science and Data (FA2)

This course is an introduction to stochastic differential equations and applications. Basic topics to be reviewed include Ito and Stratonovich integrals, Ito formula, SDEs and their integration. The course will focus on diffusion processes and diffusion theory, with topics include Markov properties, generator, Kolmogrov’s equations (Fokker-Planck equation), Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will briefly introduce applications, with topics include statistical inference of SDEs, filtering and control.

Distribution Area: Quantitative and Mathematical Sciences

This course prepares the student to solve practical engineering flow problems and concentrates on the kinematics and dynamics of viscous fluid flows. Topics include the control volume and differential formulations of the conservation laws, including the Navier-Stokes equations. Students examine vorticity and circulation, dynamic similarity, and laminar and turbulent flows. The student is exposed to analytical techniques and experimental methods, and the course includes an introduction to computational methods in fluid dynamics. It also includes a programming project to develop a numerical solution to a practical fluid flow problem. Prerequisite(s): An undergraduate fluid mechanics course.

This is a three-branch course covering theory, implementation, and application of computational fluid dynamics (CFD). The theory side covers the basics of CFD, finite volume discretization schemes, time integration, solution of systems of equations, boundary conditions, error analysis, turbulence models, and meshing. On the implementation side students will implement a number of small-scale CFD solvers and pre-processing tools in order to get a working knowledge of the simulation process. The application side covers the use of fully featured, readily available CFD solver to study an array of gradually complex flow phenomena.

This course covers machine learning fundamentals (e.g., optimization, perceptron, and universal approximation), some popular and advanced machine learning techniques (e.g., Supervised, Unsupervised, Probabilistic, Convolutional, and Generative Networks), and supercomputing techniques (with a focus on MARCC) to address mechanical engineering-related machine learning problems. The course requires Python 3+ programming skills; a free 3-hour Python 3+ tutorial will be provided to those who need to learn Python.

This course is an introduction to the mathematical derivation, behavioral insight into and control of the dynamics of aerospace vehicles. The course will cover current vehicles of interest ranging from small unmanned aircraft, to hypersonic aircraft and spacecraft in earth orbit. Starting from first principles in vector math and conservation of linear and angular momentum in inertial and non-inertial (rotating) coordinate systems we will develop the fundamental equations of motion that describe the flight of these vehicles. Because understanding is best achieved through hands on experience students will develop and implement the necessary vector math, transformations, earth environment models and rigid body dynamics in MATLAB; the models you develop will directly parallel and follow the progression of the course ultimately realizing a full nonlinear 6-degree-of-freedom simulation of an aircraft that we will use to investigate and understand the nature of their dynamic motion and to discover and implement control systems to change and improve their natural dynamic response.