AMS 691  Topics in Applied Mathematics AND DATA SCIENCE

 

Varying topics selected from the list below if sufficient interest is shown. Several topics may be taught concurrently in different sections: Advanced Operational Methods in Applied Mathematics, Approximate Methods in Boundary Value Problems in Applied Mathematics, Control Theory and Optimization Foundations of Passive Systems Theory, Game Theory, Mixed Boundary Value Problems in Elasticity, Partial Differential Equations, Quantitative Genetics, Stochastic Modeling, Topics in Quantitative Finance.

********************************

FALL 2025:

>AMS 691.01:  Recent Progress in AI/ML: Applications, Architectures, and Systems

This course will cover recent progress in AI/ML in applications, architectures, and systems. The course will be self-contained as much as possible. If you are unsure about your background, please send the instructor an email inquiry with your background.

3 credits

No course materials required

Topics (subject to change):
*Overview of recent AI/ML applications
*GenAI overview
*Techniques behind GenAI: transformer
*Systems behind GenAI: GPU clusters, accelerators
*Algorithms behind GenAI: reinforcement learning
*Other applications based on GenAI
*Survey of competitive models vs transformer
*Survey of key systems development
*Survey of algorithmic innovations
*Sustainable AI and AI for sustainability
*Other topics: responsible AI, secure AI, edge AI (depends on time)

*****************************

>AMS 691.02:  Natural Language Processing

This course will introduce fundamental concepts in natural language processing (NLP). NLP includes a range of research problems that involve computing with natural language. Some are user-facing applications, such as spam classification, question answering, summarization, and machine translation. Others serve supporting roles, such as part-of-speech tagging and syntactic parsing. Solutions to these challenges are derived from a combination of machine learning (especially deep learning) techniques, algorithms, and principles from linguistics. NLP also provides fundamental building blocks to large language models (LLMs), which powers an even more diverse set of applications (including but not limited to language) in the era of generative AI.

Prerequisites:  Basic knowledge of calculus, linear algebra, and probability; programming proficiency (no specific language required but Python is preferred); a machine learning course is recommended but not required.

3 credits

Course Materials (recommended ONLY):

* SLP3: Daniel Jurafsky and James H. Martin. Speech and Language Processing. This is the main reference for the course. Drafts of most chapters of the 3rd edition are freely available online.

*INLP: Jacob Eisenstein. Introduction to Natural Language Processing. 2019. A 2018 draft is available online.

*LFNLP: Emily M. Bender. Linguistic Fundamentals for Natural Language Processing, 100 Essentials from Morphology and Syntax. 2013.

*LFNLP II: Emily M. Bender. Linguistic Fundamentals for Natural Language Processing II, 100 Essentials from Semantics and Pragmatics. 2020.

*NLTK book: Steven Bird, Ewan Klein, and Edward Loper. Natural Language Processing with Python - Analyzing Text with the Natural Language Toolkit. Free version online, with Python programming practices.

Topics:
*Words: definition, tokenization, morphology, word senses
*Lexical semantics: distributional semantics, word embeddings, word clustering
*Text Classification: classifiers, linear models, features, naive Bayes, training linear classifiers via loss function optimization, loss functions, stochastic gradient descent
*Neural networks: MLP, CNN, RNN and Transformers, fine-tuning
*Language Modeling: n-gram models, smoothing, neural network--based language modeling
*Sequence Labeling: part-of-speech tagging, named entity recognition, hidden Markov models, conditional random fields, dynamic programming, Viterbi
*Syntax: weighted context-free grammars, dependency syntax, inference algorithms
*Semantics: compositionality, semantic role labeling, frame semantics, lambda calculus, semantic parsing, grounded semantics
*Pragmatics: phenomena, rational speech act model
*Cross-lingual NLP: translation, decoding, lexicon induction, unsupervised translation
*Large language models (LLMs): background, challenges, prompting

 
Learning Objectives:
*Understand key challenges of computing with natural language
*Understand and apply solutions to standard NLP tasks*
*Be able to implement basic neural network architectures for core NLP tasks using deep learning toolkits
*Be able to derive dynamic programming algorithms to perform inference in structured output spaces, and to analyze their computational properties
*Understand common types of syntactic and semantic analysis, and how they are used in downstream applications
*Recognize and characterize the errors made by NLP systems
*Understand how modern large language models (LLMs) work, along with how to use them effectively

*****************************

>AMS 691.03: Fundamentals of Reinforcement Learning

Deep understanding of reinforcement learning (RL) is essential for machine learning researchers, data scientists and practicing engineers working in areas such as artificial intelligence, machine learning, data/network science, natural language processing, computer vision, among others. RL has found its applications in our everyday life such as AlphaGo, AlphaFold, autonomous driving, healthcare, etc. This course will provide an introduction to the field of RL, and emphasize on hands-on experiences. Students are expected to become well versed in key ideas and techniques for RL through a combination of lectures, written and coding assignments. Students will advance their understanding and the field of RL through a course project. The topics that will be covered (time permitting) include but not limited to:

• Markov Decision Processes (MDPs);
• Value Functions;
• Policy Iteration and Value Iteration;
• Monte Carlo Methods;
• Temporal Difference (TD) Learning;
• SARSA and Q-Learning;
• TD(λ);
• (Linear) Function Approximation;
• Policy Gradient Algorithms;
• Other topics (e.g., Multi-Agent RL, RL Theory; Deep RL)

Prerequisites:
• Calculus and Linear Algebra (You should be comfortable taking derivatives and
understanding matrix vector operations and notation.)
• Basic Probability and Statistics (You should know basics of probabilities, mean, standard
deviation, etc.)
• Python: All programming in the assignments and the project will be in Python (e.g.,
using numpy and Tensorflow). There will be roughly two programming problems in the
assignments. You are expected to be efficient in Python or eager to learn it by yourself.
This course will NOT teach programming.
• We will be formulating cost functions, taking derivatives and performing optimization with
gradient descent.
• Have heard of Markov decision process and RL before in an AI or ML course, but we will
quickly cover the basics.

3 credits

Course Materials:
There is no official textbook for the class, but some supporting readings will be based on:

Reinforcement Learning: An Introduction, Sutton and Barto (Available at Sutton’s university
website). There is no requirement to own the book, however, it is a fantastic introduction to the topic. Some other additional references that may be useful are listed below:

• Artificial Intelligence: A Modern Approach, Stuart J. Russel and Peter Norvig. [Link]
• Deep Learning. Ian Goodfellow, Yoshua Bengio, and Aaron Courville. [Link]
• Reinforcement Learning: State-of-the-Art, Marco Wiering and Martijin van Otterlo, Eds.
[Link]
• Dynamic Programming and Optimal Control, Vol. I, 3rd Edition, D. P. Bertsekas, 2005.
• Dynamic Programming and Optimal Control, Vol. II: Approximate Dynamic
Programming, 4th Edition, D. P. Bertsekas, 2012.
• Applied Probability Models with Optimization Applications, S. M. Ross, 1992.
• Optimization and Control, R. Weber, 2016. (Course notes available on Prof. Weber’s
website at Cambridge University.)

 

*****************************

>AMS 691.04:  Topics in Data Science 

AMS 691 is a one credit seminar course designed to support DS students in meeting internship requirements, fostering a sense of community, and providing opportunities for interaction and professional development. The course will feature seminars led by invited speakers and student presentations on their internship projects. It also serves as a platform for students to engage in discussions and activities together.

1 credit

No course materials required

*****************************

>AMS 691.05:   Deep Learning

An introduction to the field of deep learning, including basic machine learning, logistic regression, loss functions, neural networks, optimization, error back-propagation, regularization and generalization, unsupervised learning and auto-encoders, convolutional neural networks, recurrent neural networks, graph neural networks, attention models, generative models, advanced topics of deep learning, applications to natural language processing, computer vision, and basic sciences.

Prerequisites:
• Basic machine learning (e.g., supervised learning, linear regression, logistic regression, support vector machines);
• Linear algebra (e.g., matrix computations, eigenvalues and eigenvectors, singular value decomposition);
• Multivariate calculus;
• Access to GPU;
• Proficiency in Python programming.

3 credits

Learning Outcomes:
Student learning outcomes include:
•Understanding the foundation, major techniques, applications, and challenges of deep learning;
•The ability to apply basic deep learning algorithms for solving real-world problems.
•The learning outcomes will be assessed based on a combination of homework assignments, exams, and projects.

Course Materials (recommended ONLY):
•NNDL: "Neural Networks and Deep Learning (Second Edition)," by Charu C. Aggarwal; Springer, July 2023. http://www.charuaggarwal.net/neural.htm

•LFD: "Learning from Data" by Y.S. Abu-Mostafa, M. Magdon-Ismail, H.T. Lin, only Chapters 3 and 7, January 1, 2012; ISNB: 9781600490064. http://amlbook.com/ https://www.amazon.com/Learning-Data-Yaser-S-Abu-Mostafa/dp/1600490069

SPRING 2025:

This course explores the fundamental principles and algorithms used in medical image processing and analysis. Key topics include interpolation, registration, enhancement, feature extraction, classification, segmentation, quantification, shape analysis, motion estimation, and visualization, including traditional and machine learning techniques. Both anatomical and functional image analysis will be covered, using data from common medical imaging modalities. Through projects and assignments, students will gain hands-on experience working with real medical imaging data.  No prerequisites.

0-3 credits; ABCF grading

No course materials

Topics:

Interpolation, registration, enhancement, feature extraction, classification, segmentation, quantification, shape analysis, motion estimation, and visualization

Learning Outcomes:

Explores the fundamental principles and algorithms used in medical image processing and analysis

******************************

AMS 691.02:  Large Language Models

No course materials

 

 

******************************

AMS 691.03:   Data Management

This course teaches how to manage databases, which are important tools for organizations. Students will learn about the different parts of a database system, how data is stored, and how to get information from databases. The course will also cover how to design and build large databases, including creating logical structures, handling multiple processes at the same time, distributing data, and managing databases.

There are no prerequisites for this course, and we will start learning R and SQL from the very beginning.  However, having some prior experience would be helpful.

3 credits; ABCF grading

Course materials will be supplied by the instructor via Brightspace

Topics:

Students will also explore topics like data warehousing (storing large amounts of data), cleaning data, and data mining (finding useful information from data). The course includes hands-on practice with R and SQL, where students will clean, organize, and combine raw data to prepare it for analysis. This practical experience will help students develop key skills in getting data ready for use in projects.

Learning Outcomes:

By the end of this course, students will understand
--the basics of managing databases, which are important tools for organizations;
--receive practical experience in designing and building databases, cleaning up raw data, and preparing it for analysis using R and SQL.