Courses

Prerequisites: MATH S1102, or the equivalent. Columbia College students who aim at an economics major AND have at least the grade of B in Calculus I may take Calculus III directly after Calculus I. However, all students majoring in engineering, science, or mathematics should follow Calculus I with Calculus II. Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers.

Linear algebra with a focus on probability and statistics. The course covers the standard linear algebra topics: systems of linear equations, matrices, determinants, vector spaces, bases, dimension, eigenvalues and eigenvectors, the Spectral Theorem and singular value decompositions. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in probability and statistics. The topics covered in the probability theory part include conditional probability, discrete and continuous random variables, probability distributions and the limit theorems, as well as Markov chains, curve fitting, regression, and pattern analysis. The course contains applications to life sciences, chemistry, and environmental life sciences. No a priori background in the life sciences is assumed.

This course is best suited for students who wish to focus on applications and practical approaches to problem solving. It is recommended to students majoring in engineering, technology, life sciences, social sciences, and economics.

Math majors, joint majors, and math concentrators must take MATH UN2010 Linear Algebra or MATH UN1207 Honors Math A, which focus on linear algebra concepts and foundations that are needed for upper-level math courses. MATH UN2015 (Linear Algebra and Probability) does NOT replace MATH UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students may not receive full credit for both courses MATH UN2010 and MATH UN2015. Students who have taken MATH UN2015 and consider taking higher level Math courses should contact a major advisor to discuss alternative pathways.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Prerequisites: MATH UN1101 (Calculus I) or equivalent courses. This course introduces students to mathematical modeling through hands-on, project-based learning. Topics include fundamental concepts from linear algebra, multivariable calculus, differential equations, probability and statistics, and introductory machine learning.

Prerequisites: MATH V1102-MATH V1201 or the equivalent and MATH V2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups.

Prerequisites: MATH S1202, MATH S2010, or the equivalent. Students must have a current and solid background in the prerequisites for the course: multivariable calculus and linear algebra. Elements of set theory and general topology. Metric spaces. Euclidian space. Continuous and differentiable functions. Riemann integral. Uniform convergence.

This course is meant to provide an introduction to regression and applied statistics for the social sciences, with a strong emphasis on utilizing the Python software language to perform the key tasks in the data analysis workflow. Topics to be covered include various data structures, basic descriptive statistics, regression models, multiple regression analysis, interactions, polynomials, Gauss-Markov assumptions and asymptotics, heteroskedasticity and diagnostics, data visualization, models for binary outcomes, models for ordered data, first difference analysis, factor analysis, and cluster analysis. Through a variety of lab assignments, students will be able to generate and interpret quantitative data in helpful and provocative ways. Only relatively basic mathematics skills are assumed, but some more advanced math will be introduced as needed. A previous introductory statistics course that includes linear regression is helpful, but not required.

Social scientists need to engage with natural language processing (NLP) approaches that are found in computer science, engineering, AI, tech and in industry. This course will provide an overview of natural language processing as it is applied in a number of domains. The goal is to gain familiarity with a number of critical topics and techniques that use text as data, and then to see how those NLP techniques can be used to produce social science research and insights. This course will be hands-on, with several large-scale exercises. The course will start with an introduction to Python and associated key NLP packages and github. The course will then cover topics like language modeling; part of speech tagging; parsing; information extraction; tokenizing; topic modeling; machine translation; sentiment analysis; summarization; supervised machine learning; and hidden Markov models. Prerequisites are basic probability and statistics, basic linear algebra and calculus. The course will use Python, and so if students have programmed in at least one software language, that will make it easier to keep up with the course.

Artificial intelligence (AI) and generative AI – like ChatGPT, MidJourney, and Gemini – are poised to change the world for everyone. It is critical that students understand (and utilize) this new technology at several levels. In this class – through readings and a dozen hands-on activities – students will come to deeply understand AI. Specifically, students will construct (using Python) some of the basic building-blocks of AI, like machine learning (like recommendation systems), natural language processing (like word embeddings) and chatbots. They will test out AI’s capabilities and refine prompts in real-world settings, whether in art, video, writing or Internet-of-Things. They will learn about how generative AI fits into the history of technology adoption and the diffusion of innovation, answering questions like: Will AI be able to replace whole jobs? And if so, when? They will use the lenses of psychology and economics to explore the impact of AI in people’s lives, including in the context of algorithmic fairness, regulation and intellectual property. They will be pushed to take human creativity in new directions, augmented by AI’s “weirdness.” Lastly, students will be pushed to further develop their own uniquely-human skills – like in critical thinking and empathy – in response to the power of generative AI to mimic humans. As best-selling author Seth Stephen-Davidowitz has recently (Dec. 2023) written, “So far, [my newest book] has higher ratings than either of my previous two books -- even though it was written in 1/36th of the time, thanks to AI. AI is wild!" By the end of this class, students will feel empowered technically and philosophically to handle all new generative AI developments. There are no specific prerequisites for this class.