Prerequisites: Calculus through multiple integration and infinite sums. A calculus-based tour of the fundamentals of probability theory and statistical inference. Probability models, random variables, useful distributions, conditioning, expectations, law of large numbers, central limit theorem, point and confidence interval estimation, hypothesis tests, linear regression. This course replaces SIEO 4150.
Part of an accelerated consideration of the essential chemical engineering principles from the undergraduate program, including topics from Reaction Kinetics and Reactor Design, Chemical Engineering Thermodynamics, I and II, and Chemical and Biochemical Separations. While required for all M.S. students with Scientist to Engineer status, the credits from this course may not be applied toward any chemical engineering degree.
Part of an accelerated consideration of the essential chemical engineering principles from the undergraduate program, including topics from Reaction Kinetics and Reactor Design, Chemical Engineering Thermodynamics, I and II, and Chemical and Biochemical Separations. While required for all M.S. students with Scientist to Engineer status, the credits from this course may not be applied toward any chemical engineering degree.
This course is a Black Studies research and publication design studio. Here, the integrity of “an imagined moral-intellectual community” and the practicalities of writing and publication are held in balance, toward two functional objectives:
(1) To revise and submit (for publication) an original seminar paper, thesis, or conference paper.
Following Laura Belcher’s,
Writing Your Journal Article in Twelve Weeks
, “…(t)he goal of this course is to aid participants in taking their papers from classroom quality to journal quality and in overcoming anxiety about academic publishing in the process.
(2) To critically survey the field of Black Studies through close reading of its journals—resulting in the production of a consulting report focused on the student’s chosen area.
This year
Souls: Souls: A Critical Journal of Black Culture, Politics and Society
returns to Columbia University, where it was founded by Manning Marable in 1999. This course is designed to prepare IRAAS/AAADS to receive custodianship of the journal and reshape its Black radical intellectual project in response to the current moment and prospective futures.
“Journal work is not only not an arbitrary undertaking, and it is certainly not simply the practice of putting competent scholarly articles into print (though that is avowedly its formal function); rather, it is distinctive for being an intellectual undertaking that is pursued with a certain horizon in mind, namely, the collective constitution of an imagined moral-intellectual community.” (David Scott, Small Axe 50 July 2016)
Facilitating reflexive, critically engaged, and sustainable writing practices; and focusing on the past and future of
Souls: A Critical Journal of Black Politics, Culture, and Society;
our hybrid colloquy sessions will typically combine theoretical and methodological discussions, close readings of assigned journals, and ‘workshop’ elements— including sharing of work and occasional short in class writing. We will meet in-person and virtually. Small group work will be required.
This course is customized for 1st-year PhD and MA students in the biomedical informatics graduate program and also
open to other interested students at Columbia. It provides a detailed overview of symbolic methods.
Introduction to the economic evaluation of industrial projects. Economic equivalence and criteria. Deterministic approaches to economic analysis. Multiple projects and constraints. Analysis and choice under risk and uncertainty.
Introduction to the economic evaluation of industrial projects. Economic equivalence and criteria. Deterministic approaches to economic analysis. Multiple projects and constraints. Analysis and choice under risk and uncertainty.
adding zero credit recitation to STAT 4203UN PROBABILITY THEORY for Fall 2026
adding zero credit recitation to STAT 4203UN PROBABILITY THEORY for Fall 2026
A graduate course only for MS&E, IE, and OR students. This is also required for students in the Undergraduate Advanced Track. For students who have not studied linear programming. Some of the main methods used in IEOR applications involving deterministic models: linear programming, the simplex method, nonlinear, integer and dynamic programming.
A graduate course only for MS&E, IE, and OR students. This is also required for students in the Undergraduate Advanced Track. For students who have not studied linear programming. Some of the main methods used in IEOR applications involving deterministic models: linear programming, the simplex method, nonlinear, integer and dynamic programming.
A graduate course only for MS&E, IE, and OR students. This is also required for students in the Undergraduate Advanced Track. For students who have not studied linear programming. Some of the main methods used in IEOR applications involving deterministic models: linear programming, the simplex method, nonlinear, integer and dynamic programming.
adding zero credit recitation to STAT 4204GU STATISTICAL INFERENCE for Fall 2026
adding zero credit recitation to STAT 4204GU STATISTICAL INFERENCE for Fall 2026
Prerequisites: Students should have completed at least one course in ecology, evolution or conservation biology. The purpose of this course is to arm emerging scientists with an understanding of conservation policy at the city, state, federal and international levels. Our focus will be on understanding the science that informs conservation policy, evaluating the efficacy of conservation policies for achieving conservation goals, and learning about the role that scientists play in forming policy.
adding zero credit recitation to STAT 4205GU for Fall 2026
adding zero credit recitation to STAT 4205GU for Fall 2026
zero credit recitation for STAT GU4206
Fundamentals of Linear Algebra including vector and Matrix algebra, solution of linear systems, existence and uniqueness, gaussian elimination, gauss-jordan elimination, the matrix inverse, elementary matrices and the LU factorization, computational cost of solutions. Vector spaces and subspaces, linear independence, basis and dimension. The 4 fundamental subspaces of a matrix. Orthogonal projection onto a subspace and solution of Linear Least Squares problems, unitary matrices, inner products, orthogonalization algorithms and the QR factorization, applications. Determinants and applications. Eigen problems including diagonalization, symmetric matrices, positive-definite systems, eigen factorization and applications to dynamical systems and iterative maps. Introduction to the singular value decomposition and its applications.
Fundamentals of Linear Algebra including vector and Matrix algebra, solution of linear systems, existence and uniqueness, gaussian elimination, gauss-jordan elimination, the matrix inverse, elementary matrices and the LU factorization, computational cost of solutions. Vector spaces and subspaces, linear independence, basis and dimension. The 4 fundamental subspaces of a matrix. Orthogonal projection onto a subspace and solution of Linear Least Squares problems, unitary matrices, inner products, orthogonalization algorithms and the QR factorization, applications. Determinants and applications. Eigen problems including diagonalization, symmetric matrices, positive-definite systems, eigen factorization and applications to dynamical systems and iterative maps. Introduction to the singular value decomposition and its applications.
Fundamentals of Linear Algebra including vector and Matrix algebra, solution of linear systems, existence and uniqueness, gaussian elimination, gauss-jordan elimination, the matrix inverse, elementary matrices and the LU factorization, computational cost of solutions. Vector spaces and subspaces, linear independence, basis and dimension. The 4 fundamental subspaces of a matrix. Orthogonal projection onto a subspace and solution of Linear Least Squares problems, unitary matrices, inner products, orthogonalization algorithms and the QR factorization, applications. Determinants and applications. Eigen problems including diagonalization, symmetric matrices, positive-definite systems, eigen factorization and applications to dynamical systems and iterative maps. Introduction to the singular value decomposition and its applications.
Linear, quadratic, nonlinear, dynamic, and stochastic programming. Some discrete optimization techniques will also be introduced. The theory underlying the various optimization methods is covered. The emphasis is on modeling and the choice of appropriate optimization methods. Applications from financial engineering are discussed.
Prerequisites: JPNS C1202 or the equivalent. Introduction to the fundamentals of classical Japanese grammar. Trains students to read Japanese historical and literary texts from the early period up to the 20th century.
.
Prerequisites: advanced calculus and general physics, or the instructors permission. Basic physical processes controlling atmospheric structure: thermodynamics; radiation physics and radiative transfer; principles of atmospheric dynamics; cloud processes; applications to Earths atmospheric general circulation, climatic variations, and the atmospheres of the other planets.
Mathematical genomics synthesizes between high resolution quantitative research and molecular measurements in the context of clinical or biological analysis. This course will provide independent research experience in this field for interested students under the direction of a faculty advisor. This course is suitable for students with background in the subject matter (e.g., biology, chemistry) who wish to learn quantitative data analysis and methods development, as well as for those with technical background (e.g., biomedical informatics) who wish to apply their skills to biological research. Instructor permission is required for registration.
Prerequisites: GREK V1201 and V1202, or their equivalent. Since the content of the course changes from year to year, it may be taken in consecutive years.
Prerequisites: LATN V3012 or the equivalent. Since the content of this course changes from year to year, it may be repeated for credit.
Prerequisites: LATN V3012 or the equivalent. Since the content of this course changes from year to year, it may be repeated for credit.
This seminar offers a sustained exploration of the fiction and nonfiction of Jamaica Kincaid, a Caribbean-American writer described by the novelist Tiphanie Yanique as “a fiction writer with a poet’s air and a journalist’s eye.” In Fall 2026, Kincaid will be a Writer-in-Residence at Columbia University, providing students with a rare opportunity to engage directly with her presence and intellectual contributions on campus. The course is organized in two parts: the first half examines Kincaid’s nonfiction—some of her essays, memoirs, and nature writing—while the second half traces her fiction from
At the Bottom of the River to See Now Then
. Central themes include colonialism and its aftermath, migration and exile, maternal relationships, ecological memory, and cultural displacement. This course emphasizes close reading, critical scholarship, and creative response. Students will be expected to contribute actively to seminar discussions and produce original work that bridges analytical and creative modes.
Introductory course is for individuals with an interest in medical physics and other branches of radiation science. Topics include basic concepts, nuclear models, semi-empirical mass formula, interaction of radiation with matter, nuclear detectors, nuclear structure and instability, radioactive decay process and radiation, particle accelerators, and fission and fusion processes and technologies.
Mathematical description of chemical engineering problems and the application of selected methods for their solution. General modeling principles, including model hierarchies. Linear and nonlinear ordinary differential equations and their systems, including those with variable coefficients. Partial differential equations in Cartesian and curvilinear coordinates for the solution of chemical engineering problems.
Mathematical description of chemical engineering problems and the application of selected methods for their solution. General modeling principles, including model hierarchies. Linear and nonlinear ordinary differential equations and their systems, including those with variable coefficients. Partial differential equations in Cartesian and curvilinear coordinates for the solution of chemical engineering problems.
Introduces modern artificial intelligence (AI), focus on applications to operations and decision-making in domains such as healthcare, finance, energy systems, and supply chain management. Topics: machine learning, neural networks, large language models (LLMs), reinforcement learning, game theory, multi-agent systems, and agentic frameworks.
Introduces modern artificial intelligence (AI), focus on applications to operations and decision-making in domains such as healthcare, finance, energy systems, and supply chain management. Topics: machine learning, neural networks, large language models (LLMs), reinforcement learning, game theory, multi-agent systems, and agentic frameworks.
Prerequisites: Intermediate reading knowledge of Spanish This course considers how language has traditionally shaped constructs of national identity in the Caribbean vis-à-vis the US. By focusing on language ‘crossings’ in Latinx Caribbean cultural production, we critically explore how various sorts of texts–narrative, drama, performance, poetry, animated TV series, and songs–contest conventional notions of mainstream American, US Latinx, and Caribbean discourses of politics and identities. Taking 20th-century social and historical context into account, we will analyze those contemporary styles and uses of language that challenge monolingual and monolithic visions of national and ethnolinguistic identities, examining societal attitudes, cultural imaginaries, and popular assumptions about the Spanish language in the Greater Caribbean and the US.
Engineering economic concepts. Basic spreadsheet analysis and programming skills. Subject to instructor's permission. Infrastructure design and systems concepts, analysis, and design under competing/conflicting objectives, transportation network models, traffic assignments, optimization, and the simplex algorithm.
A close reading of works by Dostoevsky (Netochka Nezvanova; The Idiot; A Gentle Creature) and Tolstoy (Childhood, Boyhood, Youth; Family Happiness; Anna Karenina; The Kreutzer Sonata) in conjunction with related English novels (Bronte's Jane Eyre, Eliot's Middlemarch, Woolf's Mrs. Dalloway). No knowledge of Russian is required.
A close reading of works by Dostoevsky (Netochka Nezvanova; The Idiot; A Gentle Creature) and Tolstoy (Childhood, Boyhood, Youth; Family Happiness; Anna Karenina; The Kreutzer Sonata) in conjunction with related English novels (Bronte's Jane Eyre, Eliot's Middlemarch, Woolf's Mrs. Dalloway). No knowledge of Russian is required.
This seminar course will provide a punctual survey of trends and figures in the experimental cultures of East Central Europe. Formations include the avant-gardes (first, postwar, and postcommunist); experimental Modernisms and Postmodernisms; alternative film, media, and visual culture; and formally inventive responses to exceptional historical circumstances. Proceeding roughly chronologically from early twentieth to early twenty-first centuries, we will examine expressionist/surrealistic painting and drama; zenithist hybrid genres such as cinépoetry and proto-conceptualist writing; mixed-media relief sculpture; post-conceptual art; experimental and animated film; and avant-garde classical music. In terms of theory, we will draw on regional and global approaches to artistic experimentation ranging from Marxist and other theories of value through discourses of the body and sexuality in culture to contemporary affect theory. The course will be taught in English with material drawn primarily from Poland, Yugoslavia, Czechoslovakia, and Hungary. Each session will include a lecture followed by discussion.
Course Description
This interdisciplinary course explores both the rights of Indigenous people in settler colonies as well as the complex historical and theoretical relationship between human rights and settler colonialism. We will pursue three lines of inquiry. The first critically explores how central political concepts of the international state system—sovereignty, property, territory, self-determination—entwine the histories of settler colonialism and human rights. The second charts the rise and mechanisms of the international Indigenous rights movement, in particular its activity at the United Nations leading to the Declaration on the Rights of Indigenous Peoples in 2007, and its contributions to ongoing debates on environmental and climate justice, group rights, natural resources and territorial autonomy, and cultural rights. The third unit interrogates settler state responses to the movement for Indigenous human rights, such as cooptation, recognition, and apology.
Through readings drawn from history, ethnography, political and critical theory, international relations, Native studies, law, and documents produced by intergovernmental organizations and NGOs, we will explore and deepen the tensions between human rights as a theory and practice and the political lives and aspirations of Indigenous peoples and activists. What technologies of rule—such as residential school systems and property law—do settler colonial states deploy to dispossess Indigenous peoples? How have Indigenous peoples used the international human rights regime to mobilize against such dispossession? How have these states resisted the global Indigenous rights movement? And can the human rights regime, rooted in the international state system, meaningfully contribute to anticolonial movements in liberal settler colonies? While we will touch on settler colonialism as it manifests around the globe, the course’s geographical focus will be on North America.
Course objectives
Throughout this course, you will:
Develop a historically-informed understanding of both international indigenous rights and settler colonialism as idea, practice, institution, and discourse;
Place the literature on human rights and settler colonialism into critical conversation in order to deepen existing conceptual problems and generate new ones;
Identify the main arguments in theoretical texts, legal and policy documents, and public debates
Introduces agentic AI systems for decision-making in operations research and financial engineering. Covers large language models, retrieval-augmented generation, conversational agents, semantic workflows, and streaming architectures. Emphasizes hands-on design and deployment of end-to-end agentic pipelines, integrating forecasting, optimization, and reasoning in operational problems and financial engineering applications. Applications include supply chain risk, energy systems, inventory management, and portfolio management.
Introduces agentic AI systems for decision-making in operations research and financial engineering. Covers large language models, retrieval-augmented generation, conversational agents, semantic workflows, and streaming architectures. Emphasizes hands-on design and deployment of end-to-end agentic pipelines, integrating forecasting, optimization, and reasoning in operational problems and financial engineering applications. Applications include supply chain risk, energy systems, inventory management, and portfolio management.
The Business Chinese I course is designed to prepare students to use Chinese in a present or future work situation. Students will develop skills in the practical principles of grammar, vocabulary, and cross-cultural understanding needed in today’s business world.
Introduction to urban data analytics (analysis and visualization of new types of ‘big’ urban data, statistical tools for urban data analysis, machine learning methods, data privacy). Conceptualization of cities as complex adaptive systems (coarse-graining of urban dynamics, network models for infrastructure interdependencies, agent-based urban simulation). Integrated urban infrastructure systems design (basic design solutions for coupled, people-centric, and climate-resilient civil infrastructure systems, Monte Carlo simulations for infrastructure scenario generation).
The purpose of this course is to provide a comprehensive introduction to accounting. It is designed to provide managerially oriented users of accounting products with the foundations of accounting concepts, the underlying mechanics, and the overall perspective required to become intelligent users of accounting information. The course will focus on the typical financial statements (e.g., balance sheets, income statements, and cash flow statements) along with the environment in which they are produced and used. The overarching perspective is that accounting reports provide information that is useful for various purposes.
The course design emphasizes that the information content of accounting reports is jointly determined by the accounting methods used to generate the reports and the environment in which the reports are produced and used. Thus, the course integrates the study of accounting methods and the study of the accounting environment. Accounting concerns, in turn, vary with market opportunities, instruments, and organizational arrangements at hand.
The course is organized into two interrelated modules. First, we review accounting-related institutional details, the nature of financial reports, audits, and disclosure. This includes the examination of the three main financial statements the nature of accrual measurement, and the information perspective that is central to the course. Second, we explore accounting methods in detail, focusing on revenue recognition, assets, liabilities, and equity. We explicitly consider accounting issues related to global organizations.
Throughout the course, we apply these concepts to real-world contexts, with attention to ethics, data integrity, financial statement analysis, valuation, and regulatory frameworks. In addition, we integrate contemporary topics such as the impact of climate change and ESG (environmental, social, and governance) considerations on financial reporting. Selected current events and emerging trends in ESG reporting are discussed in relation to the accounting frameworks and principles covered in class.
Finally, we explore the role of artificial intelligence—particularly large language models (LLMs)—in financial analysis. The course introduces how AI tools can assist in interpreting financial statements, while emphasizing the importance of human judgment and expertise in ensuring responsible and effective use of AI technologies.
For Mendelson Students Only.
Prerequisites: at least 3 years of intensive Chinese language training at college level and the instructor's permission. This advanced course is designed to specifically train students' listening and speaking skills in both formal and colloquial language through various Chinese media sources. Students view and discuss excerpts of Chinese TV news broadcasts, soap operas, and movie segments on a regular basis. Close reading of newspaper and internet articles and blogs supplements the training of verbal skills.
The objective of the course is to show that the orchestrators of the Harlem Renaissance intended to delineate aesthetic paradigms for African Americans to frame their subjective experiences. And that the orchestrators envisioned the Renaissance project through the formulation of standards for fiction, poetry, drama, music, and art that are cohesive with the socio-cultural experiences of African Americans during the first half of the 20th century. In the course, we will read the essays of Alain Locke, Jean Toomer, Zora Neale Hurston, Countee Cullen, Langston Hughes, and WEB DuBois. These essays were the media for the writers to present the aesthetic achievements of their time, while simultaneously promoting the ideals that fiction, poetry, drama, music, and art ought to follow. Close reading of the essays and class discussions will show how the Harlem Renaissance elaborated the criteria that African Americans may follow to derive meaning from their subjective experiences. Moreover, that the Harlem Renaissance shaped African American consciousness by providing the suitable aesthetic compass to understand its relation to the social and cultural institutions. Finally, the course explores how the Harlem Renaissance sought to define both, the aesthetic autonomy and distinctiveness of African Americans.
Prerequisites: Third Year Chinese or the equivalent The course is designed to help students master formal Chinese for professional or academic purposes. It includes reading materials and discussions of selections from Chinese media on contemporary topics, Chinese literature, and modern Chinese intellectual history. The course aims to enhance students' strategies for comprehension, as well as their written and oral communication skills in formal modern Chinese.
Following up on Black Performance Studies I (Spring 2026), this seminar explores the elasticity of historical narratives through the genius of Black performance. Inspired by the blues sung by Sammie in the groundbreaking film “Sinners,” directed and written by Ryan Coogler, this course will explore cultural portals and crossroads such as the juke joint, ocean, sycorax, stage, and studio. We will rely upon strategies of time travel and shapeshifting as key methods towards survival but also deep study. Given the current-day context of AI, propaganda, and systemic attack against public arts, cultural, and civic institutions, this course brings us back to basics in terms of writing, reading, and creating original work. In other words, we will workshop how to tell stories that compel readers and audiences to listen with intent and purpose. Students will learn how to read art and culture as social texts by studying masterworks of Black Performance vis-à-vis visual culture, media, fine arts, and performing arts, including dance and music production.
As a seminar, students should prepare to participate in in-depth discussions weekly by completing the reading and viewing assignments. Since the central subject matter is visual culture based in African-American studies, an interdisciplinary field, the course has a strong multi-media element. Students should expect to view films regularly, both online and in class. Students are also strongly encouraged to attend outside exhibitions and performances for extra credit. Syllabus is subject to change.
The class will be run as both deep study and performance workshop. In performance studies, performing is a method for analysis and synthesis. So throughout the course, we will use performance and performativity as modes of thinking through ideas including improvisation. Students will be trained in cultural studies theory, criticism but also, as cultural makers, movers and shakers.
This course reads the long Black intellectual and artistic tradition for its prescriptive stream: not only its description of Black life or its correction of the racist record, but its instruction-- in how to live a good life. Where the dominant canon of the good life runs from Aristotelian eudaimonia through the contemporary science of happiness; this tradition has theorized flourishing from music, literature, and the art of everyday life evident in the Black proverb, sermon, kitchen, and dance floor — it has largely gone unread as philosophy. Organizing that archive around five values (pleasure, duty, rest, friendship, and making) and across the diaspora from Havana to Harlem, Port of Spain to Nairobi, the course proceeds as an ethnography of an idea.
Black thought is most readily admitted into the curriculum as evidence of the wound —suffering, power, and subjection — while the normative question of how one ought to live tends to remain the European canon’s to keep: the Greeks are granted eudaimonia, and Black studies the seminar on its own subjection. This course relocates that question. Following Hurston’s account of vernacular form (“Characteristics of Negro Expression,” 1934) and Gwaltney’s ethnography of ordinary Black thought (Drylongso, 1980), it treats everyday practice as already-constituted theory and reads it as such.
The habit of mind here insists on the preciousness of Black life as a present, non-foreclosed claim —aliveness rather than impossibility. Taking seriously and departing from Hartman’s, where beauty remains shadowed by a death still arriving. The departure is temporal and paradigmatic:
Alongside Hartman’s central analytic of subjection and recovery of the “beautiful experiment” inside the wayward life (Wayward Lives, Beautiful Experiments, 2019); this course reads from inside and after the end (Sun Ra Arkestra,
It's After the End of the World, 1971
)— amid and after the levees break(ing), within and after an
epi
- ot
pan
- demic, for example— for aliveness as a totality rather than as the negative of antiblackness (Quashie, Black Aliveness, 2021), and for the tradition’s own instruction in how to live (Griffin, Read Until You Understand, 2021; Riggs, Tongues Untied, 1989). The good life is approached not as a future the revolution will deliver or the empire permit, but as a present claim. It is offered in the spirit Du Bois named whe
The fourth millennium BC was a time of tremendous innovation in monumental architecture, the organization of urban space and developments in the visual arts in southern Mesopotamia. As settlements grew into city-states, monumental architectural works transformed the landscape. New technologies of metallurgy, casting, the mechanical reproduction of images, stone sculpture and seal carvings emerged alongside the invention of writing, a technology first documented in the city of Uruk, the place of the setting of the Epic of Gilgamesh. Sculpted images and monuments began to be inscribed with texts that reveal a great deal about the ontological and agentive, the aesthetic and the order of the divine. The lecture introduces students to these extraordinary developments in early art and architecture of ancient Sumer (southern Iraq). Lectures will discuss votive statues, portraiture, image rituals, and the visual manifestation of the gods. The lectures also introduce the extraordinary developments in architecture and monuments.
Prerequisites: CHNS W4006 or the equivalent. This is a non-consecutive reading course designed for those whose proficiency is above 4th level. See Admission to Language Courses. Selections from contemporary Chinese authors in both traditional and simplified characters with attention to expository, journalistic, and literary styles.
Prerequisites: JPNS W4006 or the equivalent. Sections 1 & 2: Readings of advanced modern literary, historical, political, and journalistic texts, and class discussions about current issues and videos. Exercises in scanning, comprehension, and English translation. Section 3: Designed for advanced students interested in developing skills for reading and comprehending modern Japanese scholarship.
In the 1970s and 1980s a group of young Italian historians transformed the methods of historical inquiry and narrative. This class explores the origins, the diffusion, as well as the debate around Italian Microhistory across Europe and the United States. In particular, we will focus on “cultural” and “social” Microhistory and its evolution in Italy, France, and the US. We will read masterpieces such as Carlo Ginzburg’s
The Cheese and the Worms
, as well as Nathalie Zemon Davis’s
The Return of Martin Guerrre.
Also, we will analyze the current application of microhistorical methods to contemporary global history and the genre of biography. Topics include pre-modern popular culture and literacy, minority and marginality, the Inquisition, individual and collective identities, and the relation between the pre-modern Mediterranean, Europe and the world.
In Italian.
Prerequisites: PHYS UN3003 and PHYS UN3007 and differential and integral calculus; linear algebra; or the instructor's permission. This course will present a wide variety of mathematical ideas and techniques used in the study of physical systems. Topics will include: ordinary and partial differential equations; generalized functions; integral transforms; Green’s functions; nonlinear equations, chaos, and solitons; Hilbert space and linear operators; Feynman path integrals; Riemannian manifolds; tensor analysis; probability and statistics. There will also be a discussion of applications to classical mechanics, fluid dynamics, electromagnetism, plasma physics, quantum mechanics, and general relativity.
The biophysics of computation: modeling biological neurons, the Hodgkin-Huxley neuron, modeling channel conductances and synapses as memristive systems, bursting neurons and central pattern generators, I/O equivalence and spiking neuron models. Information representation and neural encoding: stimulus representation with time encoding machines, the geometry of time encoding, encoding with neural circuits with feedback, population time encoding machines. Dendritic computation: elements of spike processing and neural computation, synaptic plasticity and learning algorithms, unsupervised learning and spike time-dependent plasticity, basic dendritic integration. Projects in MATLAB.
Exposes scientists and engineers to the development, protection, and exploitation of intellectual property rights. Legal processes for obtaining and defending intellectual property rights are detailed. Best practices for innovation and commercialization are explored. Methods for valuation and monetization of intellectual property are illustrated through application of basic accounting and financial principles to case studies.
Exposes scientists and engineers to the development, protection, and exploitation of intellectual property rights. Legal processes for obtaining and defending intellectual property rights are detailed. Best practices for innovation and commercialization are explored. Methods for valuation and monetization of intellectual property are illustrated through application of basic accounting and financial principles to case studies.
Prerequisites: ECON UN3211 and ECON UN3213 and STAT UN1201 Topics include behavior uncertainty, expected utility hypothesis, insurance, portfolio choice, principle agent problems, screening and signaling, and information theories of financial intermediation.
Prerequisites: One semester of college-level calculus, college-level chemistry. College-level physics or geoscience. Or instructor’s permission. The accelerating climate change of the current day is driven by humanity’s modifications to the global carbon cycle. This course introduces the basic science of the global carbon cycle, focusing on large-scale processes occurring on annual to centennial timescales. Students will leave this course with an understanding of the degree to which the global carbon cycle is understood and quantified and the key uncertainties that are the focus of current research. We will build an understanding of the potential pathways and the significant challenges to limiting global warming to 2oC as intended by the 2015 Paris Climate Agreement.
The course will begin with a brief review of climate science basics and the role of CO2 in climate and climate change (weeks 1-2). Next, the fundamental processes of the ocean and the terrestrial biosphere carbon sinks will be covered (weeks 3-7). In weeks 8-10, our focus will be the drivers of current and future anthropogenic carbon emissions. In weeks 11-12, we will consider the potential for engineered sinks to contribute to global budgets. In weeks 13-14, students will present their final projects, and we will review the course content in the context of the December release of the current year’s Global Carbon Budget.
Prerequisites: PHYS UN3003 and PHYS UN3007 Formulation of quantum mechanics in terms of state vectors and linear operators. Three dimensional spherically symmetric potentials. The theory of angular momentum and spin. Identical particles and the exclusion principle. Methods of approximation. Multi-electron atoms.
This course, designed for advanced undergraduate and graduate students in the fields of East Asian Studies, Religion, and Art History, examines the principal modes, media, and contexts of visual culture in Japanese Buddhist history. The course is not designed as a survey of Japanese Buddhist art either in terms of media or chronology but rather an exploration, through the analysis of selected case studies, of the modalities of perception, materiality, and reception that distinguish the form and function of visual media in Japanese Buddhist contexts. Students are expected to have completed preliminary coursework in relevant areas of East Asian history, religion, or art history. The course is organized around selected objects in the collection of the Metropolitan Museum of Art and will include meetings at the Met for viewing and analysis of objects in the galleries and in storage.
Prerequisites: PHYS GU4021 or the equivalent. Thermodynamics, kinetic theory, and methods of statistical mechanics; energy and entropy; Boltzmann, Fermi, and Bose distributions; ideal and real gases; blackbody radiation; chemical equilibrium; phase transitions; ferromagnetism.
Prerequisites: (PHYS GU4021 and PHYS GU4022) In this course, we will learn how the concepts of quantum mechanics are applied to real physical systems, and how they enable novel applications in quantum optics and quantum information. We will start with microscopic, elementary quantum systems – electrons, atoms, and ions - and understand how light interacts with atoms. Equipped with these foundations, we will discuss fundamental quantum applications, such as atomic clocks, laser cooling and ultracold quantum gases - a synthetic form of matter, cooled down to just a sliver above absolute zero temperature. This leads us to manybody quantum systems. We will introduce the quantum physics of insulating and metallic behavior, superfluidity and quantum magnetism – and demonstrate how the corresponding concepts apply both to real condensed matter systems and ultracold quantum gases. The course will conclude with a discussion of the basics of quantum information science - bringing us to the forefront of today’s quantum applications.
Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Prerequisite: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines.
Prerequisites: genetics or molecular biology. The course covers techniques currently used to explore and manipulate gene function and their applications in medicine and the environment. Part I covers key laboratory manipulations, including DNA cloning, gene characterization, association of genes with disease, and methods for studying gene regulation and activities of gene products. Part II also covers commercial applications, and includes animal cell culture, production of recombinant proteins, novel diagnostics, high throughput screening, and environmental biosensors.
Systems biology approaches are rapidly transforming the technological and conceptual foundations of research across diverse areas of biomedicine. In this course we will discuss the fundamental developments in systems biology with a focus on two important dimensions: (1) the unique conceptual frameworks that have emerged to study systems-level phenomena and (2) how these approaches are revealing fundamentally new principles that govern the organization and behavior of cellular systems. Although there will be much discussion of technologies and computational approaches, the course will emphasize the conceptual contributions of the field and the big questions that lie ahead. Lectures and discussions of primary literature will enable students to scrutinize research in the field and to internalize systems biology thinking in their own research. To make this a concrete endeavor, the students will develop mini-NIH-style grant proposals that aims to study a fundamental problem/question using systems biology approaches. The students will then convene an in-class NIH-style review panel that will assess the strengths and weaknesses of these proposals. In addition, the students will have the opportunity to defend their proposals in a live presentation to the class. The course is open to graduate students in Biological Sciences. Advanced undergraduates in biological sciences, and other graduate students with background in biology from other disciplines, including physics, chemistry, computer science, and engineering may also attend after consulting with the instructor.
Poets, Rebels, Exiles examines the successive generations of the most provocative and influential Russian and Russian Jewish writers and artists who brought the cataclysm of the Soviet and post-Soviet century to North America. From Joseph Brodsky—the bad boy bard of Soviet Russia and a protégé of Anna Akhmatova, who served 18 months of hard labor near the North Pole for social parasitism before being exiled—to the most recent artistic descendants, this course will interrogate diaspora, memory, and nostalgia in the cultural production of immigrants and exiles.
This course is intended for graduate students or senior undergraduates interested in an introduction to state-of-the-art computational and systems biology. The course will illustrate the cycle of biology to bytes to biology using computational and systems biology approaches. It will cover various types of state-of-the-art techniques and data modalities used to measure cells and biological systems, as well as methods and software tools used for analysis and visualization of the resulting high-throughput data. The course includes a weekly lecture in addition to either a lab or a journal club. Labs are intended to support course projects and assignments and are optional with instructor approval.
Developing features - internal representations of the world, artificial neural networks, classifying handwritten digits with logistics regression, feedforward deep networks, back propagation in multilayer perceptrons, regularization of deep or distributed models, optimization for training deep models, convolutional neural networks, recurrent and recursive neural networks, deep learning in speech and object recognition.
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 UN1102 and MATH UN1202 and MATH UN2010 or the equivalent. The second term of this course may not be taken without the first. Rings, homomorphisms, ideals, integral and Euclidean domains, the division algorithm, principal ideal and unique factorization domains, fields, algebraic and transcendental extensions, splitting fields, finite fields, Galois theory.
Participants may elect to enroll in the 1-credit CWI independent study offered through the Sociology Department. CWI takes place Thursdays from 2-4pm in Knox Hall, room 509. The course ID for Fall 2025 is SOCI4043GU.
The Center for Wealth and Inequality (CWI), in this course, is a forum for students interested in social science topics broadly related to inequality. In particular, it will provide an opportunity for students to read and discuss the works presented in the weekly CWI seminar series, while also sharing and refining their own works in progress. The CWI seminar series is sponsored by Columbia University’s Institute for Social and Economic Research Policy (ISERP) and is devoted to the investigation of social and economic inequality. The seminar invites speakers from both within and outside of Columbia to present recent papers covering a wide range of topics pertaining to inequality, such as poverty, labor market behavior, education, and the family. The research topics and methodologies are at the cutting edge of the interdisciplinary study of wealth and inequality, as the CWI seminar invites speakers from multiple social science disciplines and fields. Past speakers include Annette Lareau, Adam Gamoran, Timothy Smeeding, Lisa Kahn, Mario Small, Rob Warren and Florencia Torche, among numerous others.
The primary preparation for these gatherings is to read the papers slated for presentation, whether written by an outside speaker or a student within the course. Students will meet to constructively critique these papers collectively, raising key questions and concepts for discussion and debate. One student will serve as the discussion leader each week by kicking off the dialogue. This student also will have the opportunity to prepare a review of the paper, if desired. When discussing CWI seminar papers, the product of each meeting will be a shortlist of pertinent questions for the visiting speaker, which participants will be encouraged to ask during the Q&A following each seminar. When focusing on student papers, participants will provide detailed comments geared towards supporting further development of our peers’ work. We regularly will be joined by the CWI presenters.
The spirit behind the student independent study is one of collective student growth and learning through semi-structured discussion. While the course will pose a relatively
Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups.
We are entering a new age of creativity, where generative AI is transforming how artists and creators work. This course invites undergraduate and graduate students at Columbia's School of the Arts to explore the powerful intersection of AI and creative practice. Through hands-on experimentation with AI tools—ranging from large language models (LLMs) to image and video generators—students will unlock new creative potential across ideation, visual development, and storytelling.
The course emphasizes how AI can augment existing creative methods, from generating ideas and developing personas to blending AI with traditional storytelling techniques. Students will also critically engage with the ethical implications of using AI in art. Culminating in a speculative design project, this course prepares students to envision and prototype the future of creative practice, where AI becomes a true collaborator in the artistic process.
Prerequisites: Course Cap 20 students. Priority given to graduate students in the natural sciences and engineering. Advanced level undergraduates may be admitted with the instructors permission. Calculus I and Physics I & II are required for undergraduates who wish to take this course. General introduction to fundamentals of remote sensing; electromagnetic radiation, sensors, interpretation, quantitative image analysis and modeling. Example applications in the Earth and environmental sciences are explored through the analysis of remote sensing imagery in a state-or-the-art visualization laboratory.
Required for second year Genetics and Development students. Open to all students. Prerequisite: at least one graduate-level biochemistry or molecular biology course, and instructor’s permission. Advanced treatment of the principles and methods of the molecular biology of eukaryotes, emphasizing the organization, expression, and evolution of eukaryotic genes. Topics include reassociation and hybridization kinetics, gene numbers, genomic organization at the DNA level, mechanisms of recombination, transposable elements, DNA rearrangements, gene amplification, oncogenes, recombinant DNA techniques, transcription and RNA splicing. Students participate in discussions of problem sets on the current literature.
Intensive study of a philosophical issue or topic, or of a philosopher, group of philosophers, or philosophical school or movement. Open only to Barnard senior philosophy majors.