Continuation of MATH GR6151x (see Fall listing).
Prerequisites: MATH GR6151 MATH G4151 Analysis & Probability I. Continuation of MATH GR6152x (see fall listing).
MIA and MPA Politics II Core.
Global Politics & International Organizations
introduces the actors, coalitions, institutions, and processes of global politics. It creates the conceptual foundations for understanding the role of international organizations in today’s multipolar and complex (or, ‘multiplex’) world. It sheds light on how states, non-state actors, and international bureaucracies act within international organizations and how they negotiate international agreements. The discussions will focus on formal and informal decision-making processes, working methods, and power in international relations. It will highlight processes within Bretton Woods institutions (the World Bank and the IMF), as well as at the United Nations, including bargaining processes at the UN Security Council. As a spillover from global politics, the course will also explore the role of international organizations in domestic policymaking processes.
In addition to critical scholarship on international organizations and global governance, the course relies on students’ analysis of relevant proceedings and debates at the UN, original policy documents, as well as expert testimony from a range of guest speakers, who share their extensive first-hand observations as actors of global governance processes. By these means,
Global Politics & International Organizations
offers insights into the processes, challenges, and impacts of activities by international organizations to make global governance regimes stronger, more effective, and hold actors more accountable.
MIA and MPA Politics II Core.
This course examines the evolution of American foreign policy within the context of U.S. political institutions, domestic dynamics, and historical experiences. It emphasizes the interplay between foreign and domestic policy, considering how American identity, political culture, and internal debates have shaped international engagement. While grounded in key moments in U.S. history, the course also addresses recent shifts in America's global role and examines the strategic, ideological, and institutional forces that continue to influence foreign policy decisions.
This course critically examines some of the moral and historical arguments for the justification of reparations for New World slavery. We explore the state of the debate about such historical injustices -- inquiring into questions of cultural trauma, memory, and generations. Our main concern will be to connect a moral claim about repair to an understanding of the injury of slavery in the Americas. Open to graduate students only. Permission is required.
MIA and MPA Politics II Core.
This course examines the development and dynamics of political parties in the United States, with a focus on the evolution of the two-party system and its influence on American politics and policymaking. Students will explore the historical foundations of party formation, ideological shifts over time, and the distinct roles parties play at national and subnational levels. The course also analyzes the structure and impact of party primaries, as well as recent technological and communication changes that have transformed modern electoral strategy and campaign practices.
Topics in Software engineering arranged as the need and availability arises. Topics are usually offered on a one-time basis. Since the content of this course changes, it may be repeated for credit with advisor approval. Consult the department for section assignment.
MIA and MPA Politics II Core.
This course examines the unique challenges and opportunities of the Global South, integrating theoretical frameworks, historical analysis, and contemporary case studies to develop a thorough understanding of how the region confronts and navigates some of the most significant issues shaping its politics and policies. By analyzing diverse political and policy dynamics in the Global South, it encourages students to think globally and recognize interconnectedness across political systems. Over seven weeks, we examine various challenges posed by institutional legacies of colonialism, the rise of populism, democratic backsliding, corruption, and political violence, while also highlighting innovative responses emerging from the Global South through contemporary case studies.
MIA and MPA Politics Core II.
This course explores how sudden disruptions—such as elections, economic shocks, natural disasters, and conflict—can challenge or derail long-term policy efforts. Using analytical tools from game theory, economics, management, and law, students will assess how policy responses are developed under pressure and how to design adaptive programs capable of withstanding unexpected change. The course combines discussion, interactive exercises, and real-world case studies to build the strategic, ethical, and communication skills needed to lead through uncertainty. Guest speakers with firsthand experience in policymaking during crises will offer practical insights into decision-making when the stakes are high. Students will learn to anticipate and manage change, balance competing demands, and identify opportunities to advance policy goals even amid disruption.
Archaeology is a sprawling, messy discipline and the role that theory does, should, and might play in the process of archaeological data collection, analysis, and interpretation has been highly contested. Archaeologists argue over whether there is such a thing as a stand-alone ‘archaeological theory’ and what kinds of theory from other disciplines should (or should not!) be imported. This course explores a range of recent theoretical conversations, orientations, and interventions within archaeology, with an eye to understanding what is currently at stake – and what is contested – in how archaeologists think about making archaeological knowledge in the contemporary moment. In doing so, this course encourages students to think about theory in archaeology as an important form of “practical knowledge” or “know how” for archaeologists (cf. Lucas 2018).
Prerequisites: CHEM UN2443 , or the equivalent.
This course will provide an introduction to the US Food and Drug Administration (FDA) and the drug development and approval process, often referred to as the “Critical Path”. The class will begin with a review of the history and organization of the FDA, and analysis of the principle steps along the critical path, including preclinical testing, clinical testing (drug development phase 0 thru IV), Good Laboratory Practices, Good Manufacturing Practices, Good Clinical Practices, and adverse event reporting. Different types of FDA submissions (IND, NDA, ANDA, SPA, eCTD), and FDA meetings will be examined, along with accelerated drug approval strategies, orphan drug development strategies, generic drug development, and post-marketing Sponsor commitments. Throughout the class we will study the related legislation and regulations that empower FDA, and the interrelated FDA guidance documents that define FDA expectations.
This class will describe the electronic and physical structure of inorganic materials with an emphasis on colloidal quantum dots and surface science.
The scenes selected for study and practice will come from dramatic works by playwrights of the 20th and 21st centuries. For the most part these writers will be American dramatists, but exceptions may sometimes be made. The scenes being used are assigned by the instructor, sometimes by way of suggestions by the student, if the student has a particular interest in a specific writer or character. Three scenes are presented each class. Each scene will be able to work with the teacher for approximately 50 minutes. The emphasis of the working session is on process, methods of rehearsal, engagement of body and voice, employment of principles of craft, and self-analysis.
Continuation of MATH GR6175x (see Fall listing).
Human–computer interaction (HCI) studies (1) what computers are used for, (2) how people interact with computers, and (3) how either of those should change in the future. Topics include ubiquitous computing, mobile health, interaction techniques, social computing, mixed reality, accessibility, and ethics. Activities include readings, presentations, and discussions of research papers. Substantial HCI research project required.
Prerequisites: ANTH G6352 Museum Anthropology: history and theory / ANTH G6353 Politics and Practice of Museum Exhibitions; G9110, G9111 and the instructors permission. Corequisites: ANTH G6353. This course addresses the practical challenges entailed in the process of creating a successful exhibition. Developing an actual curatorial project, students will get an opportunity to apply the museum anthropology theory they are exposed to throughout the program. They will be given a hands-on approach to the different stages involved in the curation of a show, from the in-depth researching of a topic to the writing, editing and design of an exhibition that will be effective for specific audiences.
This course introduces the legal frameworks, institutions, and advocacy strategies that underpin the international human rights system. With a practitioner’s lens, students will explore civil, political, economic, social, cultural, and environmental rights through treaties, customary law, and jurisprudence. Emphasis is placed on understanding where and how the law offers avenues for redress, and the evolving role of human rights advocacy in confronting modern challenges, including corporate accountability, gender discrimination, and climate justice.
Students will examine the structure and operation of key international and regional human rights mechanisms, the limits and opportunities of legal enforcement, and the relationship between international human rights law and international humanitarian law. The course integrates doctrinal learning with applied analysis through case studies, reflections, and simulations.
Attendance in the first class session is mandatory.
This course introduces the legal frameworks, institutions, and advocacy strategies that underpin the international human rights system. With a practitioner’s lens, students will explore civil, political, economic, social, cultural, and environmental rights through treaties, customary law, and jurisprudence. Emphasis is placed on understanding where and how the law offers avenues for redress, and the evolving role of human rights advocacy in confronting modern challenges, including corporate accountability, gender discrimination, and climate justice.
Students will examine the structure and operation of key international and regional human rights mechanisms, the limits and opportunities of legal enforcement, and the relationship between international human rights law and international humanitarian law. The course integrates doctrinal learning with applied analysis through case studies, reflections, and simulations.
Attendance in the first class session is mandatory.
MPA Financial Management Core I and II.
This course introduces the principles and practices of financial reporting, with the goal of enabling students to become informed users of financial information in both public and private sector contexts. Emphasis is placed on understanding the three primary financial statements: the balance sheet, income statement, and statement of cash flows; and the accounting concepts and rules that shape them. Students will examine how financial information is prepared, disclosed, and interpreted, and will develop analytical tools to assess an organization’s financial health and operational effectiveness. Real-world financial statements will be used throughout the course to build fluency in the language and application of accounting.
MPA Financial Management Core I and II.
This course introduces the principles and practices of financial reporting, with the goal of enabling students to become informed users of financial information in both public and private sector contexts. Emphasis is placed on understanding the three primary financial statements: the balance sheet, income statement, and statement of cash flows; and the accounting concepts and rules that shape them. Students will examine how financial information is prepared, disclosed, and interpreted, and will develop analytical tools to assess an organization’s financial health and operational effectiveness. Real-world financial statements will be used throughout the course to build fluency in the language and application of accounting.
Prerequisites: STAT GR6201 Continuation of STAT G6201
This course is designed to prepare social workers for clinical work with bereaved families. We emphasize the idea that grief therapy focuses importantly on active listening linked to interventions that provide validation, support or guidance. We provide a way of understanding grief using an attachment theory model that explains a big-picture framework for understanding grief and adaptation to loss. The centerpiece of the course is a presentation of an approach to grief therapy derived from our efficacy-tested treatment for complicated grief and incorporating attention to self-care, racism, ethical obligations and dilemmas and using peer or experienced supervision in doing grief therapy.
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
Prerequisites: the instructors permission prior to registration. Please contact the instructors for more information. This graduate student field survey provides an overview of the scholarly study of American politics. The course has been designed for students who intend to specialize in American politics, as well as for those students whose primary interests are comparative politics, international relations, or political theory, but who desire an intensive introduction to the ;American; style of political science.
This course introduces students to the literature on globalization and the diffusion of culture and institutions. It covers literatures in sociology and political science as well as some anthropology and history. This course will not discuss economic, financial, or migratory globalization in depth. In the first part, we will survey the major theories of the global diffusion of culture and institutions: world polity theory, global field theory, the policy diffusion literature, etc. In the second part, we discuss select topics, such as the role of local power relations in diffusion processes or the consequences of diffusion for patterns of cultural similarity and difference across the world.
Prerequisites: the director of graduate studies permission. Corequisites: ECON G6410. Consumer and producer behavior; general competitive equilibrium, welfare and efficiency, behavior under uncertainty, intertemporal allocation and capital theory, imperfect competition, elements of game theory, problems of information, economies with price rigidities.
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
The purpose of this course is to introduce social work students to the culture, history, policies, programs, and issues that face different generations of the United States military veterans and their families. This course will provide the essential background and knowledge base necessary to assist military veterans and their families through appropriate referrals for social and medical services. The military veteran community will be examined using both micro and macro lenses in order to provide an overarching understanding of the complex matters faced by today’s military veterans, their families, and communities.
This new class will provide Writing for Film & Television students with a foundational experience in TV Writing in the second semester of their first year at Columbia. They will have studied feature writing in their first semester, and this class will explore how TV is different from the feature form, the unique structure of TV when compared to features, plays, and novels, the key elements of a good TV show and pilot, the different worlds of network and streaming and the current marketplace, how to structure a TV pilot, and how to write and begin revising the pilot episode for an original TV show.
Prerequisites: the director of graduate studies permission. Concept of full employment. Models of underemployment and theory applicability, determinants of consumption and of investment, multiplier and accelerator analysis, an introduction to monetary macroeconomics, the supply side and inflation. Integration of macroeconomics with microeconomic and monetary analysis.
Principles of traditional and emerging sensors, data acquisition and signal processing techniques, experimental modal analysis (input-output), operational modal analysis (output-only), model-based diagnostics of structural integrity, data-based diagnostics of structural integrity, long-term monitoring and intelligent maintenance. Lectures and demonstrations, hands-on laboratory experiments.
Few modern writers have been as adored, reviled, translated, or adapted as Fyodor Dostoevsky. In this seminar, we explore the “afterlife/survival” (
Überleben
) of Dostoevsky’s writings, with a particular emphasis on his reception and transformations during the decades of global modernism. We will ask: when and how was the dominant 20th-century image of Dostoevsky made? How is this image reflected, and refracted, in the later theory and practice of the novel? What resonances has it found across the political spectrum, from the pre-revolutionary Russian Empire through interwar Europe, the mid-century United States, and our present era of resurgent technocracy, imperialism, and nationalism? The first half of the course focuses on the Silver Age writers who did most to frame Dostoevsky’s legacy and artistic persona for an international readership. In the second half, we turn to a range of historical, literary, and theoretical contexts where this legacy comes into play. Students will take an active role in researching and shaping the story about Dostoevsky’s uncanny “survival” that our course tells, engaging with a range of readings in modernist literature, criticism, and novel theory. Midway through the course, each student will be responsible for a reception case study, researching either a place and time where Dostoevsky was widely influential or (by permission) a single author whose work comes into close dialogue with his.
Note: Russian-language readings will be provided in the original; many are also available in translation. Other readings will be provided and discussed in English translation, though reading in the original is always encouraged. The course is open to all graduate students by permission.
Prerequisites: the instructors permission. This course develops theory of designing markets--namely, ;mechanisms; of allocating resources--that are efficient, fair and non-manipulable. Understanding the incentives participants face under alternative mechanisms will be a central theme of the course. Specifically, the course will consist of two parts. The first part deals with environments where monetary transfers can be used, and focuses on topics such as optimal nonlinear pricing, optimal auction design, property rights assignment, dynamic mechanisms and assignment games/sponsored search auctions in Internet advertising. The second part concerns market design without monetary transfers and discusses matching theory as a primarily tool for analyzing the topic. Specifically, we shall discuss matching of agents on one side with agents on the other and matching of agents to indivisible resources/positions, and apply the theories to problems of house allocation, centralized labor market matching, and school choice.
Review of random variables. Random process theory: stationary and ergodic processes, correlation functions and power spectra, non-stationary, non-white and non-Gaussian processes. Uncertainty quantification and simulation of environmental excitations and material/media properties, even when subject to limited/incomplete data: joint time-frequency analysis, sparse representations and compressive sampling concepts and tools. Stochastic dynamics and reliability assessment of diverse engineering systems: complex nonlinear/hysteretic behaviors and/or fractional derivative modeling. Emphasis on solution methodologies based on Monte Carlo simulation, statistical linearization, and Wiener path integral. Examples from civil, marine, mechanical and aerospace engineering.
Prerequisites: ECON G6216 and G6412, or the instructor's permission. This course deals with business cycle theories and methods for evaluating such theories. The course extends the canonical real business cycle model to analyze models with cyclical variation in markups, models of endogenous fluctuations, and models of news-driven short-run fluctuations. Attention is given to numerical methods to approximate the dynamics implied by stochastic general equilibrium models, with particular emphasis given to perturbation methods. The course will also include an operational introduction to full and limited-information approaches to the estimation of DSGE models.
For whom does the dramatic chorus speak? In this graduate seminar, we track the figure of the chorus in drama, theater, and performance from antiquity to modernity, investigating how the chorus represents collectivity and enacts new social forms onstage. The chorus, as we’ll find, can be an instrument of tyranny or of transformation, depending on the members who comprise it – whether they are an assembly of elders (Sophocles’s Antigone), a gathering of survivors (Euripides’s The Trojan Women), a cadre of revolutionaries (Bertolt Brecht’s The Mother), a crowd of athletes 2 (Elfriede Jelinek’s Sports Play), or a digital network of chattering AI machines (Annie Dorsen’s Prometheus Firebringer). Our seminar attends to how the chorus shifts across historical, geographic, and cultural contexts: from ancient Athens to postwar Germany, and from American mass culture to decolonizing movements in the Caribbean, the chorus has proven to be a remarkably flexible and resilient element of live art. The course offers graduate students a broad introduction to canonical works of dramatic literature both ancient and modern, while also featuring lesser-known plays and productions by emerging artists. For the first two-thirds of the term, our sessions will juxtapose a dramatic text with a work of social theory (by thinkers including Nietzsche, Simmel, and Goffman), as well as at least one scholarly essay. Our broader goal in the course is to develop fluency in critical methods at the intersection of drama, philosophy, and social theory. In this course, that is, we approach theater as not only responsive to traditions of social thought, but also generative of new social practices and collective habits of body and mind. In addition to regular weekly assignments, students will complete one short essay and one seminar-length paper at the end of the term.
Lifestyle medicine is an area of medicine focusing on lifestyle modification for prevention and treatment of chronic illness. The course aims to describe the practice of lifestyle medicine and evidence-based understanding of 6 key principles of lifestyle medicine, including nutrition, physical activity, stress management, sleep, social connection and risk avoidance. Through the course students will learn the evidence behind the 6 principles and how to apply them in a clinical encounter. Each module will begin with asynchronous pre-class work, followed by a synchronous didactic lecture and interactive workshop, and then concluded with post-class homework on the topic.
What can we learn from anthropological and ethnographic research in and about a damaged world, a world confronted by the violence and effects of war, climate change, transnational migration, post-industrial abandonment, and the lives and afterlives of colonialism and slavery? What are the ethnographic debates that address the catastrophes produced by capitalism and the lifeforms that emerge out of its ruins? What types of anthropological critique emerge in times enunciated as ‘the end of the world’? And what comes after this end? Ethnographies at the End of the World addresses these questions by paying close attention to some of the most relevant debates in contemporary anthropological theory and anthropological critique. These debates include, among others, discussions on violence and trauma, the politics of life and death, the work of memory and oblivion, and the material entanglements between human and non-human forms of existence. The aim of this seminar is to generate a discussion around the multiple implications of these theoretical arrangements and how anthropologists deploy them in their ethnographic understandings of the world we live in. In doing so, this course provides students with a fundamental understanding and conceptual knowledge about how anthropologists use and produce theory, and how this theoretical production is mobilized as a social critique. This course is reading intensive and operates in the form of a seminar. It is intended, primarily, for MA students in the department of anthropology and graduate students in other departments.
This seminar surveys the defining political economy issues of our time. It examines the interplay between politics and economics in the key substantive issue areas of economic growth, institutional comparative advantage, financial crises, industrial relations, social welfare, and gender. The seminar surveys the most provocative, influential contributions in multiple fields of study utilizing a wide range of research methods. The course equips students with the conceptual and empirical tools to better understand current developments, provides exposure to multiple perspectives, and builds confidence in developing one’s own point of view.
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
This course provides an advanced introduction to core theories of international relations and their application to real-world policymaking in a global context. Students will examine foundational and contemporary analytical frameworks, including realism, liberalism, constructivism, game theory, selectorate theory, decision-making models, two-level game theory, and complex interdependence. Through a structured combination of lectures, discussions, and collaborative group analysis, students will apply these theories to contemporary global issues, with a focus on the U.S.–China relationship. The course culminates in a two-session policy forum, in which students present theory-based policy analyses they have developed over the semester. Emphasis is placed on practical application, clarity of analysis, and professional presentation.
This is a course for thoughtful people who wish to influence actual policy outcomes related to sustainability challenges in major cities. Its objective is not to provide a primer on urban sustainability solutions; this is readily available from textbooks and will change by the time you are in a position to act. Rather, the course’s objective is to prepare you for the kind of challenges that will face you as a policy practitioner in the field of urban sustainability. Cities are increasingly recognized as a key level of government for environmental and sustainability policy. As at all levels, politics and policy are intensely intertwined, and perhaps more so at the local level because the decisions involved often affect constituents directly and intimately --in their neighborhoods, in their homes, in their commutes. This reading-heavy colloquium explores the politics and the policy of urban sustainability from the perspective of someone who wishes to effect change. It culminates in a team project in which students act as a sustainability policy team in a mayoral (or equivalent) office in one of the world’s major cities. The course considers key components of the city itself, with the objective of understanding what shapes the city and its impact on the environment. It mainly uses case studies from the twentieth-century United States, paired with international readings to allow a comparison with other experiences. The focus on deep case studies allows the consideration of the situations, constraints, and political dynamics of specific situations. It is intended to provide students with the ability to recognize patterns in urban political and policy dynamics related to sustainability. These are paired with an overview of leading solutions and how the professor believes practitioners should evaluate them for their own cities. The course also prominently features in-class presentations and discussions of the students’ main project: a team-based memo making a specific recommendation to solve a problem in a specific major world city, which is presented twice, once for a diagnosis of the problem in a given city and a second time with a policy recommendation. This project is the major portion of the overall grade for the class, and is used to allow the students to wrestle with the challenge of turning ideas from past and present into successful urban sustainability policies that can get implemented in a political and institutional world. In order to cover issues in depth, this course is not exhaustive;
Networks help us describe the complex interactions that occur among large populations of distinct
entities. Some have argued that their incorporation into economic models represents a paradigm
change for the ?eld. This course will introduce you to the central questions in current networks
research as well as the tools and methods used to study networks in economic theory. Topics we
will cover include network games and interventions, di?usion processes, network formation, social
learning and opinion dynamics, and networked markets. The course is divided into two halves. The
rst half will consist of lectures on these topics. In the second half, we will read and discuss recent
research papers.
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
MPA Financial Management I Core.
This course provides a practical introduction to budgeting as a critical tool for planning, decision-making, and leadership across the public, private, and nonprofit sectors. Students will develop hands-on skills in budget creation, financial analysis, and cash flow management while exploring how budgets shape organizational strategy and operations.
MPA Financial Management Core II.
This course introduces students to budgeting and financial management in the public sector, with an emphasis on real-world application and analytical skill development. Drawing on current and historical challenges—including the 2008 financial crisis and the COVID-19 pandemic—students will explore the political, technical, and managerial dimensions of public budgeting in the United States.
Topics include the structure and function of operating and capital budgets, principles of revenue generation, deficit spending, and debt financing. Students will work with actual federal, state, and local budgets, gaining experience in constructing, analyzing, and presenting budgets in a government context. By the end of the course, students will be able to evaluate budget proposals and make data-informed recommendations grounded in public policy goals and financial feasibility.
Prerequisites: ECON G6211 and ECON G6212. This course provides an overview of topics in industrial organization (IO) economics. Its goals are to survey the main outlines of modern IO, to develop key theoretical ideas, to demonstrate important techniques, to link theory to empirical work, and to relate theoretical and empirical results to policy issues. Empirical two-period models. Empirical single-agent and multiple-agent dynamic models.
MPA Financial Management II Core.
This course introduces nonprofit and social enterprise finance, financial management, and budgeting. The course is practical and hands-on. The course will examine how financial management principles assist nonprofit and social enterprise leaders make operating, program, and long-term financial and strategic decisions. Students will learn underlying concepts and practical skills through readings, discussions, Case Studies, and assignments. The course is designed to give students a range of core financial and managerial skills that are especially relevant to students who want to go on to establish, manage, or work in nonprofit organizations or social enterprises.
This course introduces the students to the field of Organizational Economics. We combine theoretical and empirical methods to study the nature, design, and performance of organizations. Organizations, such as firms, bureaucracies, and political parties, live in a second-best world, where inefficiencies are inevitable. Our goal is to understand and measure these inefficiencies, study their causes and how to minimize them. This course is divided in two parts of equal length. The first part introduces a few of the main theoretical models and findings from the organizational-economics literature. The second part focuses on how to bring the models to the data. By design, the course is intended for a broad set of students: those who are theoretically inclined, those who are empirically inclined, and those who are both. Many of the tools and skills that are developed in this course will be useful not only within organizational economics but, more broadly, to other fields such as industrial organization, political economy, development economics. Our ultimate goal is to accelerate the students' transition toward conducting their own independent research.
MPA Financial Management Core II.
This course provides students with a foundational understanding of financial accounting, with an emphasis on interpreting financial statements. Students will study the three core financial statements—the balance sheet, income statement, and statement of cash flows—and learn the accounting principles and rules that shape how financial data is recorded and presented.
Through hands-on exercises and case examples from service, manufacturing, and retail operations, students will learn to apply double-entry bookkeeping, distinguish between key accounting treatments such as capitalizing and expensing, and evaluate how management’s assumptions and estimates influence reported outcomes. The course aims to build financial fluency for policy professionals and prepare students to critically assess financial information to support evidence-based decision-making.
A study of advanced cryptographic research topics such as: secure computation, zero knowledge, privacy, anonymity, cryptographic protocols. Concentration on theoretical foundations, rigorous approach, and provable security. Contents varies between offerings. May be repeated for credit.
Affine and projective varieties; schemes; morphisms; sheaves; divisors; cohomology theory; curves; Riemann-Roch theorem.
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search
This is a Law School course. For more detailed course information, please go to the Law School Curriculum Guide at: http://www.law.columbia.edu/courses/search