Columbia University

Online courses, programs, and events available through Columbia’s Columbia+ platform.

University news and multimedia content showcasing research, campus life, and impact (includes YouTube channel "Columbia Today and Beyond").

Statistical snapshot of the University — enrollment, facts, and institutional metrics maintained by the Office of Planning and Institutional Research.

University events and calendars (lectures, conferences, exhibitions, and community events) coordinated across campus units.

Graduate-level lecture notes on continuous-time Markov chains (CTMC): formal CTMC definition, exponential holding times, transition rate matrix (Q), embedded discrete-time chain, Chapman–Kolmogorov equations, Kolmogorov backward equations P'(t)=Q P(t) with solution P(t)=e^{Qt}, balance equations and stationary distributions, and worked examples (Poisson process, M/M/1, M/M/c, M/M/∞, birth-death processes, Jackson networks). (Lecture notes authored/hosted by Karl Sigman.)

Focused lecture/notes on the gambler's ruin problem (solution formula for hitting probabilities, limit behavior as N→∞, insurance and risk examples, random walk hitting probabilities and worked examples). Useful for courses in stochastic processes and applied probability.

Faculty-hosted collection of Dustin R. Rubenstein’s selected publications (research articles, book chapters, and hosted PDFs) spanning social evolution, cooperative breeding, behavioral ecology, and integrative animal behavior. The lab page provides links to many open PDFs and the lab/CV for contact and mentoring information.

Comprehensive CV for Dustin R. Rubenstein including positions, honors, grants, publications, teaching, and service (PDF hosted on Columbia servers).

Examples of research PDFs hosted on faculty pages (e.g., Rubenstein Lab PDFs such as 'Superb starlings swap helper and breeder roles with kin and non-kin' (2025 Nature PDF) and numerous peer-reviewed articles and book chapters).

Research article and method paper describing LARS-based variable selection constrained by effect heredity for the analysis of highly fractionated designed experiments. (Hosted on a Columbia departmental page.)

Research paper (Kun Soo Park and Ward Whitt) presenting continuous-time Markov chain (CTMC) models and fitting algorithms to estimate the lockup premium for extended hedge-fund lockups. The paper includes a sensitivity analysis showing the lockup premium increases with the persistence factor (γ) and the return volatility (σ), and decreases with the death probability (δ). It also examines sensitivity to measurement times (x_u, T_p, x_d) and finds that while T_p has relatively smaller impact, T_u and T_d can be significant. Numerical premiums from the new CTMC models are very similar to previous DTMC models (slightly larger when intra-year transitions from good to sick are allowed). The approach is noted as applicable to other financial settings with good-sick-dead structure (e.g., private equity lockups).

Summary of sensitivity-analysis conclusions from Park & Whitt CTMC work: lockup-premium sensitivity dominated by persistence factor (γ), volatility (σ) and performance-measure death probability (δ); measurement-timing parameters T_u and T_d can materially affect premiums; DTMC yearly-update models produce similar numerical premiums to CTMC models, indicating modest effect from intra-year transitions.

Undergraduate textbook 'Animal Behavior' (12th ed., 2022) authored by Dustin R. Rubenstein, published by Oxford University Press; listed on the Rubenstein Lab books page.

Previous edition (11th ed., 2018) of the Animal Behavior textbook coauthored by Rubenstein and Alcock, information and endorsements listed on the books page.

Edited volume 'Comparative Social Evolution' (2017) with P. Abbot (Cambridge University Press), listed on the Rubenstein Lab books page.

Review and update on how environmental conditions shape cooperative breeding and social evolution (Sheng-Feng Shen & Dustin R. Rubenstein, 2025/2026). Key points: the 'dual benefits' framework distinguishes resource-defense (RD) and collective-action (CA) grouping benefits; RD dominates in stable, patchy habitats and favors kin-based groups, whereas CA dominates in fluctuating/harsh environments and can favor more flexible kin/non-kin groups; CA-based cooperative species may show greater resilience to environmental change (larger ranges, higher abundances). The paper summarizes testable experiments and implications for conservation under climate change.

Historical computing system installed at Columbia's Watson Scientific Computing Laboratory beginning in August 1955: the IBM 650 was the first general-purpose computer to be installed and used at Columbia, supported more than 200 research projects, and served as a hands‑on teaching platform. Key technical characteristics: vacuum-tube logic, decimal arithmetic with 10-digit words plus sign, rotating drum memory (1,000/2,000/4,000 words), instruction format that included the address of the next instruction (to accommodate drum latency), optional 60‑word core memory buffer, and support for assemblers (SOAP, written at Watson Lab in 1955) and later Fortran compilers (FORTRANSIT, 1957). The Watson Lab 650s remained in operation through the 1950s and into the early 1960s.

SOAP was developed/written at Watson Scientific Computing Laboratory (Stan Poley, 1955) as a symbolic optimizing assembler for the IBM 650, helping programmers place instructions optimally on drum memory to avoid latency. SOAP was a key early software artifact created at Columbia and used in Watson Lab courses.

Watson Lab supported intensive computing courses and produced user guides and teaching materials (examples cited on the computing-history page include Mace & Alsop 'A Simplified System for the Use of an Automatic Calculator' and Jeenel's expanded book based on Watson Lab courses). These materials documented early programming practices on drum machines and are part of Columbia's computing history resources.

Degree programs across Columbia’s schools, including the Core Curriculum for undergraduates and professional and graduate programs across 17 schools.

University-wide research mission spanning basic and applied research across disciplines, including specialized centers and initiatives (e.g., Columbia Climate School, engineering research centers).

Academic medical care, research, and public-health programs delivered through Columbia’s Irving Medical Center and affiliated health schools.

A network of Columbia Global Centers and international programs spanning multiple cities worldwide to support research, teaching, and engagement (launched 2009).

Library system resources including Butler Library and Rare Book & Manuscript Library supporting teaching, research, and public programs.

Office of Government & Community Affairs, Columbia Neighbors, and programs that connect the university with surrounding communities and public policy stakeholders (including Community Benefits Agreement related to Manhattanville development).

Campus events, public lectures, conferences, and continuing-education offerings (including Columbia+ online courses).

Intercollegiate athletics, campus activities, and student services across Columbia’s campuses.

Giving channels, alumni engagement, and fundraising initiatives supporting university priorities.

Interdisciplinary school created to address the global climate crisis (announced 2020) supporting research, education, and policy engagement on climate change.

New research, teaching and administrative buildings and public spaces developed in Manhattanville beginning in 2009 (includes arts and science research facilities, e.g., Jerome L. Greene Science Center, Lenfest Center for the Arts, and The Forum).

University hub for student-facing services and initiatives — Housing, Dining, Columbia Health, University Life, student groups, and diversity, equity and inclusion initiatives.

On-campus housing and residence-life resources for undergraduate and graduate students (policies and residence information available via Housing).

Dining services and meal-plan resources for students across Columbia campuses.

Student health services and resources provided through Columbia Health.

University Life provides centralized information and initiatives for graduate and undergraduate students, including student groups and campus community resources.

Arts at Columbia programming and intercollegiate athletics (Go Columbia Lions).

Columbia hosts faculty research labs and program pages that publish publications, CVs, data and teaching materials (example: the Rubenstein Lab/DRR faculty pages with publication PDFs, CV, and lab contact info). These lab pages provide access to papers, datasets, and course materials and support research, mentoring, and field programs.

Departmental and faculty research in applied probability and stochastic processes, including lecture notes and course materials on gambler's ruin, CTMCs and related stochastic-process topics (useful for students and researchers in probability and queueing theory).

Historical computing services and intensive short courses provided through the Watson Scientific Computing Laboratory (examples: hands-on use of IBM 650 machines beginning in August 1955; SOAP assembler development and Watson Lab 650-based programming courses). The Watson Lab 650s supported more than 200 Columbia research projects and were used for computing instruction and faculty/ student research in the 1950s–1960s.

Thomas Hunt Morgan Professor of Conservation Biology; Professor, Department of Ecology, Evolution and Environmental Biology; Director / Principal Investigator, Rubenstein Lab (Columbia University)

Professor, Department of Industrial Engineering and Operations Research (author of FCLT review; coauthor of CTMC hedge-fund lockup paper)

Author (Gambler’s Ruin / Stochastic Processes lecture notes), Columbia-affiliated

20th President (named 2023; first female president) — former

Chief Executive Officer, Columbia University Irving Medical Center; appointed Interim President (Aug 2024) before returning to medical center leadership

Board of Trustees Co-Chair; appointed Acting President (Mar 28, 2025)