# Research Areas

The Courant Institute has a tradition of research which combines pure and applied mathematics, with a high level of interaction between different areas. Below we list some of the current areas of research. The choice of categories is somewhat arbitrary, as many faculty have interests that cut across boundaries, and the fields continue to evolve.

We give a very brief overview of the research in each area; more detailed information may be found on individual faculty webpages.

**Computer Science**

## Algorithms & Theory

Three key issues for an algorithm are: Is it correct? How efficient is it? Can one do better? Our strong and diverse group seeks provable answers to these questions. It focuses on problems and questions in the following areas: complexity theory, cryptography, computational geometry, computational algebra, randomness (in algorithm design and average case analysis) and algorithmic game theory.

- Algorithms & Theory: Richard Cole, Yevgeniy Dodis, Subhash Khot, Oded Regev, Victor Shoup, Alan Siegel, Joel Spencer, Chee Yap
- Research Groups: Cryptography Group, Algorithms and Geometry Collaboration
- Related Seminars: Cryptography Seminar, CS Theory Seminar, TCS+ Online Theory Seminar, Geometry Seminar

## Computational Biology

Computational Biology uses mathematical, statistical and algorithmic techniques to solve problems arising in biological research. The faculty working in this area collaborate with biologists and applied mathematicians on a broad range of problems in genomics, proteomics, molecular modeling, systems biology. The CS Department, along with a number of other departments and schools (Biology, Chemistry, Mathematics, Neuroscience, Sackler Institute of Biomedical Sciences, Mt. Sinai School of Medicine) participates in the interdisciplinary Computational Biology Program.

- Computational Biology: Richard Bonneau, Bhubaneswar Mishra, Tamar Schlick, Dennis Shasha
- Research Groups: Center for Genomics and Systems Biology, Center for Data Science, Bonneau Lab, Computational Biology, Chemistry, and Biomathematics Group

## Formal Methods & Verification

The long-term goal of the formal methods group is to increase the reliability of hardware and software systems by providing tools and techniques for the analysis of these systems. In formal analysis, a mathematical model of a system is developed, which can then be used to prove properties of the system or to discover bugs in the system when the proof fails. The activities and interests of the formal methods group cover a broad spectrum, from the study of mathematical foundations in programming languages and logic, to the implementation of verification tools and the application of these tools for proving the correctness of computer systems.

- Formal Methods & Verification: Patrick Cousot, Benjamin Goldberg, Thomas Wies
- Research Groups: Analysis of Computer Systems Group
- Related Seminars: Formal Methods Seminar

## Graphics, Vision & User Interfaces

Researchers in Computer Graphics work on computational and mathematical techniques for creating and manipulating computer representations of real and virtual objects and making images of such objects. The main directions of computer graphics research at NYU include animation, geometric modeling, physically-based simulation and computational photography.

The area of Computer Vision is concerned with algorithms and theory necessary to extract information from visual data (images, video, range scans, stereo images, 3D MRI and CAT scan data etc). There is a growing overlap between computer vision and graphics research, as the data acquired from images and video is increasingly used in computer graphics applications.

- Graphics, Vision & User Interfaces: Rob Fergus, Davi Geiger, Yann LeCun, Daniele Panozzo, Kenneth Perlin, Claudio Silva, Denis Zorin
- Research Groups: Media Research Lab, VLG Lab (Vision, Learning, Graphics), CILVR Lab (Computational Intelligence, Learning, Vision, and Robotics), MAGNET (Media and Games Network)

## Machine Learning

Machine learning is concerned with developing of mathematical foundations and algorithm design needed for computers to learn, that is, to adapt their responses based on information extracted from data. For example, learning algorithms may allow a robot to navigate an unknown environment, improving its performance as it acquires more and more data, or a voice-controlled system to improve its recognition of a person's speech after analysis of a sufficient number of samples. Machine learning techniques draw on many fundamental areas from statistics to theoretical computer science, and are used in a broad variety applications: robotics, speech analysis, health care, finance, computer games, handwriting recognition to name just a few.

- Machine Learning: Kyunghyun Cho, Rob Fergus, Davi Geiger, Yann LeCun, Mehryar Mohri, Foster Provost
- Research Groups: Center for Data Science, CILVR Lab (Computational Intelligence, Learning, Vision, and Robotics), Clinical Machine Learning Group,
- Related Seminars: CILVR Lab Seminar

## Natural Language, Speech Processing, & Knowledge Representation

The amount of text which is available in electronic form is growing at an explosive rate. In addition to the web, large quantities of text are being collected for medical, legal, commercial, and scientific applications. But the tools for getting the information we need out of this text are still quite primitive. Our research groups in natural language processing are building systems to to extract specific information from large text collections, and to present it in the user's preferred language. A closely related area, speech processing, deals with coding, synthesis and extraction of information from speech signals.

Natural language processing has a long history at NYU. The Linguistic String Project was one of the pioneers in natural language processing research in the United States. The Proteus Project focuses on automatically learning the linguistic knowledge needed for information extraction and machine translation. It has developed extraction systems in English and Japanese, and a series of language-independent translation models. It also conducts a wide range of basic research, and develops large-scale dictionaries and other resources for natural language processing.

- Natural Language, Speech Processing, & Knowledge Representation: Kyunghyun Cho, Ernest Davis, Ralph Grishman, Adam Meyers, Mehryar Mohri, Satoshi Sekine
- Research Groups: The Proteus Project

## Networks, Operating & Distributed Systems

Systems and networking research explores how to structure the basic software running on individual computers and how to coordinate between different computers. Significant challenges include how to support increasing numbers of processors in modern computer systems, leverage the many embedded and mobile computing devices, and build services that scale to a global audience.

- Networks, Operating & Distributed Systems: Zvi Kedem, Jinyang Li, Theodore Rappaport, Dennis Shasha, Lakshminarayanan Subramanian, Michael Walfish
- Research Groups: Systems Research Group, Center for Technology and Economics Development, NYU Wireless

## Scientific Computing (Computer Science)

Scientific Computing has a long tradition at the Courant Institute, which was founded by Richard Courant at the dawn of the computer era. Computers were invented in the late 1940's and early 1950's for exactly one purpose: solving hard scientific and engineering problems which required too much numerical computation to do by hand. Now, virtually all branches of science and engineering rely heavily on computing. Several areas of scientific computing, especially linear algebra and optimization, are important in data science, machine learning and physics-based graphics. Many faculty at Courant, both in the Computer Science and Mathematics Departments, have strong interests in Scientific Computing, both in specific application areas and in general techniques and analysis that have broad applicability.

- Scientific Computing: Marsha Berger, Leslie Greengard, Michael Overton, Olof Widlund, Margaret Wright, Denis Zorin
- Research Groups: Center for Data Science, Courant Mathematics and Computing Laboratory
- Related Seminars: Numerical Analysis and Scientific Computing Seminar, Applied Math / Applied Math Lab Seminar

## Algebraic Geometry

The research focus of the Algebraic geometry group at Courant lies at the interface of geometry, topology, and number theory. Of particular interest are problems concerning the existence and distribution of rational points and rational curves on higher-dimensional varieties, group actions and hidden symmetries, as well as rationality, unirationality, and hyperbolicity properties of algebraic varieties.

- Algebraic Groups and Representation Theory: Fedor Bogomolov, Yuri Tschinkel
- Arithmetic Geometry: Fedor Bogomolov, Alena Pirutka, Yuri Tschinkel
- Complex Algebraic Geometry: Fedor Bogomolov, Sylvain Cappell, Yuri Tschinkel
- Holomorphic Symplectic Geometry: Fedor Bogomolov, Yuri Tschinkel
- Number Theory: Alena Pirutka, Yuri Tschinkel
- Related Seminars: Algebraic Geometry Seminar, Geometry Seminar, Number Theory Seminar

## Analysis & PDE

Most, if not all, physical systems can be modeled by Partial Differential Equations (PDE): from continuum mechanics (including fluid mechanics and material science) to quantum mechanics or general relativity. The study of PDE has been a central research theme at the Courant Institute since its foundation. Themes are extremely varied, ranging from abstract questions (existence, uniqueness of solutions) to more concrete ones (qualitative or quantitative information on the behaviour of solutions, often in relation with simulations). The study of PDE has strong ties with analysis: methods from Fourier Analysis and Geometric Measure Theory are at the heart of PDE theory, and theory of PDEs often suggest fundamental questions in these domains.

- Calculus of Variations: Scott Armstrong, Bob Kohn, Sylvia Serfaty, Guido de Philippis
- Elliptic and Parabolic Problems: Scott Armstrong, Fengbo Hang, Bob Kohn, Fang-Hua Lin, Nader Masmoudi, Sylvia Serfaty, Vlad Vicol
- Geometric Measure Theory: Fang-Hua Lin, Guido de Philippis
- Harmonic Analysis: Sinan Gunturk
- Integrable Systems: Percy Deift
- Mathematical Elasticity: Bob Kohn
- Mathematical Fluid Mechanics: Jonathan Goodman, Nader Masmoudi, Jalal Shatah, Vlad Vicol
- Nonlinear Waves: Nader Masmoudi, Jalal Shatah, Vlad Vicol
- Spectral Theory: Percy Deift
- Related Seminars: Analysis Seminar, Applied Math / Applied Math Lab Seminar, Harmonic Analysis and Signal Processing Seminar

## Computational & Mathematical Biology

Biological applications of mathematics and computing at Courant include genome analysis, biomolecular structure and dynamics, systems biology, embryology, immunology, neuroscience, heart physiology, biofluid dynamics, and medical imaging. The students, researchers and faculty who work on these questions are pure and applied mathematicians and computer scientists working in close collaboration with biological and medical colleagues at NYU and elsewhere.

- Computational Neuroscience: David McLaughlin, Charles Peskin, Aaditya Rangan, John Rinzel, Lai-Sang Young
- Computational Biology: Leslie Greengard, Bhubaneswar Mishra, Alex Mogilner, Mehryar Mohri, Aaditya Rangan
- Mathematical Biology: Alex Mogilner, Charles Newman, Jerome Percus, Charles Peskin, Lai-Sang Young
- Research Groups: Research and Training Group in Mathematical Modeling and Simulation
- Related Seminars: Applied Math / Applied Math Lab Seminar, Biomathematics / Computational Biology Colloquium, Computational Neuroscience and Biology Seminars

## Dynamical Systems & Ergodic Theory

The subject of dynamical systems is concerned with systems that evolve over time according to a well-defined rule, which could be either deterministic or probabilistic; examples of such systems arise in almost any field of science. Ergodic theory is a branch of dynamical systems concerned with measure preserving transformation of measure spaces, such as the dynamical systems associated with Hamiltonian mechanics. The theory of dynamical systems has applications in many areas of mathematics, including number theory, PDE, geometry, topology, and mathematical physics.

- Dynamical Systems: Yuri Bakhtin, Lai-Sang Young
- Related Seminars: Dynamical Systems Seminar

## Geometry

Geometry research at Courant blends differential and metric geometry with analysis and topology. The geometry group has strong ties with analysis and partial differential equations, as there are many PDE's and techniques of interest to both groups, such as Einstein's equations, the minimal surface equation, calculus of variations, and geometric measure theory.

- Convex Geometric Analysis: Erwin Lutwak, Deane Yang, Gaoyong Zhang
- Symplectic Geometry: Sylvain Cappell, Mikhael Gromov
- Geometric Analysis: Jeff Cheeger, Mikhael Gromov, Fengbo Hang, Bruce Kleiner, Fang-Hua Lin, Robert Young, Valentino Tosatti, Chao Li
- Geometric Group Theory: Mikhael Gromov, Bruce Kleiner, Robert Young
- Metric Geometry: Jeff Cheeger, Mikhael Gromov, Bruce Kleiner, Robert Young
- Geometric Topology: Sylvain Cappell
- Related Seminars: Geometric Analysis and Topology Seminar

## Physical Applied Mathematics

A central theme at the Courant Institute is the study of physical systems using advanced methods of applied mathematics. Currently, areas of focus include fluid dynamics, plasma physics, statistical mechanics, molecular dynamics and dynamical systems. The tradition at the Institute is to investigate fundamental questions as well as to solve problems with direct, real-world applications. In doing so, the people looking into these questions build on the strong synergies and fresh ideas that emerge in the frequent collaboration with analysis and PDE specialists as well as experts in scientific computing at the institute.

- Fluid Dynamics: Marsha Berger, Oliver Bühler, Aleksandar Donev, Edwin Gerber, Jonathan Goodman, David Holland, Richard Kleeman, Michael O'Neil, Olivier Pauluis, Charles Peskin, Leif Ristroph, Michael Shelley, K. Shafer Smith, K.R. Sreenivasan, Esteban Tabak, Jun Zhang
- Statistical Mechanics: Aleksandar Donev, Charles Newman, Eric Vanden-Eijnden, Daniel Stein
- Molecular Dynamics: Aleksandar Donev, Eric Vanden-Eijnden
- Dynamical Systems: Lai-Sang Young
- Materials Science: Bob Kohn, Georg Stadler
- Research Groups: Magneto-Fluid Dynamics Division, Research and Training Group in Mathematical Modeling and Simulation
- Related Seminars: Applied Math / Applied Math Lab Seminar, Atmosphere Ocean Science Colloquium, Atmosphere Ocean Science Student Seminar, Magneto-Fluid Dynamics Seminar

## Probability Theory

Domains of interest range from stochastic processes to random discrete structures to statistical physics (percolation, random matrices…), which has become more and more central in recent years. Probability theory has natural connections with a number of fields (computational methods, financial mathematics, mathematical physics, dynamical systems, graph theory) since a great number of phenomena can be best modeled or understood by probabilistic means.

- Large Deviations: Gerard Ben Arous, Charles Newman, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Markov Chain Mixing Times: Eyal Lubetzky
- Motion in Random Media: Scott Armstrong, Yuri Bakhtin, Gerard Ben Arous, Eyal Lubetzky, Charles Newman, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Random Graphs: Eyal Lubetzky
- Random Matrices & Spectral Analysis: Gerard Ben Arous, Paul Bourgade, Percy Deift, Sylvia Serfaty, Ofer Zeitouni
- Schramm-Loewner evolutions: Nina Holden
- Statistical Physics, Percolation & Interacting Particle Systems: Yuri Bakhtin, Gerard Ben Arous, Eyal Lubetzky, Charles Newman, S. R. Srinivasa Varadhan, Ofer Zeitouni, Nina Holden
- Stochastic Analysis, PDEs & Diffusions: Scott Armstrong, Yuri Bakhtin, Gerard Ben Arous, Percy Deift, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Related Seminars: Probability and Mathematical Physics Seminar, Student Probability Seminar

## Scientific Computing (Mathematics)

Courant faculty have interests in stochastic modeling in statistical and quantum mechanics, nonlinear optimization, matrix analysis, high-dimensional data analysis, and numerical solutions of the partial differential equations that lie at the heart of fluid and solid mechanics, plasma physics, acoustics, and electromagnetism. Central to much of this work is the development of robust and efficient algorithms. As these algorithms are applied to increasingly complex problems, significant attention is being devoted to the design of effective and supportable software.

- Computational Physics: Marsha Berger, Aleksandar Donev, Jonathan Goodman, Leslie Greengard, Michael O'Neil, Michael Shelley, Georg Stadler, Margaret Wright, Denis Zorin
- Numerical Analysis: Michael Overton, Aaditya Rangan, Olof Widlund
- Scattering Theory: Yu Chen, Leslie Greengard
- Research Groups: Research and Training Group in Mathematical Modeling and Simulation
- Related Seminars: Applied Math / Applied Math Lab Seminar, Numerical Analysis and Scientific Computing Seminar