# Research

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.

- Algebraic Geometry
- Analysis and PDE
- Computational and Mathematical Biology
- Dynamical Systems and Ergodic Theory
- Geometry
- Physical Applied Mathematics
- Probability Theory
- Scientific Computing

#### 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 AND 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
- Elliptic and Parabolic Problems: Scott Armstrong, Fengbo Hang, Bob Kohn, Fang-Hua Lin, Nader Masmoudi, Sylvia Serfaty
- Geometric Measure Theory: Fang-Hua Lin
- Harmonic Analysis: Pierre Germain, Sinan Gunturk
- Integrable Systems: Percy Deift, Henry McKean
- Mathematical Elasticity: Bob Kohn
- Mathematical Fluid Mechanics: Pierre Germain, Jonathan Goodman, Nader Masmoudi, Jalal Shatah
- Nonlinear Waves: Pierre Germain, Nader Masmoudi, Jalal Shatah
- Spectral Theory: Percy Deift
- Related Seminars: Analysis Seminar, Applied Mathematics Seminar, Harmonic Analysis and Signal Processing Seminar

#### COMPUTATIONAL AND 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, Daniel Tranchina
- Computational Biology: Leslie Greengard, Bhubaneswar Mishra, Alex Mogilner, Mehryar Mohri, Aaditya Rangan, Daniel Tranchina
- Mathematical Biology: David Cai, Alex Mogilner, Charles Newman, Jerome Percus, Charles Peskin
- Related Seminars: Applied Mathematics Seminar, Biomathematics / Computational Biology Colloquium, Computational Neuroscience and Biology Seminars

#### DYNAMICAL SYSTEMS AND 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: Tim Austin, 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.

- Symplectic Geometry: Sylvain Cappell, Mikhael Gromov
- Geometric Analysis: Jeff Cheeger, Mikhael Gromov, Fengbo Hang, Bruce Kleiner, Fang-Hua Lin, Robert Young
- Geometric Group Theory: Tim Austin, Mikhael Gromov, Bruce Kleiner, Robert Young
- Metric Geometry: Jeff Cheeger, Mikhael Gromov, Bruce Kleiner, Robert Young
- Geometric Topology: Sylvain Cappell, Bruce Kleiner
- 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, David Cai, Aleksandar Donev, Edwin Gerber, Jonathan Goodman, David Holland, Miranda Holmes-Cerfon, Richard Kleeman, Andrew Majda, Michael O'Neil, Olivier Pauluis, Charles Peskin, Leif Ristroph, Michael Shelley, K. Shafer Smith, K.R. Sreenivasan, Esteban Tabak, Jun Zhang
- Plasma Physics: Harold Weitzner, Antoine Cerfon
- Statistical Mechanics: Aleksandar Donev, Andrew Majda, Charles Newman, Eric Vanden-Eijnden, Miranda Holmes-Cerfon, Daniel Stein
- Molecular Dynamics: Aleksandar Donev, Eric Vanden-Eijnden, Miranda Holmes-Cerfon
- Dynamical Systems: Lai-Sang Young
- Materials Science: Miranda Holmes-Cerfon, Bob Kohn, Georg Stadler
- Related Seminars: Applied Math Lab Seminar, Applied Mathematics 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, Joel Spencer, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Markov Chain Mixing Times: Yuri Bakhtin, Eyal Lubetzky
- Motion in Random Media: Scott Armstrong, Yuri Bakhtin, Gerard Ben Arous, Eyal Lubetzky, Charles Newman, Vladas Sidoravicius, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Random Graphs: Eyal Lubetzky, Joel Spencer
- Random Matrices & Spectral Analysis: Gerard Ben Arous, Paul Bourgade, Percy Deift, Sylvia Serfaty, Ofer Zeitouni
- Statistical Physics, Percolation & Interacting Particle Systems: Gerard Ben Arous, Eyal Lubetzky, Charles Newman, Vladas Sidoravicius, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Stochastic Analysis, PDEs & Diffusions: Scott Armstrong, Yuri Bakhtin, Gerard Ben Arous, Percy Deift, Henry McKean, S. R. Srinivasa Varadhan, Ofer Zeitouni
- Related Seminars: Probability and Mathematical Physics Seminar, Student Probability Seminar

#### SCIENTIFIC COMPUTING

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, Antoine Cerfon, Aleksandar Donev, Dimitris Giannakis, Jonathan Goodman, Leslie Greengard, Michael O'Neil, Michael Shelley, Georg Stadler, Margaret Wright, Denis Zorin
- Numerical Analysis: Michael Overton, Aaditya Rangan, Nick Trefethen, Olof Widlund
- Scattering Theory: Yu Chen, Leslie Greengard
- Related Seminars: Applied Mathematics Seminar, Numerical Analysis and Scientific Computing Seminar