The Courant Institute of Mathematical science is an independent division of New York University under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics. The Director of the Courant Institute directly reports to New York University's Provost and President and works closely with deans and directors of other NYU colleges and divisions respectively. The Courant Institute is named after Richard Courant, one of the founders of the Courant Institute and also a mathematics professor at New York University from 1936 to 1972.The Courant Institute is considered one of the most prestigious and leading mathematics schools and mathematical science research centers in the world. It is ranked #1 in applied mathematical research, #5 in citation impact worldwide, and #12 in citation worldwide. On the Faculty Scholarly Productivity Index, it is ranked #3 with an index of 1.84. It is also known for its extensive research in pure mathematical areas, such as partial differential equations, probability and geometry, as well as applied mathematical areas, such as computational biology, computational neuroscience, and mathematical finance. The Mathematics Department of the Institute has 18 members of the United States National Academy of science and five members of the National Academy of Engineering. Four faculty members have been awarded the National Medal of Science, one was honored with the Kyoto Prize, and nine have received career awards from the National Science Foundation. Courant Institute professors Peter Lax, S. R. Srinivasa Varadhan, Mikhail Gromov won the 2005, 2007 and 2009 Abel Prize respectively for their research in partial differential equations, probability and geometry. Louis Nirenberg received the Chern Medal in 2010, and Subhash Khot won the Nevanlinna Prize in 2014.The undergraduate programs and graduate programs at the Courant Institute are run independently by the Institute, and formally associated with the NYU College of Arts and Science and NYU Graduate School of Arts and Science respectively. Wikipedia.
Holmes-Cerfon M.,Courant Institute of Mathematical Sciences
Annual Review of Condensed Matter Physics | Year: 2017
Nano- and microscale particles, such as colloids, commonly interact over ranges much shorter than their diameters, so it is natural to treat them as "sticky," interacting only when they touch exactly. The lowest-energy states, free energies, and dynamics of a collection of n particles can be calculated in the sticky limit of a deep, narrow interaction potential. This article surveys the theory of the sticky limit, explains the correspondence between theory and experiments on colloidal clusters, and outlines areas where the sticky limit may bring new insight. © 2017 by Annual Reviews. All rights reserved.
Marx D.,Ruhr University Bochum |
Chandra A.,Indian Institute of Technology Kanpur |
Tuckerman M.E.,Courant Institute of Mathematical Sciences
Chemical Reviews | Year: 2010
A scientific review informed about the anomalous diffusion of hydroxide in basic solutions OH-. The discussion was initiated with a review of structural diffusion of H+ in acidic solutions, enabling a comparison to be made. The presentation was based on the formal and unifying 'presolvation concept' that was explained within the context of the H+ diffusion process. A theoretical formalism was used by the researchers to connect the qualitative predictions of the presolvation concept to actual mechanisms and the resulting kinetics in quantitative detail. The theoretical formalism allowed structural diffusion to be analyzed on the basis of appropriately defined population correlation functions. This theoretical framework led to consistent sets of various lifetimes and rates for OH- that were directly compared to H+.
Kidston J.,Princeton University |
Gerber E.P.,Courant Institute of Mathematical Sciences
Geophysical Research Letters | Year: 2015
Future climate predictions by global circulation models in the Coupled Model Intercomparison Project Phase 3 (CMIP3) archive indicate that the recent poleward shift of the eddy-driven jet streams will continue throughout the 21st century. Here it is shown that differences in the projected magnitude of the trend in the Southern Hemisphere are well correlated with biases in the latitude of the jet in the simulation of 20th century climate. Furthermore, the latitude of the jet in the models' 20th century climatology is correlated with biases in the internal variability of the jet stream, as quantified by the time scale of the annular mode. Thus an equatorward bias in the position of the jet is associated with both enhanced persistence of the annular mode, and an increased poleward shift of the jet. These relationships appear to be robust throughout the year except in the austral summer, when differences in forcing, particularly stratospheric ozone, make it impossible to compare the response of one model with another. These results suggest that the fidelity of a model's simulation of the 20th century climate may be related to its fitness for climate prediction. The cause of this relationship is discussed, as well as the implications for climate change projections. Copyright © 2010 by the American Geophysical Union.
E W.,Princeton University |
Vanden-Eijnden E.,Courant Institute of Mathematical Sciences
Annual Review of Physical Chemistry | Year: 2010
Transition-path theory is a theoretical framework for describing rare events in complex systems. It can also be used as a starting point for developing efficient numerical algorithms for analyzing such rare events. Here we review the basic components of transition-path theory and path-finding algorithms. We also discuss connections with the classical transition-state theory. Copyright © 2010 by Annual Reviews. All rights reserved.
Boureau Y.-L.,Courant Institute of Mathematical Sciences |
Dayan P.,University College London
Neuropsychopharmacology | Year: 2011
Affective valence lies on a spectrum ranging from punishment to reward. The coding of such spectra in the brain almost always involves opponency between pairs of systems or structures. There is ample evidence for the role of dopamine in the appetitive half of this spectrum, but little agreement about the existence, nature, or role of putative aversive opponents such as serotonin. In this review, we consider the structure of opponency in terms of previous biases about the nature of the decision problems that animals face, the conflicts that may thus arise between Pavlovian and instrumental responses, and an additional spectrum joining invigoration to inhibition. We use this analysis to shed light on aspects of the role of serotonin and its interactions with dopamine. © 2011 Nature Publishing Group All rights reserved.
Buhler O.,Courant Institute of Mathematical Sciences
Annual Review of Fluid Mechanics | Year: 2010
This article reviews the methods of wave-mean interaction theory for classical fluid dynamics, and for geophysical fluid dynamics in particular, providing a few examples for illustration. It attempts to bring the relevant equations into their simplest possible form, which highlights the organizing role of the circulation theorem in the theory. This is juxtaposed with a simple account of superfluid dynamics and the attendant wave-vortex interactions as they arise in the nonlinear Schrödinger equation. Here the fundamental physical situation is more complex than in the geophysical case, and the current mathematical understanding is more tentative. Classical interaction theory might be put to good use in the theoretical and numerical study of quantum fluid dynamics. Copyright © 2010 by Annual Reviews. All rights reserved.
Shelley M.J.,Courant Institute of Mathematical Sciences |
Zhang J.,Courant Institute of Mathematical Sciences
Annual Review of Fluid Mechanics | Year: 2011
The flapping or bending of a flexible planar structure in a surrounding fluid flow, which includes the flapping of flags and the self-streamlining of flexible bodies, constitutes a central problem in the field of fluid-body interactions. Here we review recent, highly detailed experiments that reveal new nonlinear phenomena in these systems, as well as advances in theoretical understanding, resulting in large part from the rapid development of new simulation methods that fully capture the mutual coupling of fluids and flexible solids. © 2011 by Annual Reviews. All rights reserved.
Laing C.,Courant Institute of Mathematical Sciences |
Schlick T.,Courant Institute of Mathematical Sciences
Current Opinion in Structural Biology | Year: 2011
RNA molecules are important cellular components involved in many fundamental biological processes. Understanding the mechanisms behind their functions requires RNA tertiary structure knowledge. Although modeling approaches for the study of RNA structures and dynamics lag behind efforts in protein folding, much progress has been achieved in the past two years. Here, we review recent advances in RNA folding algorithms, RNA tertiary motif discovery, applications of graph theory approaches to RNA structure and function, and in silico generation of RNA sequence pools for aptamer design. Advances within each area can be combined to impact many problems in RNA structure and function. © 2011 Elsevier Ltd.
Khot S.,Courant Institute of Mathematical Sciences
Proceedings of the Annual IEEE Conference on Computational Complexity | Year: 2010
This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. © 2010 IEEE.
Arsenio D.,Courant Institute of Mathematical Sciences
Archive for Rational Mechanics and Analysis | Year: 2012
We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier-Stokes-Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna-Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre-Villani solutions towards the Leray solutions. © 2012 Springer-Verlag.