A personal view of mathematical research:

• Emergent behavior in collective dynamics, 2022 AMS Josiah Willard Gibbs Lecture
• IMS Public Lecture The Mathematics of Scientific Computation
  
• Velocity averaging, kinetic formulations and regularizing effects in conservation laws and related PDEs
  
• On a multiscale representation of images as hierarchy of edges
  
• Processing discontinuous spectral data
  
• Critical thresholds and Eulerian dynamics
  
• Burgers-type equations and vanishing hyper-viscosity
  
• Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity
  
• On a new scale of borderline regularity spaces for Euler equations
  
• Detection, reconstruction and approximate evolution of piecewise smooth solutions [ICM 2002 Lecture]

Selected Publications    Other lists of publications: [by chronological order] [by subject classification] [significant publications]

• On the stability of Runge-Kutta methods for arbitrarily large systems of ODEs Communications on Pure & Appl. Math., (2024).
  
• Swarm-based gradient descent method for non-convex optimization (with J. Lu & A. Zenginoglu) Communications of the AMS (2024).
  
• Constructions of bounded solutions of div $$u=f in critical spaces (with A. Cohen & R. DeVore)
  Multiscale,Nonlinear and Adaptive Approximation II (2024) pp. 177-200 .
  
• Swarm-based gradient descent meets simulated annealing (with Z. Ding, M. Guerra & Q. Li) SINUM ArXiv2404.18015 (2024).
  
• Critical thresholds in pressureless Euler-Poisson eq. with background states (with Y.-P. Choi, D.-h. Kim & D. Koo) ArXiv2402.12839 (2024).
  
• Swarm-based optimization with random descent (with A. Zenginoglu) Acta Applicandae Mathematicae 190(2) (2024).
  
• Swarming: hydrodynamic alignment with pressure Bulletin AMS 60(3) (2023) 285-325.
  
• On the sparsity of LASSO minimizers ... (with S. Foucart & M. Zhong) Constructive Approximation 57 (2023) 901-919
  
• Geometric structure of mass concentration for pressureless Euler ... (with D. Lear, T. Leslie & R. Shvydkoy) AdvMath 401(4) (2022) 108290.
  
• On the fast spreading scenario (with S. He and A. Zlatoš) Communications of the AMS 2 (2022) 149-171.
  
• Multi-species Patlak-Keller-Segel system (with S. He) Indiana U. Math Journal 70(4) (2021) 1577-1624.
  
• On the mathematics of swarming: emergent behavior in alignment dynamics Notices of the AMS 68(4) (2021) 493-503.
  
• Multi-flocks: emergent dynamics in systems with multi-scale collective behavior (with R. Shvydkoy) MMS 19(2) (2021) 1115-1141.
  
• Anticipation breeds alignment (with R. Shu) ARMA 240 (2021) 203-241.
  
• A game of alignment: collective behavior of multi-species (with S. He) AIHP (c) Non Linear Analysis 38(4) (2021) 1031-1053.
  
• Flocking hydrodynamics with external potentials (with R. Shu) ARMA 238 (2020) 347-381.
  
• Topologically-based fractional diffusion and emergent dynamics ... (with R. Shvydkoy) SIMA 52(6) (2020) 5792-5839.
  
• Global regularity of two-dimensional flocking hydrodynamics (with S. He), Comptes rendus - Mathématique Ser. I 355 (2017) 795–805.
  
• On the computation of measure-valued solutions (with U. Fjordholm and S. Mishra), Acta Numerica 25 (2016) 567-679.
  
• Hierarchical construction of bounded solutions in critical regularity spaces CPAM 69(6) (2016) 1087-1109.
  
• Heterophilious dynamics enhances consensus (with S. Motsch), SIAM Review 56(4) (2014) 577–621 (with Introduction by D. J. Higham)
  
• On the shortcomings of the 2/3 de-aliasing method (with C. Bardos), Numerische Mathematik 129 (2014) 749-782.
• A review of numerical methods for nonlinear partial differential equations, Bull. AMS 49(4) (2012) 507-554.
  
• ENO reconstruction and ENO interpolation are stable (+errata) (with U. Fjordholm & S. Mishra), FoCM 13(2) (2012) 139-159.
  
• A new model for self-organized dynamics and its flocking behavior (with S. Motsch), J. Stat. Physics 144(5) (2011) 923-947.
  
• Multiscale hierarchical decomposition of images with applications to deblurring, ... (with S. Nezzar & L. Vese) CMS 6(2) (2008) 281-307.
  
• Recovery of edges from spectral data with noise---a new perspective (with S. Engelberg) SINUM 46(5) (2008) 2620-2635.
  
• Filters, mollifiers and the computation of the Gibbs phenomenon Acta Numerica 16 (2007) 305-378.
  

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• Velocity averaging, kinetic formulations and regularizing effects in quasilinear PDEs (with T. Tao) CPAM 60 (2007), 1488-1521.
  
• Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems
  Acta Numerica 12 (2003), 451-512.
  
• High resolution methods for time dependent problems with piecewise smooth solutions "International Congress of Mathematicians", Proceedings of the ICM02 Beijing 2002 (Li Tatsien, ed.), Vol. III: Invited lectures, Higher Education Press (2002) 747-757.
  
• On a new scale of regularity spaces with applications to Euler's equations Nonlinearity 14 (2001) 513-532.
  
• Strong stability-preserving high-order time discretization methods (with S. Gottlieb & C.-W. Shu) SIAM Review 43 (2001) 89-112.
  
• New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations (with A. Kurganov)
  JCP, 160 (2000) 241-282.
  
• Approximate solutions of the incompressible Euler equations with no concentrations (with M. Lopes Filho & H. J. Nussenzveig) Annal. de l'institut Henri Poincare (c) Non Linear Analysis, 17 (2000) 371-412.
  
• Approximate solution of nonlinear conservation laws and related equations AMS Proc. Symp. Appl. Math. 54 (1998) 325-368.
  
• A kinetic formulation of multidimensional scalar conservation laws and related eqs (with P.-L. Lions & B. Perthame) JAMS 7 (1994) 169-191.
  
• Local error estimates for discontinuous solutions of nonlinear hyperbolic equations SINUM 28 (1991) 891-906.
  
• A kinetic equation with kinetic entropy functions for scalar conservation Laws (with B. Perthame) Comm. Math. Phys. 136 (1991) 501-517.
  
• Non-oscillatory central differencing for hyperbolic conservation laws (with H. Nessyahu) JCP 87 (1990) 408-463.
  
• ... computing the eigensystem of ... tridiagonal matrices by the divide and conquer approach (with D. Gill) SISC 11 (1990) 161-173.
  
• Convergence of spectral methods for nonlinear conservation laws SINUM 26 (1989) 30-44.
  
• Numerical viscosity and the entropy condition for conservative difference schemes MathComp 43 (1984) 369-381
  
• The equivalence of L²-stability, the resolvent condition and strict H-stability LAA 41 (1981) 151-159.
  
• Scheme-independent stability criteria for ... hyperbolic initial-boundary value problems II (with M. Goldberg) MathComp 36 (1981) 603-626.
  

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