Andrea L. Bertozzi, the 2019 Ralph E. Kleinman Prize recipient

By George Em Karniadakis and Mason A. Porter
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The Ralph E. Kleinman Prize, established in 1998, is awarded every two years by SIAM to a researcher for outstanding contributions that bridge the gap between mathematics and applications [1]Professor Andrea L. Bertozzi of UCLA was the recipient of the 2019 award at ICIAM in Valencia, Spain for her “groundbreaking work on PDEs, numerical analysis, scientific computing, mathematical data science, and for applications of mathematics to problems in the physical, life and social sciences”. Professor Bertozzi is a member of the US National Academy of Sciences (2018) and the American Academy of Arts and Sciences (2010) and a Fellow of SIAM, AMS and APS. She received the Sonia Kovalevskaya Prize in 2009, and she is a 2015 and 2016 Thomson Reuters highly cited’ researcher in Mathematics. Many of Prof. Bertozzi’s top papers are coauthored with her students and postdocs. Prof. Bertozzi’s combined research and mentoring record is prolific, with over 240 publications and 37 PhD students who have graduated to date.

We summarize, by topic area, some of Prof. Bertozzi’s important contributions in the past twenty-five years that contributed to her nomination of the Kleinman prize.

(1)   Swarming and Aggregation EquationProf. Bertozzi developed the theory of kinematic and dynamic swarming motion in multiple dimensions. Prior to her work, state of the art was largely in one dimension.  Starting from pairwise physical interactions between individual members of a swarm, a key challenge is to understand how the local interaction dynamics lead to large-scale patterns, such as flocks and mills. Prof. Bertozzi’s research in this area of dynamical systems started from a statistical-physics approach, discovering the role of H-stability in nonconservative swarming models and developing a quantitative multidimensional continuum model for swarming dynamics. Prof. Bertozzi’s modeling work led to new theoretical problems in nonlinear PDEs regarding well-posedness of aggregation equations — a problem with similar structure to the unsolved Millenium Prize problem for three-dimensional Navier–Stokes Equations in fluid dynamics.  Prof. Bertozzi and her collaborators subsequently solved the main problem for smooth solutions of aggregation equations in all dimensions followed by theory for Lp well-posedness theory.  Prof. Bertozzi’s 2011 paper in Nonlinearity with her students Jacob Bedrossian (now a professor at University of Maryland) and Nancy Rodríguez (now on the faculty at University of Colorado Boulder) developed theory for a related general class of problems with nonlinear diffusion [2].

(2)   Thin-Film Fluids: Prof. Bertozzi and her collaborators have worked for over two decades in thin-film fluids, starting with fundamental research on well-posedness of solutions of the fully nonlinear thin-film equation (in joint work with Mary Pugh) [3] and blowup behavior of these higher-order nonlinear PDEs.  In the late 1990s, in collaboration with Michael Shearer and her former postdoc Andreas Münch (now a professor at University of Oxford) [4], Prof. Bertozzi developed the theory of undercompressive shocks in driven films (inspired by experiments performed at the College de France in Anne-Marie Cazabat’s group).  Very recently, she and her current students and postdocs (Claudia Falcon and Hangjie Ji) have extended that work to the ‘tears of wine’ problem (a story of popular interest), including a live demonstration at the 2019 SIAM Applied PDEs conference in La Quinta, California. Prof. Bertozzi’s group has also developed the first quantitatively correct theoretical model for particle-laden flows with a free surface by deriving a model that involves shear-induced migration balanced with hindered settling of particles in a thin layer on an incline.

(3)   Mathematics of Crime: Prof. Bertozzi has collaborated with social scientists and local law enforcement agencies for over ten years, with several notable achievements. Her first paper on this topic [5] is another of her highly cited’ works. In it, she and her collaborators developed a reaction–diffusion–advection model for residential burglaries and hotspot formation.  The bifurcation analysis of that problem led to a cover article in PNAS in 2010 [6]. Her leadership has led to over ten PhD theses in this area of research, and her recent work on this topic has examined gang recruitment and youth intervention in Los Angeles, California.

(4)  Graph Models for Machine Learning: Prof. Bertozzi, in collaboration with Navy scientist Arjuna Flenner, developed a class of algorithms for graph clustering. These algorithms were inspired by the well-known Ginzburg–Landau energy that is used in PDEs for diffuse-interface models.  Their first paper on this topic [7] won a SIAM Outstanding Paper Prize (2014) and was republished as a SIGEST paper in SIAM Review in 2016. Prof. Bertozzi and a series of her PhD students and postdocs developed many new advances of this topic, including threshold dynamics-based semi-supervised learning, multiclass semi-supervised learning, semi-supervised modularity optimization for community detection on networks, convex algorithms for several of these problems, and numerical convergence analysis of these methods. Prof. Bertozzi’s graph-clustering methods have been adapted to hyperspectral pixel segmentation for videos, classification of ego motion in body-worn video cameras, and uncertainty quantification.

In summary, Professor Andrea L. Bertozzi is a world leader in mathematics, with outstanding contributions in diverse areas of both theory and applications.  Her work — which spans analysis of PDEs, algorithm design, and theoretical models for physical-science and social-science problems — is precisely in the spirit of the prestigious Ralph E. Kleinman award.

8 in memory of Ralph E.

[2] Bedrossian J, Rodríguez N, Bertozzi AL. Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion. Nonlinearity (2011) 24(6):1683.

[3] Bertozzi AL, Pugh M. The lubrication approximation for thin viscous films: Regularity and long‐time behavior of weak solutions. Communications in Pure and Applied Mathematics (1996) 49(2):85.

[4] Bertozzi AL, Münch A, Shearer M. Undercompressive shocks in thin film flows. Physica D (1999) 134(4):431.

[5] Short MB, D'Orsogna MR, Pasour VB, Tita GE, Brantingham PJ, Bertozzi AL, Chayes LB. A statistical model of criminal behavior. Mathematical Models and Methods in Applied Science (2008) 18(supp01):1249.

[6] Short MB, Brantingham PJ, Bertozzi AL, Tita GE. Dissipation and displacement of hotspots in reaction-diffusion models of crime. Proceedings of the National Academy of Sciences of the United States of America (2010) 107(9):3961.

[7] Bertozzi AL, Flenner A. Diffuse interface models on graphs for classification of high dimensional data. Multiscale Modeling & Simulation: A SIAM Interdisciplinary Journal (2012)10(3):1090.

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