元旦放假期间上网，看到 DAMTP（Department of Applied Mathematics and Theoretical Physics http://www.damtp.cam.ac.uk/）， University of Cambridge上有不少自己感兴趣的东西，摘抄一下了。浏览了一下这所世界著名大学的应用数学与理论物理系，看着他们的教授数量不多，但Research associate很多，但研究很有活力，研究方向也很广（涉及到Applied and Computational Analysis, Astrophysics,Fluid and Solid Mechanics, Mathematical Biology, Quantum Information, High Energy Physics等几个大方向，而每个大方向下面又分几个研究小组），很能解决实际问题。想起我们大学的数学院系以十记甚至上百的教职人员（很多professor都大几十个），成果却不怎么样，研究方向很窄，很传统，很“数学”，学生数量很多，也很难就业。很困惑，也很惭愧！什么时候我们的数学院系才能将自己的研究方向转向大量的实际问题呢，让数学和其他学科如物理，化学，生物等学科一样直接为经济建设服务呢。数学只有从象牙塔走进广阔的天地，才能表现出巨大的活力和潜力！现在流行做下一个十年计划，期待下一个十年能够有很大的改观！     

       下面转载一段关于DAMTP, University of Cambrige的Applied and Computational Analysis下面的一个group Applied Partial Differential Equations研究方向，这其中有我的研究兴趣所在。

APDE: Research Topics

• nonlinear Schroedinger-type equations
  We are mainly interested in the analytical and numerical study of multiscale phenomena in linear and nonlinear Schroedinger-type equations. The corresponding asymptotic analysis is based on semiclassical expansions, homogenization limits, adiabatic approximations, etc. Applications arise in the modelling of
  • Bose-Einstein condensates,
  • electron dynamics in crystals and nanostructures,
  • nonlinear wave propagation.
• classical and quantum kinetic equations
  Main mathematical topics under investigation are mean free path (scaling) limits such as generalized energy transport systems, concentration effects in bosonic Boltzmann equations, Wigner transforms, quantum Fokker-Planck equations and open quantum systems, kinetic equations with non-standard scattering laws, homogenisation limits and long time asymptotics. The main application areas are
  • Bose-Einstein condensates (interaction of thermalised and non-thermalised particles),
  • quantum scattering in semiconductors,
  • collective behaviour of swimming mirco-organismns (squirmers) and chemotaxis,
  • kinetics and mean field models for human crowding behaviour, Pareto economics and opinion formation in human societies.
• reaction-diffusion/convection systems
  Entropy methods establishing explicit functional inequalities relating entropy and entropy dissipation for nonlinear systems are modern tools to derive a-priori estimates and quantitative rates for the large time asymptotics in model systems describing
  • reversible chemical kinetics,
  • drift and diffusion of charge carries in semiconductors.
• free boundary problems
  Research focuses on regularity and geometry of free boundaries, fully nonlinear PDEs and level set methods. Applications arise in
  • superconductivity,
  • mathematical biology.
• higher order PDES
  The study of higher-order PDEs is very challenging, analytically as well as numerically. One reason for this is the absence of maximum-minimum principles which belong to the standard toolkit of second order elliptic and parabolic equations. In particular we are interested in
  • inpainting (image interpolation). An advantage of higher order over second-order diffusions is the smooth propagation of grayvalues into the inpainting domain (avoiding the staircasing effect). A particular application which we pursue is
  • virtual fresco restoration (ongoing project in collaboration with experts from the Academy of Fine Arts in Vienna).