Beverley McKeon California Institute of Technology Following the introductory lecture on the motivation for study of turbulence in engineering applications and the properties of the mean velocity profile, this talk will focus on challenges in understanding and modeling the instantaneous (incompressible) turbulence field that develops in flow over a wall. I will review the characteristics, statistics and scaling of the fluctuations as the Reynolds number approaches values relevant to a range of applications, some mathematical tools to understand them, and outstanding problems ripe for consideration during the present program. Particular areas of focus will include 1) What do we know and how well do we know it? The state of the art in data and understanding from experiments and simulation, including a description of turbulence from both spectral and structural viewpoints and a discussion of practical limitations on accuracy of the results; 2) How well can we model wall turbulence? A sampling of existing modeling approaches ranging from the phenomenological to the mathematical; 3) How could we go forward? A (personal) perspective on successes and limitations of current approaches, and opportunities for unification of our understanding; 4) Themes for the fourth workshop. How can current understanding be leveraged towards prediction and control, what properties of wall turbulence could be (but are not being) exploited and how can mathematics enable these objectives? Presentation (PDF File)