Confirmed speakers

• Ciprian Demeter Indiana University
  
  Title: Proof of the ℓ² decoupling conjecture
  
  Abstract: We prove the ℓ² Decoupling Conjecture for hypersurfaces with nonvanishing Gaussian curvature in the expected range of L^p spaces. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which in turn implies the full range of expected L^p_x,t Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various connections with Incidence Geometry and Number Theory are also discussed. This is joint work with Jean Bourgain.
• Wilfrid Gangbo, Georgia Institute of Technology 

         Title: Analysis of the almost axisymmetric flow energy

         Abstract: (pdf)

• Allan Greenleaf, University of Rochester
  
  Title: Multilinear operators and Erdös-Falconer point configuration problems.
  
  Abstract: Configuration problems of Erdös type concern counting the number of times that geometric configurations or quantities (lengths of line segments, areas of triangles, noncongruent simplices, etc.) occur among the points of a discrete set with a large number, N, of points. Falconer type problems are analogues of Erdös type problems in the setting of continuous geometry. I will describe some of these problems and recent progress that has been made on them using estimates for multilinear operators.
• Suzanne Lenhart, University of Tennessee, Knoxville and National Institute for Mathematical and Biological Synthesis
  
  Title: Using optimal control of PDEs to investigate population questions
  
  Abstract: We use optimal control of partial differential equations to investigate conservation questions in population models. One example will address a question about resource allocation to increase population abundance with limited resources; the control represents the availability of resources. A second example is motivated by the question: Does movement toward a better resource environment benefit a population? The control is the advective coefficient in a parabolic PDE with nonlinear growth.
• Irina Mitrea, Temple University
  
  Title: Recent progress in the Riemann-Hilbert problem for Dirac operators in uniformly rectifiable domains
  
  Abstract: Historically, the development of modern Harmonic Analysis has been inexorably linked with the theory of Complex Variables in the plane. While subsequent real variables methods have permitted generalizations to higher dimensions, the interplay between Harmonic Analysis and Complex Analysis remains strong even in the higher dimensional setting. Of course, this presupposes working with functions of several complex variables or with other notions of analyticity, amenable to the higher dimensional case. In my talk I will exemplify the intricate nature of such connections by presenting progress on two problems originating in Complex Analysis which due to developments in Harmonic Analysis can be now solved in significantly more general settings than originally anticipated.
  
  In its classical form, the Riemann-Hilbert problem asks for determining two holomorphic functions defined on either side of a surface Σ, satisfying a boundary condition of transmission type along Σ involving a symbol function Φ. In this regard, I will report on recent progress with Marius Mitrea and Michael Taylor describing the Fredholm solvability in the most geometric measure theoretic setting in which such a problem is meaningfully formulated. This involves replacing a complex plane by a Riemannian manifold M, the surface Σ by a uniformly rectifiable subset of M, and the Cauchy-Riemann operator by a general Dirac operator on M with low regularity assumptions on its coefficients. This topic interfaces with Index Theory on manifolds, and as an application I will discuss the most general Bojarski index formula known to date.