We consider the problem of coherent imaging using intensity-only measurements. The main challenge in intensity-only imaging is recovering phase information that is not directly available in the data, but is essential for coherent image reconstruction.  Imaging without phases arises in many applications such as crystallography, ptychography and optics where images are formed from the spectral intensities.  

The  earliest and most widely used methods for imaging with intensity-only measurements are alternating projection algorithms. The basic idea is to project the iterates on the intensity data sequentially in both the real and the Fourier spaces. Although these algorithms are very efficient for reconstructing the missing phases in the data, and performance is often good in practice, they do not always converge to the true, missing phases. This is especially so if strong constrains or prior information about the object to be imaged, such as spatial support and non-negativity, are not reliably available.

Rather than using phase retrieval methods, we propose a different approach in which well-designed illumination strategies exploit the spatial and frequency diversity inherent in the problem. These illumination strategies allow for the recovery of interferometric data that contain relative phase information which is all that is needed to reconstruct a so-called holographic image. There is no need for phase reconstruction in this approach. Moreover, we show that this methodology leads to holographic images that suffer no loss of resolution compared with those that use full phase information. This is so when

the media through which the probing signals propagate are assumed to be homogeneous. 

We also consider inhomogeneous media where wavefront distortions can arise. In such media the incoherence in the recovered interferometric data can be reduced by restricting them to small spatial and frequency offsets. Using an efficient implementation of this restriction process we obtain holographic images with a somewhat reduced resolution compared to the homogeneous medium case.

The robustness of our approach will be explored with numerical simulations carried out in an optical (digital) microscopy imaging regime.

[Cancellation due to weather-related flight delays.  Talk to be rescheduled.]

The asymptotic expansion for the Bergman kernel has
important applications in complex analysis. Short-time asymptotic
expansion of the heat kernel played an important role in spectral
geometry. We will present our work on Feynman diagram formulas for
the coefficients in the asymptotic expansion of Bergman and heat
kernels on Kahler manifolds and their applications.

Simulating the low-Re hydrodynamics of particulate flows is an extremely challenging and important problem that arises in several disciplines. In this talk, I will present recent advances made by our group in overcoming several computational bottlenecks, especially those arising in the context of dense suspensions confined by complex geometries. In particular, a spectrally-accurate scheme to resolve the interactions of close-to-touching particles, a novel periodizing scheme for arbitrary geometries and a new boundary integral equation formulation for colloidal and active suspensions will be presented. Incorporating stable time-marching schemes, fast direct solvers based on low-rank factorizations and the fast multipole method, we were able to simulate the hydrodynamics of over 1000 deformable particles flowing through a periodic microfluidic-chip geometry in less than a minute per time-step on a laptop. I will discuss several applications and our ongoing efforts to simulate the electrohydrodynamics of vesicle suspensions, evaluate the stresses experienced by motile cells, investigate the controllability of low-Re swimmers and to design microfluidic chips for cell sorting and separation.

Given a compact Riemannian manifold with nonnegative Ricci curvature and convex boundary, it is interesting to estimate its size in terms of the volume, the area of its boundary etc.  I will discuss some open problems and present some partial results.

Joint Seminar with Analysis.