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Recently, with the enormous development of deep learning techniques, solving underdetermined linear systems (more unknowns than equations) have become one of major concerns in medical imaging. Typical examples include undersampled MRI, local tomography, and sparse view CT, where deep learning techniques have shown excellent performance. Although deep learning methods appear to overcome limitations of existing mathematical methods in handling various underdetermined problems, there is a lack of rigorous mathematical foundations which would allow us to understand reasons why deep learning methods perform that well. This talk deals with this learning causal relationship about structure of training data suitable for deep learning to solve highly underdetermined inverse problems. We examine whether or not a desired reconstruction map can be learnable from the training data and the underdetermined system. Most problems of solving underdetermined linear systems in medical imaging are highly non-linear.
In this talk, we introduce an idea of producing rational torsion points on J_0(N), which is well-known. Conjecturally, the points constructed in this way exhaust all the rational torsion points on J_0(N). So, we briefly explain how to compute the orders of such points, and prove the conjecture up to finitely many primes. (If you would like to join this online seminar, please contact Bo-Hae Im to get the Zoom link.)
The essential dimension quantifies the algebraic-geometric complexity of a class of algebraic objects (such as, but not necessarily, the class of Galois extensions with a given group): roughly speaking, it is the minimal number of parameters required to describe all objects in this class (over all fields containing a given field K). We introduce and discuss arithmetic-geometric and local analogues of this notion. These are supposed to quantify the difference in complexity between the local and global Galois theory of a given group over a given number field K. In particular, we show that the "local dimension" of a finite group is bounded by 2 - whereas arithmetic dimension remains mysterious in general. We give an application concerning solution of Grunwald problems.
In this talk, I will present a recent work in collaboration with physicists on the analysis of real time Transmission Electron Microscopy (TEM) images to understand molecular transition from crystal solid state to liquid state. Molecules are deposited on graphene with multilayer structures, which are projected and overlaid in noisy 2d TEM images. The problem is to find all the molecular centers in the extremely noisy 2d images where projected molecules are overlaid and to track the centers across the image frames. Before discussing the method that we considered, I will give a brief history in the development of image segmentation techniques with some theoretical and numerical details of old fashioned methods. Then, our method of image segmentation for molecular center identification follows.