The subject of p-adic differential equations was pioneered by Dwork in 1950’s, who investigated p-adic properties of solutions of a certain hypergeometric differential equation. This study of Dwork’s study led to extremely fascinating applications in number theory; especially, on elliptic curves and modular forms. The main goal of this colloquium talk is to explain the motivating example of the p-adic hypergeometric differential equation studied by Dwork and its link to the Legendre family of elliptic curves. If time permits, I’d like to discuss some generalization of Dwork’s study to families of abelian varieties and its potential applications.

We discuss a triangle of viewpoints for circle diffeomorphism groups: analysis, dynamics and group theory. In particular, we illustrate how the regularities (from the analytic side) of diffeomorphisms govern the dynamics and the group theoretical properties of diffeomorphisms. This line of study can be traced back to the works of Hölder, Denjoy, Tsuboi, Thurston and many more.

Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning approaches for inverse problems thanks to its excellent performance. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of deep convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of encoder-decoder CNNs. Our unified framework shows that encoder-decoder CNN architecture is closely related to nonlinear frame basis representation using combinatorial convolution frames, whose expressivity increases exponentially with the network depth and channels. We also demonstrate the importance of skipped connection in terms of expressivity and optimization landscape. We provide extensive experimental results from various biomedical imaging reconstruction problems to verify the performance encoder-decoder CNNs.