Title: Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle
Abstract: For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + \epsilon}$, $0<\epsilon \ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.


Title: Historical Review of Fluid Antenna and Movable Antenna
Abstract: Recently, significant attention has been drawn to the development of two antenna technologies known as "Fluid Antenna" and "Movable Antenna" in wireless communication research community, owing to their flexibility and reconfigurability for improving the wireless system performance in various applications. However, some confusions/concerns have also ensued on their nomenclature. In fact, both "Fluid Antenna" and "Movable Antenna" are not newly-made terms, while they have a longstanding presence in the field of antenna technology. This article thus aims to review the historical evolution of these technologies for fostering a clear understanding of their origins and recent development in the realm of wireless communication. It is hoped that this article will help dispel any confusion, concern or even dispute on the appropriate use of their names in the literature and motivate more research endeavors to focus on resolving their technical issues in the future.


Title: Zero-shot Active Learning Using Self Supervised Learning
Abstract: Deep learning algorithms are often said to be data hungry. The performance of such algorithms generally improve as more and more annotated data is fed into the model. While collecting unlabelled data is easier (as they can be scraped easily from the internet), annotating them is a tedious and expensive task. Given a fixed budget available for data annotation, Active Learning helps selecting the best subset of data for annotation, such that the deep learning model when trained over that subset will have maximum generalization performance under this budget. In this work, we aim to propose a new Active Learning approach which is model agnostic as well as one doesn't require an iterative process. We aim to leverage self-supervised learnt features for the task of Active Learning. The benefit of self-supervised learning, is that one can get useful feature representation of the input data, without having any annotation.


Title: The six-vertex model on random planar maps revisited
Abstract: We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-M\'elou and the first author.