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# Upcoming Seminars

For a Nash manifold X and a Nash vector bundle E on X, one can form the topological vector space of Schwartz sections of E, i.e. the smooth sections which decay fast along with all derivatives. It was shown by Aizenbud and Gourevitch, and independently by Luca Prelli, that for a Nash manifold X, th complex of Schwartz sections of the de Rham complex of X has cohomologies isomorphic to the compactly supported cohomologies of X.

In my talk I will present a work in progress, joint with Avraham Aizenbud, to generalize this result to the relative case, replacing the Nash manifold M with a Nash submersion f:M-->N. Using infinity categorical methods, I will define the notion of a Schwartz section of a Nash bundle E over a complex of sheaves with constructible cohomologies, generalizing the notion of Schwartz section on an open semialgebraic set. I will then relate the Schwartz sections of the relative de Rham complex of a Nash submersion f:M-->N with the Schwartz functions on N over the derived push-forward with proper support of the constant sheaf on M. Finally, I will coclude with some applications to the relation between the Schwartz sections of the relative de Rham complex and the topology of the fibers of f.

Zoom meeting: https://weizmann.zoom.us/j/98304397425

Robust recovery of lost colors in underwater images remains a challenging problem. We recently showed that this was partly due to the prevalent use of an atmospheric image formation model for underwater images and proposed a physically accurate model. The revised model showed: 1) the attenuation coefficient of the signal is not uniform across the scene but depends on object range and reflectance, 2) the coefficient governing the increase in backscatter with distance differs from the signal attenuation coefficient. Here, we present the first method that recovers color with our revised model, using RGBD images. The Sea-thru method estimates backscatter using the dark pixels and their known range information. Then, it uses an estimate of the spatially varying illuminant to obtain the range-dependent attenuation coefficient. Using more than 1,100 images from two optically different water bodies, which we make available, we show that our method with the revised model outperforms those using the atmospheric model. Consistent removal of water will open up large underwater datasets to powerful computer vision and machine learning algorithms, creating exciting opportunities for the future of underwater exploration and conservation. (Paper published in CVPR 19).

Zoom link: https://weizmann.zoom.us/j/99608045732

Consider the function field $F$ of a smooth curve over $\FF_q$, with $q\neq 2$.

L-functions of automorphic representations of $\GL(2)$ over $F$ are important objects for studying the arithmetic properties of the field $F$. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.

I will present a conceptual proof that the two families coincide, by categorifying the question. This correspondence will necessitate comparing two very different sets of data, which will have significant implications for the representation theory of $\GL(2)$. In particular, we will obtain an exotic symmetric monoidal structure on the category of representations of $\GL(2)$

It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Zoom meeting: https://weizmann.zoom.us/j/98304397425

Single image super resolution (SR) has seen major performance leaps in recent years. However, existing methods do not allow exploring the infinitely many plausible reconstructions that might have given rise to the observed low-resolution (LR) image. These different explanations to the LR image may dramatically vary in their textures and fine details, and may often encode completely different semantic information. In this work, we introduce the task of explorable super resolution. We propose a framework comprising a graphical user interface with a neural network backend, allowing editing the SR output so as to explore the abundance of plausible HR explanations to the LR input. At the heart of our method is a novel module that can wrap any existing SR network, analytically guaranteeing that its SR outputs would precisely match the LR input, when downsampled. Besides its importance in our setting, this module is guaranteed to decrease the reconstruction error of any SR network it wraps, and can be used to cope with blur kernels that are different from the one the network was trained for. We illustrate our approach in a variety of use cases, ranging from medical imaging and forensics, to graphics.

Zoom link: https://weizmann.zoom.us/j/96418208541