Geodesically embedding hyperbolic manifolds
Abstract: A hyperbolic n-manifold M geodesically embeds if is realized as a totally geodesic embedded submanifold in an n+1-dimensional hyperbolic manifold. Understanding which hyperbolic manifolds embed geodesically is an interesting problem: on one hand we know no general obstruction to the fact that a hyperbolic manifolds geodesically embeds, on the other hand showing explicitly that a hyperbolic manifold geodesically embeds is often quite hard. This can be done essentially in two ways: either with explicit constructions using the geometry of hyperbolic polytopes, or employing arithmetic techniques. We will give examples of both approaches and review the main results available in the literature.
Deep learning: why all the hype?
Abstract: In recent years, deep learning methods have been responsible for astonishing breakthroughs in computer vision, speech recognition, natural language processing, and robotics—among other applications. In this talk, I give an introduction to the topic of deep learning, illustrating with examples one of the greatest benefits of deep learning methods: the ability to learn directly from raw unstructured data (such as images and text) given large amounts of labeled data and computational power. I will show that a side effect of this learning process is that we are now able to use the internal representations learned by the model to map symbolic concepts into useful numerical representations (also known as “embeddings”) that can be processed by other algorithms. Finally, I will talk about generative adversarial networks (GANs), a more recent deep learning architecture that allows the generation of realistic synthetic data when trained on a set of real data and show an application developed by our team of researchers at IBM Research Brazil that uses GANs to generate realistic seismic data from geoscientist’s sketches.