Title: Turing degrees, intuitionistic logic, and the Muchnik topos.
Speaker: Sankha Basu, Assistant Professor, IIIT-Delhi
Date: Sep 09, 2019 (Monday)
Abstract: Kolmogorov, in 1923, proposed a model for intuitionistic propositional logic called the Calculus of Problems. Although simple and natural, this model was not rigorous. A rigorous version of this was given by Medvedev and Muchnik in the 1950’s and 60’s using the concept of Turing oracles. Thus started the study of mass problems and the reducibility notions between these. Sheaves over topological spaces as models for higher-order intuitionistic logic were studied independently. These models are also examples of elementary topoi. In this work, we have extended the Kolmogorov/ Medvedev/Muchnik line of work to a model of higher-order intuitionistic logic that we call the Muchnik topos. The Muchnik topos may be described in brief as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We have also introduced, within the Muchnik topos, a class of intuitionistic real numbers, different from the Dedekind and Cauchy reals. We call these the Muchnik reals. I will explain the basic notions on the way to the development of the Muchnik topos and the Muchnik reals and hence the talk should be accessible to anyone who is familiar with the concept of a topological space.

Title: Special Values of the Riemann Zeta function.
Speaker: Tapas Chatterjee, Indian Institute of Technology, Ropar
Date: Aug 26, 2019 (Monday)
Abstract: We will start our discussion with some classical results involving arithmetic and tran- scendental nature of special values of the Riemann Zeta function and then link this transcendence problem with a folklore conjecture of John Milnor motivated by a con- jecture of P. Chowla and S. D. Chowla. Finally, if time permits we will discuss my results related to the arithmetic nature of these special values.

Title: ∂P: A Differentiable Programming System to Bridge Machine Learning and Scientific Computing
Speaker: Viral B. Shah, Co-founder and CEO, Julia Computing
Date: August 3, 2019 (Saturday)
Abstract: Scientific computing is increasingly incorporating the advancements in machine learning andthe ability to work with large amounts of data. At the same time, machine learning models are becoming increasingly sophisticated and exhibit many features often seen in scientific comput- ing, stressing the capabilities of machine learning frameworks. Just as the disciplines of scientific computing and machine learning have shared common underlying infrastructure in the form of numerical linear algebra, we now have the opportunity to further share new computational infras- tructure, and thus ideas, in the form of Differentiable Programming. We describe a Differentiable Programming (∂P) system that is able to take gradients of Julia program making Automatic Dif- ferentiation a first class language feature. Our system supports almost all language constructs (control flow, recursion, mutation, etc.) and compiles high-performance code without requiring any user intervention or refactoring to stage computations. This enables an expressive program- ming model for deep learning and, more importantly, it enables users to utilize the existing Julia ecosystem of scientific computing packages in deep learning models. We discuss support for ad- vanced techniques such as mixed-mode, complex and checkpointed differentiation, and present how this leads to efficient code generation. We then showcase several examples of differentiat- ing programs and mixing deep learning with existing Julia packages, including differentiable ray tracing, machine learning on simulated quantum hardware, training neural stochastic differential equation. Link to paper: https://arxiv.org/abs/1907.07587.