Potential-Function Proofs for First-Order Methods

This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants

FoCaLiZe: Inside an F-IDE

For years, Integrated Development Environments have demonstrated their usefulness in order to ease the development of software. High-level security or safety systems require proofs of compliance to standards, based on analyses such as code review and, increasingly nowadays, formal proofs of conformance to specifications. This implies mixing computational and logical aspects all along the development, which naturally raises the need for a notion of Formal IDE. This paper examines the FoCaLiZe environment and explores the implementation issues raised by the decision to provide a single language to express specification properties, source code and machine-checked proofs while allowing incremental development and code reusability. Such features create strong dependencies between functions, properties and proofs, and impose an particular compilation scheme, which is described here. The compilation results are runnable OCaml code and a checkable Coq term. All these points are illustrated through a running example.Comment: In Proceedings F-IDE 2014, arXiv:1404.578

Understanding the Random Displacement Model: From Ground-State Properties to Localization

We give a detailed survey of results obtained in the most recent half decade which led to a deeper understanding of the random displacement model, a model of a random Schr\"odinger operator which describes the quantum mechanics of an electron in a structurally disordered medium. These results started by identifying configurations which characterize minimal energy, then led to Lifshitz tail bounds on the integrated density of states as well as a Wegner estimate near the spectral minimum, which ultimately resulted in a proof of spectral and dynamical localization at low energy for the multi-dimensional random displacement model.Comment: 31 pages, 7 figures, final version, to appear in Proceedings of "Spectral Days 2010", Santiago, Chile, September 20-24, 201

Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers