Abstract: In this presentation we will see two of the most famous and classic results in reduction theory: the Reduction of Symplectic Manifolds with Symmetry Theorem, formulated by Marsden & Weinstein in 1974 and the Reduction of Poisson Manifolds Theorem (using distributions), presented by Marsden & Ratiu in 1986. Marsden & Ratiu claims that the Marden & Weinstein’s theorem follows from their reduction theorem for Poisson manifolds as a particular case. We will prove that statement, but without using one of their argument, because… you will see why!

Abstract: Number theory is one of the oldest fields of mathematics whose primary focus is the study of the integers. Often, it is necessary to go to higher dimensions to study them and their properties. In this seminar, we begin by looking into Quaternion Algebras, a generalization of the usual quaternions and continue with the definition of an Order of a Quaternion Algebra. Afterwards, we will see how quaternions, and in particular orders, are useful to prove a generalization of Lagrange's four-square theorem. 

Abstract: In this work we investigate a mathematical model to reconstruct the mechanical properties of an elastic medium, for the optical coherence elastography imaging modality. To this end, we propose machine learning tools by exploring neural networks to solve the inverse problem of elastography. In our framework, we analyze the theoretical relative error between the exact solution and the neural network for the case of noise free data and noisy data. Our algorithm updates the parameters combining the backpropagation technique with the ADAM optimizer to minimize a cost function which is defined using the fully discretized scheme of the direct problem. We report several computational results using fabricated data with and without noise.

Keywords: Deep learning, linear elasticity, inverse problem, mechanical properties reconstruction, neural networks, finite element method, optical coherence elastography

Abstract: In this seminar, I will start by very lightly reviewing important concepts in Optimization (linear programming, duality, integer programming), as well as some classic problems - knapsack and cutting stock. These problems are widely studied on their own, but here will be used to motivate column generation. Afterwards, we will introduce Dantzig-Wolfe decomposition, a method to reformulate difficult problems into ones that can be solved with column generation. All of this to reach my work, on production-maintenance scheduling problems, very difficult nonlinear integer problems, whose structure is amenable to a Dantzig-Wolfe reformulation which is (hopefully) much faster.

Abstract: Numerical semigroups are the cofinite subsemigroups of the natural numbers containing 0. One can verify that every numerical semigroup has a unique set of minimal generators, also refered to as its primitives.  Let A(n) denote the set of numerical semigroups whose maximum primitive is n. We discuss the properties of the sequence (|A(n)|), and show that it has some surprising connections with the inverse problem of the Frobenius coin-exchange problem. Moreover we discuss some connections with the area of additive combinatorics. We also discuss the Wilf's conjecture, and show that it is asymptotically true- in the sense that almost all semigroups in A(n) satisfy Wilf's conjecture as n tends to infinity.