The aim of this talk is to understand the qualitative properties that emerge from a PDE model inspired from neurosciences, in order to understand what are the key processes that lead to mathematical complex patterns for the solutions of this equation. 

In this talk I will discuss the development and justification of linearised shock-capturing for aeronautical applications such as flutter, forced response and design optimisation.  At its core is a double-limiting process, reducing both the viscosity and the size of the unsteady or steady perturbation to zero. The design optimisation also requires the consideration of the adjoint equations, but with shock-capturing this is best done at the level of the numerical discretisation, rather than the PDE.

We  report  on the theory of evolution equations that combine a strongly nonlinear parabolic character with the presence of fractional operators representing long-range interaction effects, mainly of fractional Laplacian type. Examples include nonlocal porous media equations and fractional p-Laplacian operators appearing in a number of variants. 

Recent work concerns the time-dependent fractional p-Laplacian equation with parameter p>1 and fractional exponent 0<s<1. It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on a delicate analysis. The superlinear and sublinear ranges involve different analysis and results. 
 

Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

This talk will be about the stable boundary seen from different recent points of view.

Free fermion chains are particularly simple exactly solvable models. Despite this, typically one can find closed expressions for physically important correlators only in certain asymptotic limits. For a particular class of chains, I will show that we can apply Day's formula and Gorodetsky's formula for Toeplitz determinants with rational generating function. This leads to simple closed expressions for determinantal order parameters and the characteristic polynomial of the correlation matrix. The latter result allows us to prove that the ground state of the chain has an exact matrix-product state representation.

Randomized NLA methods have recently gained popularity because of their easy implementation, computational efficiency, and numerical robustness. We propose a randomized version of a well-established FEAST eigenvalue algorithm that enables computing the eigenvalues of the Hermitian matrix pencil $(\textbf{A},\textbf{B})$ located in the given real interval $\mathcal{I} \subset [\lambda_{min}, \lambda_{max}]$. In this talk, we will present deterministic as well as probabilistic error analysis of the accuracy of approximate eigenpair and subspaces obtained using the randomized FEAST algorithm. First, we derive bounds for the canonical angles between the exact and the approximate eigenspaces corresponding to the eigenvalues contained in the interval $\mathcal{I}$. Then, we present bounds for the accuracy of the eigenvalues and the corresponding eigenvectors. This part of the analysis is independent of the particular distribution of an initial subspace, therefore we denote it as deterministic. In the case of the starting guess being a Gaussian random matrix, we provide more informative, probabilistic error bounds. Finally, we will illustrate numerically the effectiveness of all the proposed error bounds.

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