# PII Networking Industrial Workshops - Medical Imagery

# Epi-convergent Smoothing with Applications to Convex Composite Functions

Smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems. In this note we synthesize and extend recent results due to Beck and Teboulle on infimal convolution smoothing for convex functions with those of X.Chen on gradient consistency for nonconvex functions.

We use epi-convergence techniques to define a notion of epi-smoothing that allows us to tap into the rich variational structure of the subdifferential calculus for nonsmooth, nonconvex, and nonfinite-valued functions. As an illustration of the versatility and range of epi-smoothing techniques, the results are applied to the general constrained optimization for which nonlinear programming is a special case.

# TBA

# Managing design-time uncertainty in software engineering

*Every software system is the accumulated result of a myriad of design decisions. But what happens when developers are uncertain about how to make these decisions? The best developer teams are those that are experts at keeping possible options open, juggling multiple design alternatives, and avoiding premature commitments. However, existing tools, languages and methodologies rarely, if ever, take design-time uncertainty into account. I will present a formal but practical framework that supports deferring design decisions while uncertainty persists, allowing development and analysis to continue. This requires drawing from diverse areas of software engineering to create novel abstractions, notations and automation approaches to seamlessly "lift" existing operations to correctly and efficiently handle sets of possible solutions to open design decisions.*

# Que peut-on dire sur un système dynamique hamiltonien qu'on ne connaît pas ?

*Les systèmes dynamiques hamiltoniens modélisent la vaste majorité des systèmes « fondamentaux » de la physique classique et, de manière plus méconnue, de la physique quantique. Ce faisant, leur étude mathématique est centrale à notre compréhension du monde. Malheureusement, il est en général très difficile, voire impossible, d'exprimer précisément l'évolution temporelle d'un système hamiltonien. Qui plus est, faute d'information sur un système physique réel, on ne sait pas quel système hamiltonien le modélise « véritablement ».*

# Three-dimensional superintegrable systems in a static electromagnetic field

*We consider a charged particle moving in a static electromagnetic field described by the vector potential $vec{A}(vec{x})$ and the electrostatic potential $V(vec{x}).$ We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely $({P}_{1},{P}_{2}),$ $({L}_{3},{P}_{3})$ and $({L}_{1},{L}_{2},{L}_{3}).$ For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.*

# Interactions of forward-and backward-time isochrons

In the 1970s Winfree introduced the concept of an isochron as the set of all points in the basin of an attracting periodic orbit that converge to the periodic orbit in forward time with the same asymptotic phase. It has been observed that in slow-fast systems, such as the FitzHugh-Nagumo model, the isochrons of such systems can have complicated geometric features; in particular, regions with high curvature that are related to sensitivity in the system. In order to understand where these features come from, we introduce backward-time isochrons that exist in the basin of a repelling periodic orbit, and we consider their interactions with the forward-time isochrons. We show that a cubic tangency between the two sets of isochrons is responsible for creating high curvature features. This study makes use of a boundary value problem formulation to compute isochrons accurately as parametrised curves.

# Harry Potter's Cloak via Transformation Optics

Resume/Abstract :

Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc., including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last decade or so, there have been several scientific proposals to achieve invisibility. We will introduce some of these in a non-technical fashion, concentrating on the so-called "transformation optics" that has received the most attention in the scientific literature.

# Ricci curvature and geometric analysis on graphs

Abstract: Ricci curvature lower bound play very important rule for geometric analysis on Riemannian manifold. So it is very interesting to introduce similar concept on discrete setting especially on graphs. We will talk about the Ricci curvature lower bound on graphs where the original idea comes from the Bochner formula on Riemannian geometry. Given the Ricci curvature lower bound on graphs, we will imply some classic results from Riemannian manifold for eigenvalue estimate, gradient estimate, Harnack inequality and heat kernel estimate and so on.

# Solyanik Estimates in Harmonic Analysis

Let $mathcal{B}$ be a collection of open sets in $mathbb{R}^n$. Associated to $mathcal{B}$ is the geometric maximal operator $M_{mathcal{B}}$ defined by $$M_{mathcal{B}}f(x) = sup_{x in R in mathcal{B}}int_R|f|;.$$ For $0 < alpha < 1$, the associated emph{Tauberian constant} $C_{mathcal{B}}(alpha)$ is given by $$C_{mathcal{B}}(alpha) = sup_{E subset mathbb{R}^n : 0 < |E| < infty} rac{1}{|E|}|{x in mathbb{R}^n : M_{mathcal{B}}chi_E(x) > alpha}|;.$$ A maximal operator $M_mathcal{B}$ such that $lim_{alpha ightarrow 1^-}C_{mathcal{B}}(alpha) = 1$ is said to satisfy a emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates.

This research is joint with Ioannis Parissis.