In this seminar, I will present a survey of micro, meso and macroscopic models where repulsion and attraction effects are included through pairwise potentials. I will discuss their interesting mathematical features and applications in mathematical biology and engineering. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. I will showcase the breadth of possible applications with three different phenomena in applications: segregation, phase transitions, and consensus.

This dissertation explores the uncertainty propagation in space fractional diffusion equations in one and multiple dimensions with variable diffusivity and order parameters.

This talk will be devoted to an overview of recent results in understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics, and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel-type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential.

For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it seems necessary to impose a nondegeneracy condition on the constitutive equation for the stress. The talk will explain this and show how in some cases, the nondegeneracy condition can be proved using the monodromy group of a holomorphic map. This is joint work with Inna Capdeboscq and Yasemin Şengül. John Ball is Professor of Mathematics at Heriot-Watt University, Edinburgh, and formerly Sedleian Professor of Natural Philosophy at Oxford. He is the current President of the Royal Society of Edinburgh, and a former President of the International Mathematical Union. He specializes in the applications of nonlinear analysis to problems of materials science, liquid crystals and computer vision. Among various awards he received the 2018 King Faisal Prize for Science, the 2018 Leonardo da Vinci Award of the European Academy of Sciences and the 2022 De Morgan Medal of the London Mathematical Society.

Liquid crystals are materials whose properties are intermediate between normal fluids and solid crystals, and have widespread use as the working substance for computers, TV, and watch displays. The lecture will introduce these materials and what mathematics can say about them, and in particular, discuss how different theories of liquid crystals describe orientational defects in different ways.

A Bose gas at zero temperature is described by a mean field which satisfies the cubic nonlinear Schr¨odinger equation (NLS) otherwise known as the Gross- Pitaevski equation. The mean field describes the evolution of the condensate in an average sense. I will describe a technique that introduces pair correlations in the evolution of the condensate. The resulting approximation tracks the evolu- tion of the condensate in norm provided that the pair wave-function satisfies an interesting system of coupled NLS equations. I will discuss the nonlinear struc- ture of the NLS system as well as a novel approach to the question of global existence of solutions of the system.

In this work we present multigrid solvers for high-order finite element discretizations of these Riesz maps with optimal complexity in polynomial degree, i.e. With the same time and space complexity as sum-factorized operator application.

Biological systems are distinguished by their enormous complexity and variability. That is why mathematical modelling and computational simulation of those systems is very difficult, in particular thinking of detailed models which are based on first principles. The difficulties start with geometric modelling which needs to extract basic structures from highly complex and variable phenotypes, on the other hand also has to take the statistic variability into account.

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. Explicit examples show that the singular set could be, in general, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension n ≤ 4. The aim of this talk is to give an overview of these results.

In this paper, we propose a new methodological framework for performing extreme quantile regression using artificial neural networks, which are able to capture complex non-linear relationships and scale well to high-dimensional data.

The Wave Map system describes the evolution of waves constrained on a (Riemannian)  manifold. For the 2 + 1 dimensional problem, when the target manifold is a sphere, the solution collapses in finite time. The Analysis is due to the pioneering work of Merle, Paphael and Rodnianski. Motivated by their work I will present a somewhat novel approach of the collapsing mechanism which is based on a view of the equations as a nonlinear gauge system. This is joint work with Dan Geba.

Sobolev spaces are centrally important objects in PDE theory. Consequently, to understand how deep neural networks can be used to numerically solve PDEs a necessary first step is to determine now efficiently they can approximate Sobolev functions. In this talk we consider this problem for deep ReLU neural networks, which are the most important class of neural networks in practical applications.

Infinity-harmonic functions have recently found application in Semi-Supervised Learning, in the context of the so-called Lipschitz Learning. With this application in mind, we will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions and its several drawbacks, leading to the notion of AMLE (Absolutely Minimising Lipschitz Extension).

In this talk, I will first give a convergence analysis of gradient descent (GD) method for training neural networks by relating them with finite element method. I will then present some acceleration techniques for GD method and also give some alternative training algorithms

The talk will give an overview of recent results for models of collective behavior governed by functional differential equations. It will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role.

In this seminar, we will present our work on Virtual Element Method (VEM) approximations. The Virtual Element Method is a recent numerical technique for solving partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. This work aims to develop effective linear solvers for general order VEM approximations of three-dimensional scalar elliptic equations in mixed form and Stokes equations. To this end, we consider block algebraic multigrid preconditioners and balancing domain decomposition by constraints (BDDC) preconditioners. The latter allows us to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the differential problems are indefinite, ill-conditioned, and of saddle point nature.

Surface water waves are a physically important phenomenon with which we all have some experience. They are also surprisingly complex and interesting from a mathematical perspective. I will discuss two recent projects in water wave modeling. The first deals with ocean waves, such as tsunamis, passing over the continental slope. It has long been known that the amplification of such waves is greater than what the traditional transmission coefficient would predict.

The aim of this course is to familiarize the students with the usage of Computer Simulation tools for complex problems. The course will introduce the basic concepts of computation through modeling and simulation that are increasingly being used in industry and academia. The basic concepts of Discrete Event Simulation will be introduced along with the reliable methods of random variate generation and variance reduction. Later in the course, the concept of simulation-based optimization and output analysis will be discussed. The example of simulation (and optimization) applied to design an optimal organ allocation policy in the US will be discussed.

In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.

We propose a novel approach of numerically approximate McKean-Vlasov SDEs that avoids the usual interacting particle approximation and Propagation of Chaos results altogether.

The aim of this talk is to investigate the following non local p-Laplacian problem with data a bounded Radon measure.

Applied Complexity aims to understand the physical origin of these behaviors and transform them into sustainable technologies that tackle global problems of global interest. These range from energy harvesting to clean water production, the design of smart materials, biomedical applications, information security, artificial intelligence, and global warming. In this talk, I will summarize my group's recent research, discussing present results and future challenges of Applied complexity both as a science and engineering.

Models for flows on networks arise in the study of traffic and pedestrian crowds. These models encode congestion effects, the behavior and preferences of agents, such as aversion to crowds and their attempts to minimize travel time. We will present the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary mean-field game.

In this work, we proved statistical inference for the common Z-estimator based on adaptively sampled data. Adaptive sampling methods, such as reinforcement learning (RL) and bandit algorithms, are increasingly used for the real-time personalization of interventions in digital applications like mobile health and education. As a result, there is a need to be able to use the resulting adaptively collected user data to address a variety of inferential questions, including questions about time-varying causal effects.

Reinforcement Learning provides an attractive suite of online learning methods for personalizing interventions in Digital Behavioral Health. However, after a reinforcement learning algorithm has been run in a clinical study, how do we assess whether personalization occurred? We might find users for whom it appears that the algorithm has indeed learned in which contexts the user is more responsive to a particular intervention. But could this have happened completely by chance? We discuss some first approaches to addressing these questions.