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The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm that iteratively samples from the conditional distributions of the probability measure of interest. Under the assumption of log-concavity, for its random scan version, we provide a sharp bound on the speed of convergence in relative entropy. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations required for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. This contrasts with gradient-based methods such as overdamped Langevin or Hamiltonian Monte Carlo (HMC), whose mixing time typically increases with the dimension. Our techniques also allow us to analyze Metropolis-within-Gibbs schemes, as well as the Hit-and-Run algorithm. This is joint work with Filippo Ascolani and Giacomo Zanella.

The goal of these lectures is to give a simple and direct introduction to some of the most basic concepts and techniques in Deep Learning. We will start by reviewing the fundamentals of Linear Regression and Linear Classifiers, and from there we will find our way into Deep Dense Neural Networks (aka multi-layer perceptrons). Then, we will introduce the theoretical and practical minimum to train such neural nets to perform the classification of handwritten digits, as provided by the MNIST dataset. This will require, in particular, the efficient computation of the gradients of the loss wrt the parameters of the model, which is achieved by backpropagation. Finally, if time permits, we will briefly describe other neural network architectures, such as Convolution Networks and Transformers, and other applications of deep learning, including Physics Informed Neural Networks, which apply neural nets to find approximate solutions of Differential Equations. The lectures will be accompanied by Python code, implementing some of these basic techniques.

The goal of these lectures is to give a simple and direct introduction to some of the most basic concepts and techniques in Deep Learning. We will start by reviewing the fundamentals of Linear Regression and Linear Classifiers, and from there we will find our way into Deep Dense Neural Networks (aka multi-layer perceptrons). Then, we will introduce the theoretical and practical minimum to train such neural nets to perform the classification of handwritten digits, as provided by the MNIST dataset. This will require, in particular, the efficient computation of the gradients of the loss wrt the parameters of the model, which is achieved by backpropagation. Finally, if time permits, we will briefly describe other neural network architectures, such as Convolution Networks and Transformers, and other applications of deep learning, including Physics Informed Neural Networks, which apply neural nets to find approximate solutions of Differential Equations. The lectures will be accompanied by Python code, implementing some of these basic techniques.

This work presents a novel numerical technique for addressing a category of distributed-order fractional partial differential equations, with particular emphasis on time-fractional wave-diffusion equations. The approach involves the development of an extended form of Bernoulli wavelet functions and the derivation of an exact expression for their Riemann-Liouville integral. By employing the Gauss-Legendre quadrature rule, a carefully selected set of collocation points, and approximations for the unknown function and its derivatives, the original problem is converted into a system of algebraic equations. An error analysis is performed to evaluate the precision of bivariate function approximation using fractional-order Bernoulli wavelets. To validate the method's efficacy, some illustrative examples are solved, with the numerical outcomes demonstrating its accuracy and reliability.

Mathematical modelling of infectious diseases involves a series of mathematical techniques and methods that make it possible to describe the dynamics of their transmission in populations. The incorporation of biological and epidemiological events related to these diseases into models, taking into account their intrinsic uncertainty, is essential to explain and predict their dynamics. This seminar introduces dynamic modelling and calibration techniques with uncertainty in two areas of epidemiology on real-world case studies: antibiotic resistance, specifically in the case study of colistin-resistant Acinetobacter baumannii, and vaccination strategies, in particular against influenza and human papillomavirus (HPV). Combining deterministic and stochastic mathematical modelling techniques, parameter analysis and calibration strategies, we can explain the observed epidemiological scenarios, predict their evolution and evaluate the efficacy of preventive public health interventions.

In this talk, we investigate the existence of infinitely many non-constant weak solutions to the quasilinear elliptic equation
\[ -\operatorname{div}(A(x,u)∇ u) + \frac{1}{2} D_u A(x,u)∇ u \cdot ∇ u = g(x,u) -λ u, \quad u∈ H^1(Ω), \] for every \( λ ∈ \mathbb{R} \), where \(Ω \) is a bounded domain with a Lipschitz continuous boundary.

Our approach is variational; however, the associated energy functional is differentiable only along directions \( v ∈ H^1(Ω) ∩ L^\infty(Ω) \).

We discuss the notion of weak slope and the critical point theory for continuous functionals, and we apply these concepts to our problem to establish the existence of multiple solutions.

Algebraic topology provides a natural framework for realizability problems, as it explores the interplay between algebraic structures and topological spaces. These questions have been around since the 1970's, with Steenrod asking when an algebra is the cohomology of a space and Kahn asking which groups are the group of self-homotopy equivalences of a simply-connected space. Addressing such questions deepens our understanding of both spaces and their associated algebraic structures, making them quite interesting.

In this talk, we present two recent realizability results concerning group actions. First, for any action of a finite group on a finitely presented abelian group, there exists a space that realizes this action as the canonical action of the group of self-homotopy equivalences on the first homology group. Second, we establish that any action of a finite group on a permutation module is the action of the group of self-homotopy equivalences of a space on its homology groups. Additionally, we show that any simplicial complex can be perturbed in a way that reduces the automorphism group to any chosen subgroup without changing the homotopy type.

(joint work with Cristina Costoya and Antonio Viruel)

Survival models with cure fractions, known as long-term survival models, are widely used in epidemiology to account for both immune and susceptible patients regarding a failure event. In such studies, it is also necessary to estimate unobservable heterogeneity caused by unmeasured prognostic factors. Moreover, the hazard function may exhibit a non-monotonic shape, specifically, an unimodal hazard function. In this article, we propose a long-term survival model based on a defective version of the Dagum distribution, incorporating a power variance function frailty term to account for unobservable heterogeneity. This model accommodates survival data with cure fractions and non-monotonic hazard functions. The distribution is reparameterized in terms of the cure fraction, with covariates linked via a logit link, allowing for direct interpretation of covariate effects on the cure fractionan uncommon feature in defective approaches. We present maximum likelihood estimation for model parameters, assess performance through Monte Carlo simulations, and illustrate the models applicability using two health-related datasets: severe COVID-19 in pregnant and postpartum women and patients with malignant skin neoplasms.

In the past decade, there has been extensive research on the nonlinear Schrödinger equation (NLS) on metric graphs, driven by both the physical and mathematical communities. Metric graphs, in essence, are one-dimensional objects that can model network-like structures. The first goal of this talk, particularly for those who are new to metric graphs, is to provide an introduction to these structures and present the appropriate functional framework for studying the NLS equation on them.

It is well-established that both the metric (size) and topological (shape) properties of the graphs can impact the existence of solutions to the NLS. As a result, no general theory currently exists for analyzing the NLS equation on metric graphs. A common approach is to focus on specific classes of graphs. In this talk, we focus on two such graphs: the $\mathcal{T}$-graph and tadpole graphs. We then discuss, using techniques from the theory of ordinary differential equations (specifically, parts of the period function), how to approach questions related to the existence, uniqueness, and multiplicity of positive solutions on these graphs.

Time permitting, we will demonstrate how this careful analysis leads to a series of existence and uniqueness/multiplicity results for a class of graphs known as single-knot graphs.

This talk is based on joint work with Simão Correia and Hugo Tavares.

In this work, we deal with the symmetric exclusion process with k slow bonds equally spaced in the torus with kn sites, where the strength of a slow bond is $\alpha n^{-\beta}$, where $\beta>1$. For k fixed, it was known (T. Franco, P. Gonçalves, A. Neumann, AIHP'13) that the hydrodynamic limit in the diffusive scaling of this process is given by the heat equation with Neumann boundary conditions, meaning that the system does not allow flux through a slow bond in the limit. In this joint work with Tiecheng Xu and Dirk Erhard, we obtain another three superdiffusive scalings for this system. If $k$ is fixed and the (time) scaling is $n^{\theta}$, where $2< \theta<1+\beta$, the system reaches equilibrium instantaneously at each box between two consecutive slow bonds, being constant in time and space. If k is fixed and $\theta=1+\beta$, the density is spatially constant inside each box, and evolves as the discrete heat equation. And if $\theta=1+\beta$ and $k$ goes to infinity, we recover the continuous heat equation (on the torus).

Cells make fate decisions in response to dynamic environments, and multicellular structures emerge from multiscale interplays among cells and genes in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells for all their genes. While those measurements provide high-dimensional gene expression profiles for all cells, it requires fixing individual cells that lose many important spatiotemporal information. Is it possible to infer temporal relationships among cells from single or multiple snapshots? How to recover spatial interactions among cells, for example, cell-cell communication? In this talk I will present our newly developed computational tools to study cell fate in the context of single cells as a system. In particular, I will show dynamical models and machine-learning methods, with a focus on inference and analysis of transitional properties of cells and cell-cell communication using both high-dimensional single-cell and spatial transcriptomics, as well as multi-omics data for some cases. Through their applications to various complex systems in development, regeneration, and diseases, we show the discovery power of such methods in addition to identifying areas for further method development for spatiotemporal analysis of single-cell data.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.