Modern Perspectives on Partial Differential Equations: Theory and Applications

Speaker : Abner SALGADO (Department of Mathematics,University of Tennessee,United States)

Elliptic problems in nondivergence form appear in weather and climate modelling, stochastic optimal control, determining the initial shape of the universe, optimal reflector design, differential geometry, optimal transport, mathematical finance, image processing and mesh generation. In fact, their understanding is a steppingstone towards the treatment of fully nonlinear elliptic equations. Despite their importance in these application areas, and in contrast to the existing treatment for variational problems, the finite element approximation of elliptic problems in nondivergence form is still an underdeveloped topic. The reasons for this are plentiful, and they begin with the mere fact that the notions of solution that are suitable for these equations is not a variational one, which the finite element method naturally mimics. In this course we survey some of the recent developments regarding the approximation, mostly with finite elements, of this type of problems. Topics may include: Approximation of strong solutions: The Cordes condition, conforming discretization of the linear problem, nonconforming discretization of the linear problem, discretization of nonlinear problems, Howard’s algorithm. Approximation of viscosity solutions: The Barles-Souganidis theorem, The discrete Alexandrov-Bakelman-Pucci estimate, monotone schemes for linear problems, monotone schemes for nonlinear problems. Number of Sessions: 5 sessions Instructor: Abner Salgado, 5 sessions each 1 hr.

Speaker : Marta LEWICKA (Department of Mathematics,University of Pittsburgh,United States)

This course will concern the analytical and geometrical questions emerging from the study of thin elastic films exhibiting residual stress at free equilibria. Prestressed thin films are present in many contexts and applications, ranging from growing tissues, through plastically strained sheets, engineered swelling or shrinking gels, to petals and leaves of flowers, atomically thin graphene layers, etc. The related mathematical questions may be seen as a variation on the classical themes: in differential geometry - that of isometrically embedding a shape with a given metric in an ambient space of possibly different dimension; and in calculus of variations - that of minimizing non-convex energy functionals parametrized by a quantity in whose limit the functionals become in some sense degenerate. One major research avenue here is directed to the flexibility and rigidity of solutions of the Monge-Amp'ere equation and system, which are fully nonlinear PDEs studied with the techniques of convex integration. The course can be taught without requiring too much background beyond the following standard content of undergraduate Calculus of Several Variables: taking partial derivatives of functions on \R^n, integration of multivariable functions, matrix operations (product, inverse, determinant). The course will be self-contained and will define all the necessary mathematical objects, mentioned in the course description, for example: Gamma-convergence, Monge-Ampere equation, system of isometric immersions The course will be self-contained, with outline as follows: • Lectures 1-2: Motivation from elasticity and calculus of variations. Rigidity theorems. Gamma convergence and dimension reduction. Hierarchy of limiting theories of prestressed thin films. • Lectures 3-4: Motivation from differential geometry. Constrained theories obtained as Gamma-limits of energy functionals. The geometric Monge-Ampere equation and Monge-Ampere system. The isometric immersion system. Non-uniqueness of solutions in the absence of notion of curvature. • Lectures 5: Convex integration for Monge-Ampere and isometric immersion systems. The Nash-Kuiper scheme and the Kallen iterations. Nash's spirals and Kuiper's corrugations. Rigidity results. Relation to Euler's equations, the role of the commutator estimate and the degree lemma. Number of Sessions: 5 sessions Instructor: Marta Lewicka, 5 sessions each 1 hr.

Speaker : Qiang DU (Department of Mathematics,Columbia University,United States)

This intensive short course introduces the fascinating world of integro-differential and nonlocal equations, which extend beyond the classical framework of Partial Differential Equations (PDEs). Unlike local PDEs, where the rate of change at a point depends only on its immediate vicinity, nonlocal equations describe systems where the behavior at a given point is influenced by points across an extended neighborhood or the entire domain. This inherent "action-at-a-distance" feature makes them powerful tools for accurately modeling diverse phenomena, from physics and finance to biology and traffic flow. We will explore how these equations are used in modeling problems such as anomalous diffusion in turbulent fluids, long-range interactions in peridynamics, and modern applications in population and traffic dynamics. The course will also cover recent theoretical advances, investigating the fundamental differences between nonlocal and local problems, including unique regularity results and analytical challenges. We will discuss the role of fractional calculus and the fractional Laplacian as a canonical example of a nonlocal operator. Number of Sessions: 5 sessions Instructor: Qiang Du, 5 sessions each 1 hr.

Speaker : Abba GUMEL (Department of Mathematics,University of Maryland,United States)

This course explores the critical role of PDEs and Dynamical Systems in modeling complex biological phenomena. We’ll cover a range of frontier topics, including the PDEs governing animal movement and population biology models with moving boundaries, the spatial spread of infectious diseases, and models describing tumor growth and cell motility. Beyond simple modeling, a significant focus will be placed on the bifurcation aspects of these equations, investigating how small changes in parameters can lead to dramatic shifts in system behavior, such as pattern formation (Turing patterns), tipping points, and the onset of chaos. This interdisciplinary perspective offers students powerful tools for analyzing biological stability and complexity. Number of Sessions: 5 sessions Instructor: Abba Gumel, 5 sessions each 1 hr.

Speaker : Abba GUMEL (Department of Mathematics,University of Maryland,United States)

To complement the theoretical framework of the "PDEs in Math Biology" course, the exercise sessions are designed as a dynamic mix of hands-on lab work and a flipped-classroom approach. These sessions focus on interactive case studies specifically centered on infectious diseases, allowing students to apply partial differential equations to real-world epidemiological patterns. By analyzing disease spread through these practical simulations, students transition from abstract mathematical models to clinical and biological applications, fostering a deeper understanding of how PDEs can predict and manage public health crises.

Speaker : Xiaochuan TIAN (Department of Mathematics,University of California San Diego,United States)

This course explores the powerful connection between PDEs and modern data science, an expanding field at the intersection of applied mathematics and artificial intelligence. We examine a two-way street: first, how PDEs provide a fundamental modeling framework for machine learning and image processing problems, and second, how machine learning techniques are used to solve highly complex and nonlinear PDEs. On the modeling side, you'll learn to formulate image processing tasks like denoising, inpainting, and segmentation as PDE problems, moving from the classic heat equation to sophisticated models like the Perona-Malik equation. We will also introduce Optimal Transport Theory, a field deeply rooted in PDEs, and its critical applications in comparing probability distributions for generative models and data analysis. Instructor: Xiaochuan Tian, 5 sessions each 1 hr.

Speaker : Xiaochuan TIAN (Department of Mathematics,University of California San Diego,United States)

This interactive activity offers a hands-on introduction to Physics-Informed Neural Networks (PINNs), a cutting-edge, highly relevant research area that effectively merges computational physics with deep learning. Structured as a guided coding session, students will learn to construct and train a neural network to solve a basic PDE problem without relying on traditional numerical discretization. The session will ideally use Python with a deep learning library like PyTorch or TensorFlow (and a PINN-specific library like DeepXDE if time allows for quicker implementation), demonstrating how the neural network is penalized by a "loss function" that incorporates both the boundary conditions and the residuals of the PDE itself. Instructor: Xiaochuan Tian, 2 sessions each 1 ¼ hour.

Speaker : Abner SALGADO (Department of Mathematics,University of Tennessee,United States)

The goal of this interactive activity is for students to learn about open source and freely available software libraries and packages for the numerical solution of PDEs. Examples of these tools include, but are not limited to, FEniCS, FreeFem++, VisualPDE, and others. This activity will complement the theoretical developments of the courses on variational theory of PDEs, and it will help students illustrate and push the limits of the developed theory. Students will be able to explore the effects of changing problem parameters like initial or boundary conditions, and model coefficients (such as diffusion rates or reaction terms). In addition, students will be able to demonstrate visually how the regularity or geometry of the boundary (e.g., a smooth circular domain versus a highly irregular, fractal-like domain) impacts quantitative and qualitative aspects of the solution to a particular PDE. While knowledge of a computer programming language will be of great benefit, we will keep the background requirements to a minimum. Number of Sessions: 2 sessions Instructor: Abner Salgado, 2 sessions each 1 ¼ hour.

Speaker : Qiang DU (Department of Mathematics,Columbia University,United States)

There will be discussions on current areas of research in nonlocal modeling. This complements the integro-differential and nonlocal equations course. Number of Sessions: 1 session Instructor: Qiang Du, 1 session, 1 ¼ hour.

Speaker : Marta LEWICKA (Department of Mathematics,University of Pittsburgh,United States)

In these problem sessions, students are organized into small groups to foster a collaborative learning environment focused on solving specific exercises assigned during the lecture. This format encourages meaningful peer-to-peer dialogue while providing a direct channel for students to seek clarification, receive hands-on guidance, and obtain instant feedback on their problem-solving approaches. The sessions will also encourage students to take an active role in the material by presenting segments of lecture proofs. This is expected to solidify their logical reasoning and technical communication skills. Number of Sessions: 2 sessions Instructor: Marta Lewicka, 2 sessions, 1.5 hour each.

Speaker : Lars DIENIG (Department of Mathematics,University of Bielefeld ,Germany)

This course provides a rigorous and modern foundation to the analysis and approximation of both linear and nonlinear Partial Differential Equations (PDEs). Instead of relying on explicit solution formulas that are often limited to a few specific equations, we will use the powerful tools of functional analysis to establish fundamental properties that hold for a broad class of problems. We will focus on Hilbert spaces and the notion of weak solutions, which allows us to find solutions that may not be differentiable in the classical sense but are still physically meaningful. The cornerstone of the course will be the introduction and study of Sobolev spaces, the natural setting for analyzing PDEs. Students will learn how to formulate PDEs in a way that allows for the application of abstract theorems from functional analysis, such as the Lax-Milgram theorem, to prove the existence and uniqueness of solutions for linear problems. The notion of weak solutions immediately provides a framework for the approximation of solutions via so-called Galerkin techniques, the most famous of which is the finite element method. The course will also describe the fundamentals behind the numerical analysis of weak solutions via finite element methods, and show how to obtain convergence, with rates, for this method. The theory and numerics will focus on the most important classical linear PDEs: the Laplace, heat, and wave equations. However, we will also briefly mention how our analytic and numerical tools extend to a select number of nonlinear PDEs, such as the nonlinear heat equation. This will include a brief introduction to monotone operator theory, which provides a powerful method for proving the existence of solutions for a wide class of nonlinear problems. Once again, the numerical analysis with finite element methods will be presented and students will once again see the fine interplay between theory and numerics. This course will be greatly complemented by Activities 1 and 2, which will focus on practical aspects and the computer implementation of the methods that will be theoretically discussed here. Number of Sessions: 5 sessions Instructor: Lars Diening, 5 sessions each 1 hr.

Speaker : Lars DIENIG (Department of Mathematics,University of Bielefeld ,Germany)

This interactive activity will build upon the courses on the approximation of variational and nonvariational problems and the programing session, and it will challenge students to efficiently solve, numerically, various stationary and time-dependent PDE systems, ranging from purely mathematical examples (like the nonlinear Schrödinger equation) to core systems in mathematical physics and biology (like reaction-diffusion models). This activity will give students hands-on experience in tackling problems beyond the classical textbook examples and can serve as a basis for a more in-depth exploration of computational tools for PDE modeling and the development of scientific software. Number of Sessions: 2 sessions Instructor: Lars Diening, 2 sessions each 1 ¼ hour.