4EU+ Flagship 3 “Digitisation - Modelling - Transformation” Variational Approaches to PDE with Local and Nonlocal Interactions
Masterclass in Mathematics
October 5-9, 2026 • Mathematikon, Heidelberg University
Overview
This block course is designed for select Master's and PhD students from the 4EU+ European University Alliance. In three specialized modules, participants will explore advanced topics in nonlinear analysis with a focus on applications in materials science and phase field models. The schedule balances theoretical depth with practical application exercises in the afternoons.
- Date: October 5-9, 2026
- Venue: Mathematikon, Heidelberg University • Im Neuenheimer Feld 206, 69120 Heidelberg, Germany
This event is organized by Flagship 3 of the 4EU+ Alliance.
Course Components
- Welcome Meeting & Joint Discussion: An introductory session with all three lecturers to discuss goals, and to discuss interconnections between the course topics.
- Main lectures: Analytical Aspects of Nonlocal operators, Energy-Driven Pattern Formation in Nonlocal systems, Modern Variational Methods in Continuum Mechanics of Solids
- Exercise Sessions: Afternoon sessions dedicated to problem sets with lecturer guidance.
- Concluding discussion: Can used for feedback and to share achieved goals and define further questions.
All lectures are scheduled in 90-minute blocks and include a 20-minute break at the end.
Schedule
Tabelle
| Time | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|
| Time | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
09:00 - 10:30 | Arrival / Individual Meetings | Martin Kružík Lecture 1 | Mouhamed M. Fall Lecture 3 | Hans Knüpfer Lecture 4 | Martin Kružík Lecture 4 |
10:30 - 12:00 | Hans Knüpfer Lecture 1 | Mouhamed M. Fall Lecture 2 | Martin Kružík Lecture 2 | Mouhamed M. Fall Lecture 4 | Concluding Discussion |
12:00 - 14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |
14:00 - 15:30 | Welcome Meeting & Discussion | Hans Knüpfer Lecture 2 | Hans Knüpfer Lecture 3 | Martin Kružík Lecture 3 | Final Test |
15:30 - 17:00 | Mouhamed M. Fall Lecture 1 | Exercise Session 1 | Exercise Session 2 | Exercise Session 3 |
Lecturer & Topic Details
Analytical Aspects of Nonlocal operators
Lecturer: Prof. Mouhamed Moustapha Fall • African Institute for Mathematical Sciences (AIMS) in Senegal
Nonlocal differential operators have recently become fundamental objects in modern analysis and partial differential equations. Their study combines classical harmonic analysis, functional analysis, and geometric methods with novel phenomena arising from nonlocality. Problems involving these operators appear naturally in probability, physics, finance, and geometry. These examples, among many others, have stimulated the development of a rigorous mathematical framework for nonlocal differential operators in the last decade.
The fractional Laplacian (− Δ )s has emerged as a canonical model. We start with the basic properties of the fractional Laplacian, including mapping properties in H ̈older spaces, maximum principles and their connection with fractional Sobolev spaces. Then we study existence of weak solutions using variational methods and derive some key qualitative properties.
Energy-Driven Pattern Formation in Nonlocal systems
Lecturer: Prof. Hans Knüpfer • Institute for Mathematics & IWR, Heidelberg University
Models from Material Science exhibit a rich variety of fascinating patterns whose analysis leads to challenging mathematical questions. A key difficulty comes from the interplay between nonlocal interactions and the nonconvex nature of the underlying energy functionals. We discuss how variational methods provide a natural framework for studying such problems. In particular, we introduce tools such as Γ-convergence and demonstrate how these methods can be used to study qualitative properties of these systems and to derive effective and simplified models.
Modern Variational Methods in Continuum Mechanics of Solids
Lecturer: Prof. Martin Kružík • Institute of Information Theory and Automation, Czech Academy of Sciences & Mathematical Institute of Charles University
In 1830, B. Bolzano observed that continuous functions attain their extreme values on compact intervals of the real line. This foundational insight was later significantly generalized by D. Hilbert and S. Banach, who developed the framework now known as the direct method in the calculus of variations. This method provides a powerful tool for proving the existence of minimizers and maximizers of nonlinear functionals, with semicontinuity playing a central role.
A major advancement came in 1965 with the work of N.G. Meyers, who extended existing results on the lower semicontinuity of integral functionals depending on maps and their gradients. In this series of lectures, we trace the development of this theory from Meyers’ contribution onward, with a particular focus on its applications to the continuum mechanics of solids. Special attention is given to problems in nonlinear elasticity, emphasizing the importance of various notions of convexity and the role of subdeterminants in matrix-valued gradients.
Travel Support Through 4EU+
4EU+ Flagship 3 sponsors one seat per 4EU+ partner university. The selected students will receive a travel allowance and a daily allowance via their local 4EU+ offices. Some universities offer a second funded seat.
The funding rate will align with or be similar to the 4EU+ rates for Blended Intensive Programs (local regulations may vary).
By applying for the course, you are automatically considered for funding.
Tabelle
Daily allowance | per day | € 79 |
Travel allowance | Paris -> Heidelberg Heidelberg <-> Milan Heidelberg <-> Prague Geneva -> Heidelberg Geneva -> Milan | € 285
(€ 211 if not an environmentally friendly means of transportation is being used) |
All other trips between 4EU+ member universities | € 417 (€309 if not an environmentally friendly means of transportation is being used) | |
Additional funding for students with fewer opportunities | € 100 |
Application
Applications for the 4EU+ Masterclass in Mathematics at Heidelberg University are open to Master's and PhD students from the 4EU+ European University Alliance. Please note that participation is limited. Registration confirmation will be based on the evaluation of your application.
To apply, please submit one PDF with the following content:
- Letter of Motivation (one DIN A4 page max.)
- Current transcript
Application Deadlines:
- External / other 4EU+ Universities: August 16, 2026
- Local / Heidelberg University: September 4, 2026
After your participation is confirmed we will contact you regarding the next steps.
Organizer
This event is organized by Flagship 3 of the 4EU+ European University Alliance. This thematic unit, dedicated to research and education with a focus on digitization, modeling, and transformation, is led by Heidelberg University and the University of Copenhagen, and administrated at the Interdisciplinary Center for Scientific Computing (IWR) and the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp).
Head of Organization: Dr. Michael Winckler (IWR, Heidelberg University)
