(in Polish) Zaawansowany rachunek wariacyjny 1000-1S26ZRW
The calculus of variations is concerned with the study of extremal problems for functionals, which arise in many models of the natural sciences. It is a central area of modern analysis. The seminar aims to present the key ideas and tools of this theory from the perspective of current research directions, with particular emphasis on applications in the theory of partial differential equations.
The course is intended for students and PhD candidates interested in engaging with contemporary research topics. A major focus will be on optimal transport theory and gradient flows in spaces of measures, which provide an evolutionary viewpoint on variational problems. Selected topics of active research will also be discussed, including gamma-convergence, metric spaces together with their connections to analysis on manifolds.
Course coordinators
Learning outcomes
Participants will:
* understand the fundamental concepts of the calculus of variations and their role in modern mathematical analysis,
* be familiar with the basic tools of optimal transport theory, gradient flows, and their applications in the theory of partial differential equations,
* be able to present research results and engage in discussions.
Assessment criteria
Participants are required to give two presentations during the academic year and actively participate in discussions.
Bibliography
* W. Górny and J.M. Mazón, Weak Solutions to Gradient Flows in Metric Measure Spaces, Cambridge Tracts in Mathematics, vol. 235, Cambridge University Press, 2026.
* E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
* F. Santambrogio. Optimal transport for applied mathematicians, volume
87 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling.
* L. Ambrosio, E. Brué, and D. Semola. Lectures on optimal transport, volume 169 of Unitext. Springer, Cham, second edition, [2024] ©2024. La Matematica per il 3+2.
* L. Ambrosio, N. Gigli, and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, second edition, 2008.
* G. Dal Maso. An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser, Boston, 1993.
* K.-T. Sturm. On the geometry of metric measure spaces. I & II. Acta Math., 196(1):65-131 & 133-177, 2006.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: