** Lecture 1
Date and time: 26 September 2024, 16:00 -17:30
Title: **The Onsager theorem and beyond

**Abstract:**In 1949 Onsager conjectured the existence of Hoelder continuous solutions of the incompressible Euler equations which do not conserve the kinetic energy. A rigorous proof of his statement has been given by Isett in 2017, crowning a decade of efforts in the subject. Onsager's original statement is however motivated by anomalous dissipation in the Navier-Stokes equations: roughly speaking it would be desirable to show that at least some dissipative Euler flow is the ``vanishing viscosity limit''. In these lectures I will review the basic ideas of the first iteration invented by La'szlo' Sze'kelyhidi Jr. and myself to produce continuous solutions which dissipate the total kinetic energy. I will then review the developments which lead Isett to solve the Onsager conjecture and touch upon the new challenges which lie ahead.

** Lecture 2
Date and time: 27 September 2024, 11:00 -12:30
Title: **The Euler equations as a differential inclusion

**Abstract:**In this lecture I will explain the first work of La'szlo' and myself, which introduced for the first time ideas from differential inclusions in the study of the incompressible Euler equations. These ideas allowed to produce far-reaching generalizations of pioneering results by Scheffer and Shnirelman, showing the abundance of counterintuitive bounded solutions.

** Lecture 3
Date and time: 30 September 2024, 11:00 -12:30
Title: **Nash's C^1 isometric embedding theorem and the Borisov-Gromov problem.

**Abstract:**In this lecture I will explain the ideas of Nash's surprising construction, in the 1950s, of many C^1 isometric embeddings of Riemannian manifolds as hypersurfaces of the Euclidean space. We will then touch upon an open problem in the area, due to to Borisov and Gromov, about the threshold Hoelder regularity for which the Nash phenomenon is possible. This problem turns out to be intimately linked to the Onsager conjecture and we will survey the results proved so far about it.

** Lecture 4
Date and time: 01 October 2024, 11:00 -12:30
Title: **C^0 convex integration for incompressible Euler

**Abstract:**In this lecture I will explain a second type of iteration, introduced by La'szlo' and myself, which produces continuous solutions of incompressible Euler in a fashion which shares a lot of similarities with the theorem of Nash explained in Lecture 3.

__Lecture 5__

Date and time: 03 October 2024, 11:00 -12:30

**Title: **The Onsager theorem and beyond

**Abstract: **In this lecture I will cover the developments in the area which followed our work and finally lead Isett to the resolution of the Onsager conjecture. If time allows I will give a glimpse of the challenges that lie ahead if one would like to give a rigorous justification that some dissipative solutions of the Euler equations are indeed limit of classical solutions of Navier-Stokes.

**About the ****speaker:**

Camillo De Lellis was born in 1976 in San Benedetto del Tronto, Italy. After earning his undergraduate degree in mathematics at the University of Pisa in 1999, he wrote his doctoral dissertation in 2002 under the supervision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa. He joined the faculty of the University of Zürich in 2004 as Assistant Professor of Mathematics, and he was appointed Full Professor in 2005. In 2018 he moved to the Institute for Advanced Study in Princeton, where he holds the IBM von Neumann Professorship. He is active in the fields of calculus of variations, geometric measure theory, hyperbolic systems of conservation laws and fluid dynamics. He has been a plenary speaker at the European Congress of Mathematics at Krakow in 2012 and is a member of the Academia Aeuropea, of the German Academy of Sciences, and of the American Academy of Arts and Sciences. He is the recepient of the 2009 Stampacchia Medal, 2013 SIAG/APDE Prize (jointly with La'szlo' Sze'kelyhidi Jr.), 2013 Fermat Prize (jointly with Martin Hairer), 2014 Caccioppoli prize, 2015 Amerio Prize, 2020 Bocher Prize (jointly with Larry Guth and Laure Saint-Raymond), 2020 Feltrinelli prize, and 2021 Myriam Mirzakhani prize.

This lecture series is part of the program "Deterministic and Stochastic Analysis of Euler and Navier-Stokes Equations"