Seminar in Analysis and Geometry

Explore advanced topics in mathematical analysis and geometry through this comprehensive seminar series featuring 21 specialized lectures from leading researchers at the Institute for Advanced Study. Delve into cutting-edge research spanning fluid dynamics, differential geometry, and topological methods, including investigations of the h-principle in foliation theory, shock development problems, and manifold embeddings. Examine critical developments in the Navier-Stokes equations, including 3D blow-up dynamics and non-uniqueness of Leray solutions, while exploring Euler flows with energy dissipation and weak solutions to magnetohydrodynamic equations. Study geometric analysis topics such as constant Chern scalar curvature metrics, totally nonparallel immersions, and level sets of weakly Lipschitz functions. Investigate symplectic and contact geometry through discussions of Floer homology for Hamiltonians and contact convexity, alongside explorations of Arnold's formula for energy variations in 2D vorticity systems. Discover applications of stochastic methods in fluid flow analysis, wave localization phenomena, and the long-time dynamics of 2D Euler equations, providing comprehensive exposure to current research frontiers in mathematical analysis and geometric theory.

Syllabus

Shades of h-principle in foliation theory - Gael Meigniez
The Shock Development Problem - Tristan Buckmaster
On embeddings of manifolds - Dishant Mayurbhai Pancholi
3D Navier-Stokes equations: the dynamics of a blow-up - Alexey P Cheskidov
Building linear homotopies in metric topology with scalable spaces - Aleksandr Berdnikov
On Arnold's formula for the second variation of energy on orbits of 2d vorticities - Vladimír Sverák
h-principle without pre-conditions for ridgy Lagrangians transverse to a distr... - Yakov Eliashberg
The landscape law and wave localization - Svitlana Mayboroda
Floer homology of Hamiltonians supported on subsets - Shira Tanny
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations - Dallas Albritton
Honda-Huang's work on contact convexity revisited - Dishant Mayurbhai Pancholi
Euler flows with local energy dissipation - Hyunju Kwon
Totally nonparallel immersions - Michael Harrison
H1/2− weak solutions of the 3D Euler equations - Matthew Novack
Weak solutions to MHD equations - Daniel Faraco
Level Sets of Weakly Lipschitz Functions - Bobby Wilson
Singularity formation for reduced models of fluid equations - Mimi Dai
Global well-posedness for the 2D one-phase Muskat equations -Hongjie Dong
Metrics of constant Chern scalar curvature - Xi Sisi Shen
Positive Lyapunov exponents and mixing in stochastic fluid flow. Part III - Samuel Punshon-Smith
Remarks on the long-time dynamics of 2D Euler - Theodore Dimitrios Drivas