Analysis Seminar

An extension of Gromov compactness theorem ensures that any family of manifolds with convex boundaries, uniform bound on the dimension and uniform lower bound on the Ricci curvature is precompact in the Gromov-Hausdorff topology. In this talk, we...

In the theory of turbulence, a famous conjecture of Onsager asserts that the threshold Hölder regularity for the total kinetic energy conservation of (spatially periodic) Euler flows is 1/3. In particular, there are Hölder continuous Euler flows...

Motivated by some work of Thurston on defining a Teichmuller theory based on best Lipschitz maps between surfaces, we study infinity-harmonic maps from a manifold to a circle. The best Lipschitz constant is taken on on a geodesic lamination...

Given a set $E$ of Hausdorff dimension $s > d/2$ in $\mathbb{R}^d$ , Falconer conjectured that its distance set $\Delta(E)=\{ |x-y|: x, y \in E\}$ should have positive Lebesgue measure. When $d$ is even, we show that $\dim_H E>d/2+1/4$ implies $|...

We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s (49) conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of vanishing viscosity. Although over the past 60 years...

Lévy matrices are symmetric random matrices whose entries are independent alpha-stable laws. Such distributions have infinite variance, and when alpha is less than 1, infinite mean. In the latter case these matrices are conjectured to exhibit a...

The study of nodal sets of Laplace eigenfunctions has intrigued many mathematicians over the years. The nodal count problem has its origins in the works of Strum (1936) and Courant (1923) which led to questions that remained open to this day. One...

Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in $\mathbb R^n$ which is only about 1.5 times as hard to fill twice as it is to fill...

Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming...

The construction of pure phases from ground states is performed for $ u > u_*(d)$ for all values of $d$ except for 39 special ones. For values $d$ with a single equivalence class all periodic ground states generate the corresponding pure phase which...