司靈得 (Daniel Spector)

The talk presents constructions of blowup solutions to the Keller-Segel system in \( \mathbb{R}^d \). d=2 (\(L^1\)-critical): There exist finite time single blowup solutions that are of Type \(II\) with finite mass. Blowup rates are quantized according to a discrete spectrum of a linearized operator around the rescaled stationary solution in the self-similar setting. There is also \textit{multiple collapsing blowup solutions} formed by a collision of multiple single solutions with self-similarity that provides a brand new mechanism of singularity formation.

 \(d \geq 3\)(\(L^1\)-supercritical): For \(d \geq 3\), there exist finite time blowup solutions having the form of collapsing-ring which consists of an imploding, smoothed-out shock wave moving towards the origin to form a Dirac mass at the singularity. For \(d = 3,4\), we found blowup solutions with infinite mass that are asymptotically self-similar with a log correction to their profile.

The constructions rely on a spectral approach for multiple-scale problems, renormalization technique, and refined energy estimates. The talk is based on a series of joint works with C. Collot (Paris Cergy), T. Ghoul (NYU Abu Dhabi), N. Nouaili (Paris Dauphine), N. Masmoudi (NYU) and H. Zaag (Paris Nord).

The weak-type(1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.  

Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily power-type are allowed. The local boundedness of the relevant minimizers is established under a suitable balance between the lower and the upper bounds. Classical minimizers, as well as quasi-minimizers are included in our discussion. Functionals subject to so-called p,q -growth conditions are embraced as special cases and the corresponding sharp results available in the literature are recovered.

The Wigner function of a quantum state is a way of describing the phase space distribution of a quantum particle. The uncertainty principle from Fourier analysis places some restriction on the allowable decay of a Wigner function. In this talk I will give an introduction to the Wigner function and show that rapidly decaying Wigner functions must also be Schwartz, which can also be interpreted as a type of uncertainty principle. This is based on joint work with Jess Riedel.

We identify necessary and sufficient conditions on \(k\) th order linear differential operators \(\mathbb{A}\) in terms of a fixed halfspace \(H \subset \mathbb{R}^n \)  such that the Gagliardo--Nirenberg--Sobolev inequality

\[\|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R}^{n},V)\]

holds. This comes as a consequence of sharp trace theorems on \(\partial H\). Strong estimates on lower order derivatives are the best possible due to the failure of Calder\'on--Zygmund theory in \(L^1\).

Joint work with Franz Gmeineder and Jean Van Schaftingen.

We consider maps between spheres S^n to S^\ell that minimize the Sobolev-space energy W^{s,n/s} for some s \in (0,1) in given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group \pi_{n}(\S^\ell). In some situations explicit examples are known if n/s = 2 or s=1.

In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exists can be chosen stable. We also discuss that the minimum W^{s,n/s}-energy in homotopy classes is continuously depending on s.

Joint work with K. Mazowiecka (U Warsaw)

During the talk I will discuss applications of elementary additive combinatorics to dimensional estimates of PDE- and Fourier-constrained measures. My main tool will be a simple certainty principle of the following form: a set \(S \subset \mathbb{R}^N\) contains a given finite linear pattern if \(S\) is a spectrum of a sufficiently singular measure.

My talk is based on two recent joint papers with Paweł Goldstein.

Jacek Gałęski in 2017, in the context of his research in geometric measure theory, formulated the following conjecture:

Conjecture. Let \(1\leq m< n\)  be integers and let  \(\Omega\subset\mathbb{R}^n\)  be open. If \(f\in C^1(\Omega,\mathbb{R}^n)\) satisfies \(\operatorname{rank} Df\leq m\) everywhere in \(\Omega\), then \(f\) can be uniformly approximated by smooth mappings \(g\in C^\infty(\Omega,\mathbb{R}^n)\)  such that \(\operatorname{rank} Dg\leq m\) everywhere in \(\Omega\).

One can also modify the conjecture and ask about a local approximation: smooth approximation in a neighborhood of any point. These are very natural problems with possible applications to PDEs and Calculus of Variations. However, the problems are difficult, because standard approximation techniques like the one based on convolution do not preserve the rank of the derivative. It is a highly nonlinear constraint, difficult to deal with.

In 2018 Goldstein and Hajłasz obtained infinitely many counterexamples to this conjecture. Here is one:

Example. There is \(f\in C^1(\mathbb{R}^5,\mathbb{R}^5)\) with \(\operatorname{rank} Df\leq 3\) that cannot be locally and uniformly approximated by mappings \(g\in C^2(\mathbb{R}^5,\mathbb{R}^5)\) satisfying \(\operatorname{rank} Dg\leq 3\).

This example is a special case of a much more general result and the construction heavily depends on algebraic topology including the homotopy groups of spheres and the Freudenthal suspension theorem.

More recently Goldstein and Hajłasz proved the conjecture in the positive in the case when \(m=1\). The proof is based this time on methods of analysis on metric spaces and in particular on factorization of a mapping through metric trees.

The method of factorization through metric trees used in the proof of the conjecture when \(m=1\) is very different and completely unrelated to the methods of algebraic topology used in the construction of counterexamples. However, quite surprisingly, both techniques have originally been used by Wenger and Young as tools for study of Lipschitz homotopy groups of the Heisenberg group, a problem that seems completely unrelated to problems discussed in this talk.