L4

Positively folded galleries arise as images of retractions of buildings onto a fixed apartment and play a role in many areas of maths (such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne-Lusztig varieties). In this talk, we will define positively folded galleries, and then look at how these can be used to study affine flag varieties. We will also look at a new recursive description of the set of end alcoves of folded galleries with respect to alcove-induced orientations, which gives us a combinatorial description of certain double coset intersections in these affine flag varieties. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.

We show that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on smoothness in a neighborhood of a solution. Given that the problem is convex, our method will converge even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including matrix factorization and training of ResNet-18.

I will present a $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems  in the Heisenberg group. In our  case, both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply.  All the results apply to the more general case of Carnot groups. Joint with Nicolas Dirr, Paola Mannucci and Claudio Marchi.

I will talk about my work arxiv:1911.05038. We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $ \omega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for $ \omega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $s>2, \, s'>\frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.
 

The asymptotically locally Euclidean Ricci-flat self-dual 4-manifolds were classified and constructed by Kronheimer as hyperkahler quotients. Each belongs to a finite-dimensional family and a particularly interesting subfamily consists of manifolds with a circle action which can be identified with the minimal resolution of a quotient singularity C^2/G where G is a finite subgroup of SU(2). The resolved singularity is a configuration of rational curves and there is a distinguished one which is pointwise fixed by the circle action. The talk will give an explicit description of the induced metric on this central sphere, and involves twistor theory and the geometry of the lines on a cubic surface.
 

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.