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The 2016 British Mathematical Colloquium (BMC) is taking place at the University of Bristol from Monday 21st March until Thursday 24th March.

On the afternoons of Tuesday 22nd and Wednesday 23rd March there will be an analysis workshop / special session, organised by John Mackay and Michiel van den Berg.

All talks will be in MV1.11 on Tuesday, and Chemistry Lecture Theatre 2 on Wednesday.

**Schedule**

Tuesday 22nd March | Wednesday 23rd March | |

2:00 - 2:30 | Fabio Cavalletti | Katrin Fässler |

2:40 - 3:10 | Taryn Flock | Ben Sharp |

3:20 - 3:50 | Virginie Bonnaillie-Noël | Maria Carmen Reguera |

4:00 - 4:30 | Tobias Huxol | Enrico Le Donne |

**Analysis workshop speakers**

*Spectral minimal partitions*

To any*k*-partition (D_{1}, …, D_{k}) of a domain Ω is associated an energy which is the maximum of the first Dirichlet-Laplacian eigenvalue on each subdomain D_{j}:sup{ λ _{1}(D_{j}), 1≤ j≤ k}.

The optimization problem consists then in determining the minimizer among any k-partitions. If k=2, the minimal partition is given by the second eigenfunction of the Dirichlet-Laplacian on Ω. As soon as k≥ 3, the situation is more complicated and there is no result for k≥3 when Ω is a square, a disk or the equilateral triangle. Nevertheless we have some partial results using the nodal partitions.

Instead of consider the infinite norm, we can consider the p-norm, (that is to say the sum when p=1). In that case, the spectral approach is no more available.

In this talk, we present main results about the infinite norm and some numerical simulations for the p-norm.

This is a joint work with B. Bogosel, B. Helffer, C. Léna, G. Vial.

*Lower bounds for fractional Riesz transforms and the Wolff energy*

In this talk we study estimates from below for the L^{2}norm of the s-dimensional Riesz transform, with kernel x/|x|^{s+1}for s∈(d−1,d), of general Borel measures in ℝ^{d}. These estimates allow to establish an equivalence between the capacity γ_{s}associated with the s-dimensional Riesz kernel and the capacity Ċ_{2/3(d−s),3/2}from non-linear potential theory associated to the Wolff potential.

This is joint work with B. Jaye, F. Nazarov and X. Tolsa.

*Lévy-Gromov Isoperimetric inequality for metric measure spaces*

Using an L^{1}localization argument, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely RCD^{*}(K,N) or more generally essentially non branching CD^{*}(K,N)) the classical Lévy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements.

*Curve packing and modulus estimates*

The modulus of curve families is a powerful tool in the study of quasiconformal and related mappings. Quantitative lower bounds for the modulus can often be obtained if the considered curve family is of a special form, for instance if it constitutes a nice foliation, or if it consists of all possible curves connecting two nondegenerate continua. In this talk I will discuss a collection of curves that is not of this form, namely a family of planar curves that contains an isometric copy of every rectifiable curve in R^{2}of length one.

The "worm problem" of L. Moser asks for the least area covered by the curves in such a family. J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family has always area at least c for some small absolute constant c>0. We strengthen Marstrand’s result by showing that for p>4, the p-modulus of a Moser family of curves is at least c_{p}>0. This is joint work with T. Orponen.

*Stability of the Brascamp-Lieb Constant*

The Brascamp-Lieb inequality generalizes many important inequalities in analysis, including the Höolder, Loomis-Whitney, and Young convolution inequalities. Sharp constants for such inequalities have a long history and have only been determined in a few cases. We investigate the stability and regularity of the sharp constant as a function of the implicit parameters. The focus of the talk will be a local-boundedness result with implications for certain nonlinear generalizations arising in PDE. This is joint work with Jonathan Bennett, Neal Bez, and Sanghyuk Lee.

*Limiting Behaviour of the Teichmüller Harmonic Map Flow*

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. It was introduced by M. Rupflin and P. Topping in 2012. The objective of the flow is to find branched minimal immersions on a given surface.

I will give some background on the flow and then describe some recent work with Rupflin and Topping. In particular we show that if the flow exists for all times then in a certain sense the maps (sub-)converge to a collection of branched minimal immersions with no loss of energy (even when allowing for degeneration of the metric at infinity). We also construct an example of a smooth flow where the image of the limit maps is disconnected.

*Besicovitch Covering Property on graded groups and applications to measure differentiation*

*Minimal hypersurfaces with bounded index*

A conjecture of Yau states that infinitely many distinct minimal hypersurfaces exist inside any closed ambient Riemannian manifold. A result of Marques and Neves (2014) has confirmed this to be the case when the ambient manifold has positive Ricci curvature, moreover that each of these hypersurfaces comes with a natural bound on its index. Therefore it seems reasonable to try to classify minimal hypersurfaces with respect to index. We will give an overview of some recent results concerning the relationship between topology, geometry and the Morse index of minimal hypersurfaces in Riemannian manifolds.