# Analysis Seminar 2016-17

### Term 1 2016/17

Organiser: Jose Rodrigo

6th October Charles Fefferman (Princeton)

Title: Mathematics of Graphene and its Photonic Analogues
Abstract: TBA

13th October Herbert Koch (Bonn)

Title: A continuous family of conserved energies for Nonlinear Sch"rodinger and Korteweg-de Vries
Abstract: TBA

20th October Wenshuai Jiang (Warwick)

Title: L^2 curvature bounds on manifolds with bounded Ricci curvature
Abstract: In this talk, we will discuss the L^2 curvature estimates on manifolds with bounded Ricci curvature and noncollapsing volume, which is a conjecture of Cheeger-Naber. First, we will talk about the background, then we introduce the concept of neck region which appears everywhere in our proof. After that, we would sketch the whole proofs.At last, we would focus on the technical details and our new observations on neck region. One is the local L^2 curvature estimate on the regular ball with harmonic radius lower bound. The other key new ingredient is a superconvexity estimate of the hessian of harmonic functions. All the estimates are on neck region. This is joint work with Prof. Aaron Naber from Northwestern University. One can find the paper in arxiv ( http://arxiv.org/abs/1605.05583 ).

27th October No Seminar (EPSRC visit)

Title: TBA
Abstract: TBA

3rd November Tobias Barker (Oxford)

Title: Regularity Criteria for the Navier-Stokes System and the Connection with Notions of Global Solutions with Non-Energy Initial Data
Abstract: In the first part of the talk, we will discuss the behaviour of critical Lorentz norms of solutions to the initial boundary value problem for the Navier-Stokes system, as time approaches possible blow up. The first part is a joint work with G.Seregin.
Time permitting, we will then explain how this scenario motivates a notion of 'global weak solutions' of the Navier Stokes Cauchy problem, with initial data in critical spaces. We will discuss the case of weak L_3 initial data, as well as corresponding global existence, uniqueness and regularity results.
The second part is a joint work with G.Seregin and V.Sverak.

10th November Lashi Bandara (Chalmers)

Title: Geometric singularities and a flow tangential to the Ricci flow
Abstract: In 2012, Gigli and Mantegazza introduced a new geometric flow via heat kernels. They demonstrated that this flow is tangential to the
Ricci flow in a suitable weak sense for smooth, compact Riemannian manifolds. A salient feature of this flow is that it can be given
meaning for compact RCD metric spaces by interpreting the equation distributionally as a flow of the distance metric. Gigli and Mantegazza
further show that the two formulations agree for the smooth, compact manifold case. As a consequence, this flow can be successfully defined
for spaces containing certain singularities. An important question is to understand regularity - do singularities disappear along the flow,
or are they retained? The quintessential example has been to study manifolds with conical singularities.

In our work, we partially address this regularity question by studying spaces with "geometric singularities", by which we mean a smooth
manifold but with a non-smooth metric. When such spaces are also RCD metric spaces with singularities on a closed subset, we obtain a metric
tensor on the open non-singular part with regularity corresponding to the regularity of the initial heat kernel. In particular, we
demonstrate that a manifold with a finite number of geometric conical singularities remains a smooth manifold away from the cone points for
all time along the flow. For "rough" initial metrics, where we expect only continuity of the flow, we demonstrate connections between
regularity of the flow and homogeneous Kato square root estimates.

17th November Sara Daneri (Friedrich-Alexander University Erlangen-Nürnberg)

Title: The Cauchy problem for dissipative Hölder Euler flows
Abstract: An abstract can be found here.

24th November Juan José López Velázquez (Bonn)

Title: Blow-up and Long time behaviour of kinetic equations with cubic nonlinearities
Abstract: In this talk I will discuss some recent results about singularity formation and long time asymptotics for two kinetic equations containing cubic nonlinearities. These equations are the Nordheim's equation for bosons and the kinetic equation for Weak Turbulence associated to the Nonlinear Schrödinger Equation. The solutions of these equations can yield singularity formation in finite time for homogeneous particle distributions. In the case of Nordheim equation the singularities are related to the formation of Bose-Einstein condensates. Issues like nonuniqueness of the solutions of this equation and self-similar behaviour for long times of the solutions of the Weak Turbulence equation will be also addressed.

1st December Daniel Peralta Salas (ICMAT)

Title: Vortex reconnection in the three-dimensional Navier-Stokes equations
Abstract: An important property of the 3D Euler equations is that the topology of the vortex structures of the fluid does not change in time as long as the solutions do not develop any singularities. To put it differently, the set of (say) vortex tubes and vortex lines of the fluid at time t is diffeomorphic to that of the initial vorticity, provided that the solution remains smooth up to this time. Of course, numerical simulations and experiments with real fluids have shown that the situation is completely different in the case of viscous fluids. In this talk I will show how vortex tubes and vortex lines, of arbitrarily complex topologies, are created and destroyed in smooth solutions to the 3D Navier-Stokes equations. This is joint work with Alberto Enciso and Renato Luca.

8th December Julian Scheuer (Freiburg)

Title: The Inverse Mean Curvature Flow and Convex Free Boundary Hypersurfaces in the Unit Ball
Abstract: An abstract can be found here.

### Term 2 2016/17

Organiser: Peter Topping

12 January Gerasim Kokarev (Leeds) Minimal surfaces and eigenvalue problems

19 January Dusa McDuff (special colloquium)

26 January Tamas Keleti (Budapest) Hausdorﬀ dimension of unions of subsets of lines

First I will talk about my line segment extension conjecture, which states that if A is the union of a family of line segments in R^n and B is the union of the corresponding lines then A and B have the same Hausdorff dimension. It will turn out that this conjecture is closely related to the Kakeya conjecture, which states that in R^n every Besicovitch set (set that contains line segments in every direction) has Hausdorff and Minkowski dimension n. With Kornelia Hera and Andras Mathe we determined the Hausdorff dimension of the above sets A and B under some conditions.

In the second part I will talk about the joint work with Alan Chang, Mairanna Csornyei and Kornelia Hera on the following type of problems. Let F be a set in R^n that contains the 1-skeleton of a cube around every point of R^n. We will distinguish three cases:

(a) we allow axis-parallel cubes of any size,
(b) we allow arbitrary rotated cubes of any size,
(c) we allow rotated unit cubes.

It turns out that the minimal Hausdorff dimension of such a set is n-1 in case (a), 1 in case (b) and 2 in case (c). The constructions are obtained using Baire category arguments.

9 February Raphael Hochard (Bordeaux) Ricci ﬂow of incomplete initial data, and application to short-time existence of the ﬂow of non-collapsed 3-manifolds with Ricci curvature bounded from below

16 February Huy Nguyen (QMUL) Mean curvature ﬂow of codimension two surfaces

23 February Andrew Lorent (Cincinnati) The Aviles Giga functional. A history, a survey and some new results

2 March Alessio Figalli (ETHZ) Free boundary regularity in the parabolic fractional obstacle problem

9 March Kevin Beanland (W&L University) Isomorphic theory of Tsirelson-like Banach spaces

16 March Alix Deruelle (Paris VI) Ricci expanders coming out of metric cones

### Term 3 2016/17

Organiser: Ben Sharp

27th April Richard Aron (Kent State University)

Title: Bishop-Phelps-Bollobas type theorems
Abstract: TBA

4th May Yoshikazu Giga (University of Tokyo) - Lecture will be in B3.03

Title: Approximation by Cahn-Hoffman facets and the crystalline mean curvature flow
Abstract: We are interested in approximation of a general compact set in an Euclidean space by nicer sets. In fact, we show that every compact set can be monotonically approximated by a set admitting a certain vector field called the Cahn-Hoffman vector field. Such a set is called a Cahn-Hoffman facet. If the divergence of the minimal Cahn-Hoffman vector field is constant such a set is often called a Cheeger set, which has been widely studied by B. Kawohl, V. Caselles and others.

More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner.

It turns out that this approximation is useful to construct suitable test functions necessary to establish comparison principle for level-set crystalline mean curvature flow equations. As a consequence, we obtained the well-posedness in arbitrary dimension. For a total variation flow of non-divergence type such a comparison result has been established by a joint work with M.-H. Giga and N. Pozar (2014). This lecture is based on my joint work with Norbert Pozar of Kanazawa University.

11th May Fritz Hiesmayr (Cambridge)

Title: Index and spectrum of the Allen--Cahn hypersurfaces
Abstract: The Allen--Cahn method of constructing minimal hypersurfaces has recently produced a new proof of the classical theorem that any
closed Riemannian manifold of dimension at least 3 contains a minimal hypersurface embedded away from a singular set of codimension 7.

In my talk I will first give an overview of the Allen--Cahn construction, and then present my work on the variational properties of two-sided hypersurfaces arising from it. Specifically, I will show that the Morse index of these Allen--Cahn hypersurfaces can be bounded above, and the spectrum of their Jacobi operator below, by the corresponding quantities for the Allen--Cahn functional.

18th May José M. Manzano (UCL)

Title: Some geometric estimates for entire minimal graphs in Heisenberg space
Abstract: TBA

25th May Elena Mäder-Baumdicker (Karlsruhe Institute of Technology)

Title: Willmore Klein bottles in Euclidean space
Abstract:
This talk concerns immersed Klein bottles in Euclidean $n$-space with low Willmore energy.

We will first discuss the existence of a smooth Klein bottle minimising the Willmore energy among immersed Klein bottles in $\mathbb{R}^n$ for $n\geq 4$. The expected minimiser is Lawson's bipolar $\tilde\tau_{3,1}$-Klein bottle, a minimal Klein bottle in $\mathbb S^4$.

Then we will see how to minimise the Willmore energy in regular homotopy classes of Klein bottles. In $\mathbb R^4$, there are three distinct homotopy classes of immersed Klein bottles that are regularly homotopic to an embedding. One contains the above mentioned minimiser. The other two are characterised by the property of having Euler normal number $+4$ or $-4$. The infimum of the Willmore energy in these two classes is $8\pi$. Furthermore, there are infinitely many distinct embedded surfaces attaining this infimum. The proof is based on the twistor theory of the Euclidean four-space.

The results are based on joint work with Jonas Hirsch and Patrick Breuning.

1st June TBA

Title: TBA
Abstract: TBA