Geometric Analysis Reading Seminar
Overview
This is an informal reading seminar on topics in geometric analysis organised by geometers at UCL, which is also of interest to analysts, and is attended by researchers at Imperial, King's, Queen Mary and UCL, as well as Cambridge and ULB (Brussels).
The topics we cover range from the elementary to the advanced and the seminar is meant to be informal and a chance to learn about the subject,
so suggestions for future topics are greatly encouraged. We particularly encourage participation by PhD students.
We keep talks between 1 and 2 hours, and we will often follow a theme for a few weeks. Our current agenda and plans for future topics include:
 AllenCahn equation and minmax theory
 Einstein metrics
 YangMills flow

Schedule
The schedule is usually Wednesdays 13pm at UCL, with some Tuesday or Thursday afternoon meetings. The room at UCL this term is usually Room 500 in the UCL Mathematics Department, 25 Gordon Street.
Please check the schedule below for the exact room and directions.

Thur 27 April 2017
13:0015:00 UCL Maths Department Room 500 
Desingularizing Einstein metrics
Speaker: Michael Singer (UCL)
Abstract: The goal will be to present Biquard's paper on desingularizing 4dimensional Einstein orbifolds via gluing, including discussions of
the obstructions.

Thurs 11 May 2017
15:3017:30 UCL Maths Department Room 505 
An overview of YangMills flow
Speaker: Casey Kelleher (UC Irvine)
Abstract: The aim is to provide a discussion of the basics of YangMills flow, including the analogy with harmonic map heat flow, the work of Donaldson and Struwe, and potentially singularities and removal of singularities.

Wed 17 May 2017
13:0015:00 UCL Maths Department Room 500 
Uhlenbeck gauge construction
Speaker: Yang Li (Imperial/LSGNT)
Abstract: This talk will focus on the construction of an appropriate gauge in which to study YangMills connection, given by work of Uhlenbeck.

Wed 24 May 2017
13:0015:00 UCL Roberts Building Room 508 
YangMills flow in dimension 4: parabolic gauge
Speaker: Huy Nguyen (QMUL)
Abstract: In this first of three talks discussing Waldron's preprint on YangMills flow in dimension 4, the goal will be to discuss the modification of the gauge construction from the previous talk to this parabolic setting.

Thurs 1 June 2017
13:0015:00 UCL Maths Department Room 500 
YangMills flow in dimension 4: energy cascade
Speaker: Reto Buzano (QMUL)
Abstract: In this second of three talks discussing Waldron's preprint on YangMills flow in dimension 4, the goal will be to go through the key energy arguments in the proof.

Tues 13 June 2017
13:0015:00 UCL Maths Department Room 500 
YangMills flow in dimension 4: completing the proof
Speaker: Kim Moore (Cambridge)
Abstract: In this final talk discussing Waldron's preprint on YangMills flow in dimension 4, the key steps from the previous talks will be brought together to summarize the proof.

Wed 1 February 2017
13:0015:00 UCL Maths Department Room 707 
Convergence of the AllenCahn equation to Brakke’s motion by mean curvature
Speaker: Felix Schulze (UCL)
Abstract: I will present the main results of Ilmanen’s paper on convergence of solutions to the AllenCahn equation to Brakke’s motion by mean curvature.

Wed 8 February 2017
13:0015:00 UCL Maths Department Room 707 
The minmax construction of AllenCahn critical points
Speaker: Fritz Hiesmayr (Cambridge)
Abstract: In my talk I will discuss the paper by Guaraco entitled
"Minmax for phase transitions and the existence of embedded minimal
hypersurfaces" from 2015. After situating the paper in the broader
context of the construction of minimal hypersurfaces, I will present
its results in two parts: first, the PDE minmax construction, and
then the energy upper and lower bounds. Time permitting, I might make
a short comparison between the AllenCahn construction and the
earlier AlmgrenPitts theory.

Wed 22 February 2017
13:0015:00 UCL Maths Department Room 707 
Stability and absence of classical singularities for the minimal hypersurface obtained as limit of AllenCahn stable solutions.
Speaker: Costante Bellettini (UCL)
Abstract: Two weeks ago Fritz described how to construct an index one critical point for the εAllenCahn functional: a suitable subsequence as ε→0 delivers a minimal hypersurface (integral varifold) in the limit. I will describe the contents of Tonegawa and TonegawaWickramasekera, which show respectively how the stability of AllenCahn solutions is inherited by the limit varifold and how to check that the varifold has no classical singularities. These are the properties that enable the use of Wickramasekera's regularity (codimension7 singular set).

Wed 1 March 2017
13:0015:00 UCL Maths Department Room 707 
CalabiYau metrics on Kummer surfaces
Speaker: Eleonora di Nezza (Imperial)
Abstract: After Yau proved the Calabi conjecture, showing the existence of Kähler metrics with Ricci curvature identically zero on compact
Kähler manifolds with vanishing first Chern class, there has been a lot of use of "gluing constructions" in order to give
an almostexplicit description of these metrics in some special cases.
In this talk I will present a paper of Donaldson: the goal is to explain a gluing construction for some CalabiYau metrics on K3 surfaces.

Wed 8 March 2017
13:0015:00 UCL Maths Department Room 707 
Collapsing CalabiYau metrics on K3 surfaces via gluing
Speaker: Fabian Lehmann (LSGNT)
Abstract:
After last week's gluing construction of a CalabiYau metric, this week we will look at Foscolo's recent construction of a family of hyperkahler metrics on the K3 which collapse with bounded curvature outside of finitely many points to T^{3}/Z_{2}. The geometry around points where the curvature blows up is modelled on rescaled ALF gravitational instantons.

Wed 15 March 2017
13:0015:00 UCL Maths Department Room 707 
Minimal surfaces in PoincaréEinstein manifolds
Speaker: Joel Fine (ULB)
Abstract: The talk will be loosely based on the article "Renormalized area and properly embedded minimal surfaces in hyperbolic 3manifolds" by Alexakis and Mazzeo. A PoincaréEinstein metric is an Einstein metric on the interior of a manifold with boundary which is asymptotically hyperbolic near the boundary. Such a metric induces a conformal structure on the boundary. They were first systematically studied by Fefferman and Graham as a way to investigate conformal structures. A central problem is to find a PoincaréEinstein metric filling in a given conformal infinity and even count the number of such solutions. Minimal surfaces give a simpler version of this boundary problem: for a fixed PoincaréEinstein metric, given a curve on the boundary, how many minimal surfaces can one find in the interior which meet the boundary at right angles in the given curve? Alexakis and Mazzeo show that, when the ambient manifold has dimension 3, the boundary value map for minimal surfaces is proper, Fredholm of index 0 and so one can define its degree, which “counts” the number of solutions to this Dirichlet type problem. I will explain their proof and then discuss why I hope a similar result should hold for certain ambient manifolds of dimension 4. If there is time, I will try and outline why this may be of interest for the original question of finding a PoincaréEinstein metric with given conformal structure at infinity.

Wed 22 March 2017
13:0015:00 UCL Maths Department Room 707 
Gluing EguchiHanson metrics and a question of Page
Speaker: Jason Lotay (UCL)
Abstract: The goal will be to present a paper by Brendle and Kapouleas which modifies the gluing construction for Ricciflat metrics on a 4manifold (the K3 surface) known as the Kummer construction, described in Eleonora's talk on 1 March. The original motivation of the paper was to try to produce the first example of a compact Ricciflat metric with full holonomy, which in particular would be the first compact nonKaehler Ricciflat 4manifold. However, the end result is not a Ricciflat metric, but instead an ancient solution to Ricci flow. I will explain the apparent obstructions to the Ricciflat gluing problem and the modification required to obtain the solution to Ricci flow.

Information
The current members of the group include:
 Costante Bellettini (UCL)
 Reto Buzano (QMUL)
 Eleonora Di Nezza (Imperial)
 Joel Fine (ULB)
 Karsten Fritzsch (UCL)
 Udhav Fowdar (UCL)
 Raul Sanchez Galan (UCL)
 Fritz Hiesmayr (Cambridge)
 Fabian Lehmann (LSGNT)
 Jason Lotay (UCL)
 Mattia Miglioranza (UCL)
 Kim Moore (Cambridge)
 Huy Nguyen (QMUL)
 Felix Schulze (UCL)
 Michael Singer (UCL)
 Giuseppe Tinalgia (KCL)
 Francesca Tripaldi (KCL)
For more information about the seminar contact Felix Schulze (f dot schulze at ucl dot ac dot uk). You can also subscribe to the seminar mailing list, if you are within the UCL network. If you are not within the UCL network, please send an email to Felix Schulze (f dot schulze at ucl dot ac dot uk) with a request to subscribe you to the mailing list.

Past seminars
Wed 9 December 2015
14:0016:00 UCL Maths Department Room 707 
Hölder continuity of tangent cones of limit spaces
Speaker: Reto Buzano (Müller) (QMUL)
Abstract: In this talk, we will focus on Colding and Naber's result that tangent cones of limit spaces of manifolds with lower bounds on the Ricci curvature vary Hölder continuously along geodesics. This will follow from the Hölder continuity of the geometry of small balls with the same radius in smooth manifolds. All results can be found in http://arxiv.org/abs/1102.5003  the aim is to give a comprehensive overview of the most important parts of this paper.

Wed 2 December 2015
14:0016:00 UCL Maths Department Room 707 
Almost volume cones and almost metric cone and the size of the singular set
Speaker: Felix Schulze (UCL)
Abstract: In this talk we will discuss that almost volume cones are almost metric cones and discuss the structure and the size of the limiting singular set.

Wed 4 November 2015
14:0016:00 UCL Maths Department Room 707 
Almost rigidity: volume convergence
Speaker: Yong Wei (UCL)
Abstract: This talk will focus on the volume convergence part of CheegerColding theory.

Wed 28 October 2015
14:0016:00 UCL Maths Department Room 707 
Almost rigidity: the almost splitting theorem
Speaker: Yang Li (LSGNT)
Abstract: This talk will be mainly about the almost splitting theorem (and depending on time I may or may not talk about the volume convergence).

Wed 21 October 2015
14:0016:00 UCL Maths Department Room 500 
Introduction to CheegerColding theory
Speaker: Panagiotis Gianniotis (UCL) and Jason Lotay (UCL)
Abstract: In this talk we will first discuss the ChengYau gradient estimate, before starting on CheegerColding theory, including quantitative maximum principles, rigidity and
almost rigidity, and the structure of limit spaces.

Wed 14 October 2015
14:0016:00 UCL Maths Department Room 707 
Introduction to spaces with lower Ricci curvature bounds
Speaker: Panagiotis Gianniotis (UCL)
Abstract: In this talk I will survey some of the basic results in Riemannian Geometry which will form the building blocks for the study of limits of spaces with
Ricci curvature bounded below. In particular, I will discuss the Bochner formula, volume and Laplacian comparision
theorems, rigidity, Gromov's compactness theorem, the strong maximum principle and the splitting theorem.

Wed 3 June 2015
14:0016:00 UCL Medical Sciences G46 HO Schild
Pharmacology Lecture Theatre 
Monopole moduli spaces and metrics (Part 4)  The AtiyahHitchin Proposition
Speaker: Karsten Fritzsch
Abstract: In this part of our mini lecture series on magnetic monopoles, I will focus on a proposition by Atiyah and Hitchin concerning a type of asymptotic decomposition of monopoles. This proposition was already stated in the last part of this series and it was explained that it can be regarded as a starting point for a route towards a compactification of the moduli space of magnetic monopoles. In this part, I will go into the details of the proof of this proposition and in particular explain the convergence results of Uhlenbeck leading to this proposition.

Wed 27 May 2015
14:0016:00 UCL Medical Sciences G46 HO Schild
Pharmacology Lecture Theatre 
The KaehlerRicci flow on Kaehler surfaces and MMP
Speaker: Eleonora di Nezza (Imperial)
Abstract: It was recently proposed by Song and Tian a conjectural picture that relates the KaehlerRicci flow (KRF) to MMP (Minimal Model Program) with scaling. Although not so much is known in high dimension, much is understood about the KRF in the case of Kaehler surfaces. We will describe the behavior of the KRF on Kaehler surfaces and how it relates to the MMP. In particular, we will show how the KRF carries out the algebraic procedure of contracting (1)curves.
(The results I will talk about are due to Song and Weinkove)

Wed 20 May 2015
14:0016:00 UCL Medical Sciences G46 HO Schild
Pharmacology Lecture Theatre 
Monopole moduli spaces and metrics (Part 3)
Speaker: Michael Singer

Wed 13 May 2015
14:0016:00 UCL Medical Sciences G46 HO Schild
Pharmacology Lecture Theatre 
Monopole moduli spaces and metrics (Part 2)
Speaker: Michael Singer

Wed 6 May 2015
14:0016:00 UCL Medical Sciences G46 HO Schild
Pharmacology Lecture Theatre 
Monopole moduli spaces and metrics (Part 1)
Speaker: Michael Singer
Abstract: We shall start with relevant definitions of monopoles, moduli spaces and the moduli space metrics. The conjectural structure of the asyptotic region of the moduli spaces will be discussed, and new progress on these metrics will be described. I will also aim to mention some important open problems within the first two weeks. After this, we will get technical with the analytic tools we use, including manifolds with corners, the relevant noncompact elliptic theory and so on. In particular I hope to explain why the use of manifolds with corners is natural and sensible for this problem. 
Wed 10 March 2015
11:001:00 UCL Taviton (16) Room 431 
An Introduction to the KaehlerRicci flow
Speaker: Eleonora di Nezza (Imperial)
Abstract: I will give an exposition of a number of wellknown results such as the maximal time of existence of the flow and the convergence on manifolds with negative and zero first Chern class. I will also discuss the regularizing properties of the KaehlerRicci flow. Finally, if time permits, I will show that the KaehlerRicci flow can be run from an arbitrary positive closed current, and that it is immediately smooth in a Zariski open subset of X.
.

Wed 25 February 2015
11:001:00 UCL Taviton (16) Room 431 
Tangent cones to twodimensional areaminimizing integral currents are unique
Speaker: Tom Begley (Cambridge)
Abstract: I will give an overview of the paper of Brian White of the same name. In this paper the author first reduces the problem of uniqueness of tangent cones to an 'epiperimetric' inequality. Then, for twodimensional areaminimizing currents, the inequality is proved by constructing explicit comparison surfaces using multiplevalued harmonic functions. As well as discussing the paper, I will start with a quick recap of the requisite geometric measure theory.

Wed 11 February 2015
11:001:00 UCL Taviton (16) Room 431 
Heat Kernel and curvature bounds in Ricci flows with bounded scalar curvature (Part II)
Speaker: Yong Wei
Abstract: I will talk about the results in sections 67 of the paper "Heat Kernel and curvature bounds in Ricci flows with bounded scalar curvature" by Richard Bamler and Qi Zhang (arXiv:1501.01291). By assuming the scalar curvature is bounded along the Ricci flow, they proved the backward pseudolocality theorem which can be coupled with Perelman's forward pseudolocality theorem to deduce a stronger \epsilonregularity theorem for Ricci flow. As an application, they derived a uniform L^2bound for the Riemannian curvature in 4dimensional Ricci flow with uniformly bounded scalar curvature and show that such flow converges to an orbifold at a singularity.

Wed 4 February 2015
11:001:00 UCL Drayton House Room B04 
Mean value inequalities and heat kernel bounds for the Ricci flow
Speaker: Panagiotis Gianniotis
Abstract: In their recent paper : "Heat Kernel and curvature bounds in
Ricci flows with bounded scalar curvature" Richard Bamler and Qi Zhang
analyse Ricci flows in which there is a bound on the scalar curvature.
The ultimate goal of this work is to understand the possible singular
behaviour of a Ricci flow when the scalar curvature remains bounded,
and find situations that this singular behaviour can be excluded.
I will talk about the first part of their paper, focusing on the
results on Sections 35, in which they prove a distance distortion
estimate (answering a question of Hamilton), construct good cutoff
functions and prove several mean value inequalities for solutions of
the heat and conjugate heat equations. These results finally lead to
Gaussian bounds for heat kernels.

Wed 28 January 2015
1:003:00 UCL 25 Gordon Street Room 707 
Asymptotic Rigidity of Selfshrinkers in Mean Curvature Flow
Speaker: Lu Wang (Imperial)
Abstract: In this talk, we discuss uniqueness of selfshrinkers with prescribed asymptotic behavior at infinity. The main tool is the Carleman type estimates.

Wed 3 December 2014
1:303:00 UCL 25 Gordon Street Room 707 
Hyperbolic AlexandrovFenchel inequality
Speaker: Yong Wei
Abstract: I will present one work in my Phd thesis. For any 2convex and starshaped hypersurface in hyperbolic space, we prove a sharp AlexandrovFenchel type inequality involving the 2nd mean curvature integral and area of the hypersurface.
I will start the talk with the motivation of the problem, including an introduction of isoperimetric and AlexandrovFenchel inequality in Euclidean space with the recent new proof and applications. Then I state our main result, recent progress and some open problems. Finally, I will give an overview of the proof: in the strictly 2convex case, the proof relies on an application of Gerhardt's convergence result of inverse mean curvature flow for strictly meanconvex hypersurfaces in hyperbolic space, and sharp Sobolev inequalities on sphere; in the general 2convex case, the proof involves an approximation argument.

Wed 26 November 2014
1:003:00 UCL 25 Gordon Street Room 707 
Eleven dimensional supergravity equations on edge manifolds
Speaker: Xuwen Zhu (MIT)
Abstract: We consider the eleven dimensional supergravity equations on
B^7xS^4 considered as an edge manifold. We compute the
indicial roots of the linearized equations using the Hoge
decomposition, and construct a realvalued generalized inverse using
different behavior of spherical eigenvalues. We prove that all the
solutions near the FreundRubin solution are prescribed by three
pairs of data on the boundary 6sphere.

Wed 19 November 2014
1:003:00 UCL 25 Gordon Street Room 707 
Uniqueness of Lawlor necks
Speaker: Jason Lotay
Abstract: I will discuss the paper by ImagiJoyceOliveira dos Santos on uniqueness of special Lagrangians and Lagrangian selfexpanders asymptotic to transverse pairs of planes. This will use the material on Fukaya categories discussed by Jonny Evans in the Symplectic Working Group Seminar on 18 Nov.

Wed 12 November 2014
1:003:00 UCL 25 Gordon Street Room 707 
Shorttime existence of Lagrangian Mean Curvature Flow
Speakers: Tom Begley and Kim Moore
Abstract: We will present recent work on the shorttime existence of Lagrangian mean curvature flow with nonsmooth initial condition. Specifically we are able to show that given a smooth Lagrangian submanifold with a finite number of isolated singularities, each asymptotic to a pair of transversally intersecting Lagrangian planes P_1 and P_2 such that neither P_1 + P_2 or P_1  P_2 are area minimising, there exists a smooth Lagrangian mean curvature flow existing for a short time, and attaining the initial condition in the sense of varifolds as t goes to 0, and smoothly locally away from singularities.
We will give an overview of the proof, which relies on a dynamic stability result for selfexpanders, a monotonicity formula for the selfexpander equation and the local regularity theorem of Brian White.

Wed 29 October 2014
12:302:00 UCL 25 Gordon Street Room D103 
Results on the Ricci flow on manifolds with boundary
Speaker: Panagiotis Gianniotis
Abstract: Despite the great progress in the study of the Ricci flow on
complete manifolds, the behaviour of the flow on manifolds with
boundary remains a mystery.
In this talk I will begin with an overview of past work on this
problem, and describe the main difficulties it poses. Then, I will
show that augmenting the Ricci flow with appropriate boundary
conditions, one obtains local existence and uniqueness of the flow,
starting from an arbitrary Riemannian manifold with boundary.
I will also describe how these boundary conditions allow derivative
estimates similar to Shi's to hold up to the boundary, although the
maximum principle arguments that are typically used to obtain such
estimates don't seem to be applicable. As a consequence we obtain a
compactness result for sequences of Ricci flows and a continuation
principle.
I will finish the talk with a discussion on some open problems and questions.

Wed 22 October 2014
12:002:00 UCL 25 Gordon Street Room 706 (121) Room 707 (12) 
Layer Potential Operators for Two Touching Domains in R^{n}
Speaker: Karsten Fritzsch
Abstract: So far, no special framework for the study of layer potential operators (or similar operators) on manifolds with corners has been developed even though both approaches, the method of layer potentials and the calculus of conormal distributions on manifolds with corners, have been proven to be very successful.
In this talk, I will demonstrate in two important special cases that the geometric viewpoint of singular geometric analysis leads to a feasible approach to the method of layer potentials: I will solve the Dirichlet and Neumann problems for Laplace's equation on the halfspace in R^{n} in spaces of functions having certain (though very general) asymptotics and then move on to study the more singular situation of two touching domains in R^{n}. Using the PushForward Theorem, on the one hand I will show that the relations between the layer potential operators and their boundary counterparts continue to hold in the singular setting, and on the other hand establish mapping properties of the layer potential operators between spaces of functions with asymptotics. We can improve these by using a local splitting of certain fibrations which arise when applying the PushForward Theorem.
In the first half of the talk, I will introduce and discuss the background material, including the method of layer potentials, polyhomogeneity and the PushForward Theorem, and the b and φcalculi of pseudodifferential operators. If time permits, I will end the talk by sketching the connection to the plasmonic eigenvalue problem on touching domains.

Wed 8 October 2014
11:001:00 UCL 25 Gordon Street Room 706 
Lagrangian mean curvature flow and symplectic topology
Speaker: Jason Lotay
Abstract: I will discuss aspects of Dominic Joyce's recent preprint in which he conjectures modified versions of the ThomasYau conjecture, linking a notion of stability arising in symplectic topology with convergence of Lagrangian mean curvature flow. I will focus on the more concrete parts of his theory and its ramifications for the study of the flow. I will start with an introduction and review of
Lagrangian mean curvature flow.

Tues 1 Apr 2014
11:001:00
UCL 25 Gordon Street Room 706

Finitetime singularities of Lagrangian mean curvature flow (part 1)
Speaker: Felix Schulze
Abstract: I will discuss some further details from the paper on the zeroMaslov class case for singularities in Lagrangian MCF by A. Neves.

Wed 12 Mar 2014 11:301:30
KCL
Strand Building Room S4.29

Singularities of Lagrangian mean curvature flow: zeroMaslov class case
Speaker: Kim Moore
Abstract: I will talk about the paper of the same name by A. Neves.

Wed 5 Mar 2014
12:0014:00
KCL
Strand Building Room S4.36

Introduction to Lagrangian mean curvature flow (part 2)
Speaker: Jason Lotay
Abstract: I will continue to describe some of the basics of Lagrangian mean curvature flow. In particular I will prove that it exists, discuss some examples and describe some more of the known results in the area.
Much of what I say is found in my paper with Pacini, a paper by Thomas and Yau,
the work of Schoen and Wolfson and in the survey by Neves.

Wed 26 Feb 2014
12:0014:00
KCL
Strand Building Room S4.36

Introduction to Lagrangian mean curvature flow (part 1)
Speaker: Jason Lotay
Abstract: I will describe some of the basics of Lagrangian mean curvature flow. In particular I will describe what it is, the relevant parts of symplectic topology required and what the goal of the flow is. I will also try to survey some of the known results in the area. Much of what I say is found in my paper with Pacini, a paper by Thomas and Yau, the work of Schoen and Wolfson and in the survey by Neves.

Wed 5 Feb 2014
12:0014:00
KCL
Strand Building Room S4.36

A local regularity theorem for mean curvature flow (part 2)
Speaker: Tom Begley
Abstract: I will continue to discuss the paper by Brian White of the same name.

Wed 29 Jan 2014
12:0013:30
KCL
Strand Building Room S4.36

A local regularity theorem for mean curvature flow (part 1)
Speaker: Tom Begley
Abstract: I will discuss the paper by Brian White of the same name.

Wed 22 Jan 2014
13:0015:00
Birkbeck Torrington Square Room 254

Introduction to mean curvature flow (part 2)
Speaker: Felix Schulze

Wed 15 Jan 2014
13:0015:00
Birkbeck Torrington Square
Room 254

Introduction to mean curvature flow (part 1)
Speaker: Felix Schulze


