## Overview

This is an informal reading seminar on topics in geometric analysis organised by J. FineJ.D. LotayM. SingerF. SchulzeL. Foscolo H.T. Nguyen, and myself, which is also of interest to analysts, and is attended by researchers at Imperial, King's, Queen Mary and UCL, as well as Cambridge, Oxford 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 topic is the study of the zero loci of Z/2 harmonic spinors, following the initial sections of the following paper by C. H. Taubes.

For more information about the seminar contact me. 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 me an email with a request to subscribe you to the mailing list.

## Schedule

We will have meetings on Thursdays 1-3pm at UCL. The venue is 222 Euston Road, room G01.

### Term 2 2022 - 2023

 Thu 2 Feb 2023 13:00-15:00 Introduction to harmonic Z/2 spinors   Speaker: Aleksander Doan (UCL) Abstract: The notion of a harmonic Z/2 spinor was introduced by Taubes as an abstraction of various degenerations of solutions to elliptic differential equations arising in geometry and gauge theory. In my talk I will discuss the motivation behind this notion and recent developments in the study of harmonic Z/2 spinors, with focus on the regularity results of Taubes and Zhang. Thu 9 Feb 2023 13:00-15:00 --- Speaker: --- Abstract: --- Thu 16 Feb 2023 13:00-15:00 --- Speaker: Thu 23 Feb 2023 13:00-15:00 --- Speaker: Thu 2 Mar 2023 13:00-15:00 --- Speaker: Thu 9 Mar 2023 13:00-15:00 --- Speaker: Thu 16 Mar 2023 13:00-15:00 --- Speaker:

## Past seminars

### Term 2 2021 - 2022

Counting associatives.
Thu 10 Feb 2022
09:00-10:00
Zoom meeting

Motivation for counting associatives; overview of the reading group schedule; introduction to calibrated geometry
Speaker: Jason D. Lotay (Oxford)

Thu 17 Feb 2022
09:00-10:00
Zoom meeting

Definition of G2 manifolds and associative submanifolds. Examples.
Speaker: Federico Trinca (Oxford)

Thu 24 Feb 2022
09:00-10:00
Zoom meeting

Infinitesimal deformations, obstructions, moduli spaces of associatives
Speaker: Corvin Paul (UCL)

Thu 3 Mar 2022
09:00-10:00
Zoom meeting

Problems with naïve counting of associatives: self-intersections, T^2 singularities and multiple covers.
Speakers: Partha Ghosh (ULB) and Jin Li (Freiburg)

Thu 10 Mar 2022
09:00-10:00
Zoom meeting

Joyce's superpotential solution to associative counting problems.
Speaker: Thibault Langlais (Oxford)

Thu 17 Mar 2022
09:00-10:00
Zoom meeting

Doan-Walpuski's Seiberg-Witten solution to associative counting problems.
Speaker: Jakob Stein (UCL)

Thu 24 Mar 2022
09:00-10:00
Zoom meeting

Counting G2-instantons and bubbling along associatives.
Speaker: Daniel Platt (KCL)

### Term 1 2021 - 2022

Stable minimal hypersurfaces in R^4.

 Thu 14 Oct 2021 15:00-17:00Zoom meeting (note time) Overview   Speaker: Huy The Nguyen (QMUL) Abstract: In this talk we will consider a new result by Chodosh-Li “Stable Minimal Hypersurfaces in R^4". This paper proves a Bernstein theorem for stable two-sided three-dimensional minimal hypersurfaces without an area growth assumption. I will give an overview of the results, the techniques as well as placing the result in the context of known results. In addition, I will introduce some of the main concepts that are used in the proof and in minimal surface theory in general such as stability and Simons identity. Thu 21 Oct 2021 14:00-16:00Zoom meeting Small Energy Curvature estimates for minimal surfaces Speaker: Artemis Vogiatzi (QMUL) Abstract: In this talk, our primary interest will be to prove Choi--Schoen's theorem about curvature estimates for minimal surfaces with small total curvature. We'll next prove some useful bounds, which will help to replace the assumptions on total curvature and area with stability. Thu 28 Oct 2021 14:00-16:00 Zoom meeting L^p bounds and Bernstein Theorems. Speaker: Kobe Marshall-Stevens (UCL) Abstract: This week we will discuss various curvature estimates; first establishing a special case of Allard’s regularity theorem for smooth surfaces before proving L^p curvature bounds due to Schoen-Simon-Yau. As a consequence of these L^p estimates we prove some Bernstein type theorems and conclude by showing how an L^3 curvature estimate is used to establish flatness in the recent paper of Chodosh-Li. Thu 4 Nov 2021 14:00-16:00Zoom meeting The Green’s function on a Stable Minimal Hypersurface Speaker: Alessio Di Lorenzo (UCL) Abstract: Green’s functions are a way to describe solutions to inhomogeneous linear differential equations Lu=f (e.g., Poisson’s equation). Indeed, we can say that they are solutions G of the distributional equation LG=δ, where δ is the Delta measure concentrated in one point, and this in general enables us to write a solution as a convolution. A manifold admitting a positive Green’s function for the Laplace operator is said to be non-parabolic. It is a known fact that stable minimal hypersurfaces (of dimension at least 3) are non-parabolic. We are going to present a proof of this result, while also checking a few properties of the level sets of the Green’s function on such hypersurfaces. Thu 18 Nov 2021 14:00-16:00Zoom meeting Stern's version of the Bochner Formula Speaker: Joshua Daniels Holgate (Warwick) Abstract: Following Section 3 of Chodosh-Li, we will discuss how Stern’s rearrangement of the Bochner formula can be combined with the stability inequality and a particular choice of test function to give an inequality that will later be used to understand the behaviour of the function F(t) as t goes to 0, and thus the geometry of the stable minimal surface M at infinity. Thu 25 Nov 2021 14:00-16:00 Zoom meeting Extension of Munteanu-Wang’s Monotonicity Formula Speaker: Lucas Lavoyer Miranda (Warwick) Abstract: In this talk we will discuss section 4 of Chodosh-Li "Stable minimal hypersurfaces in R^4", where the authors extend the work of Munteanu-Wang on the monotonicity of the F(t) functional. Our goal is to show that as t goes to 0, which can be seen as the behaviour of the manifold along its end, we have F(t)= O(t^2). As previously discussed, this will imply that our manifold is flat. Thu 2 Dec 2021 14:00-16:00 Zoom meeting Proofs of the main results Speaker: Myles Workman (UCL) Abstract: This week we finish the Chodosh-Li paper "Stable Minimal Hypersurfaces in R^4", by looking at Sections 5 and 6. The main result of the paper is contained in Section 5, "A complete, connected, two-sided, stable, minimal immersion M^3 ---> R^4 is flat." While Section 6 is concerned with what can be deduced by reducing the stability assumption to that of just finite index. It was previously known that finite total curvature bounds the index. In Section 6 Chodosh-Li extend this result to an if and only if. In proving this, we will also see how the finite index interacts with the topology of the manifold at its ends. In both cases the proofs follow from the work that has been presented in the previous weeks, with the behaviour of our function F as t goes to 0 being a crucial step in the main calculations.

### Term 3 2020 - 2021

Mean curvature flow. Isoperimetric inequalities.

We follow B. Andrew's survey and paper, and S. Brendle's paper.

 Thu 13 May 2021 14:15-16:15Zoom meeting Moduli of Continuity for Heat Equations   Speaker: Quintin Luong (QMUL) Abstract:In this talk we will present some results of the survey of B. Andrews 'Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations', concerning the moduli of continuity of functions evolving by certain heat equations. The technique used is based on applying the maximum principle to functions of two spatial points and relates the evolution of the modulus to a one-dimensional problem. Thu 20 May 2021 14:15-16:15Zoom meeting Non-collapsing in mean convex Mean Curvature Flow Speaker: Artemis Vogiatzi (QMUL) Abstract: Following this paper by Ben Andrews, we will discuss the proof of a noncollapsing estimate for compact, mean-convex hypersurfaces moving under mean curvature flow. We will see how the proof of this non collapsing result requires only the use of the maximum principle. Thu 27 May 2021 14:15-16:15 Zoom meeting The isoperimetric inequality for a minimal submanifold in Euclidean space: introduction Speaker: Kostas Leskas (UCL) Abstract: In this talk we are going to investigate one of the most famous mathematical problems, the isoperimetric problem. In a simple form that is 'among domains of fixed volume the disk has the boundary of least area'. This problem dates to ancestral times, from the queen Dido and the ancient Greeks. Mathematicians have used many different techniques to tackle this problem in its various forms, from calculus of variations to differential geometric techniques and PDE methods to geometric measure theory. We are going to see some of the ideas developed, reaching to the recent work of Brendle where he proves, using the ABP method from non-linear PDE theory, the isoperimetric problem for minimal submanifolds of R^n, up to codimension 2. Thu 3 June 2021 14:15-16:15Zoom meeting The isoperimetric inequality for a minimal submanifold in Euclidean space: I, inequality Speaker: Kobe Marshall-Stevens (UCL) Abstract: The goal of this talk is to present a sharp Isoperimetric inequality for minimal submanifolds proved by Simon Brendle in this paper. Ideas behind the Aleksandrov-Bakelman-Pucci maximum principle will be used to first prove the Isoperimetric inequality for smooth domains, which Brendle’s proof parallels, before showing the result. Thu 10 June 2021 14:15-16:15Zoom meeting The isoperimetric inequality for a minimal submanifold in Euclidean space: II, equality case Speaker: Myles Workman (LSGNT) Abstract: We continue from last week's talk on Brendle's paper. Last week the first half of the paper was presented, proving the inequality for codimension at least 2 (Theorem 1). This week we focus on the second half of the paper which is devoted to the special case of codimension 2. We will show that the equality case arises if and only if our surface is a flat round ball and the function is constant (Theorem 3). This immediately leads to a strict isoperimetric result for minimal submanifolds in Euclidean space of codimension at most 2.

### Term 2 2020 - 2021

Existence of stationary integral varifolds.

The existence, in an arbitrary compact Riemannian manifold, of stationary integral varifolds (weak notion of minimal submanifolds) of codimension 2 is proved via the abelian Higgs model, following this recent paper by Pigati-Stern.

 Thu 21 Jan 2021 16:00-18:00Zoom meeting Introduction to Abelian Yang-Mills-Higgs - differential geometry of U(1) bundles   Speaker: Daniel Platt (Imperial) Abstract: The talk will provide all definitions and facts needed from gauge theory to study the "Minimal submanifolds from the abelian Higgs model" paper. After discussing Hermitian line bundles and S^1 principal bundles, which are essentially the same thing, and connections on them, we will derive the Euler-Lagrange equations of the rescaled Yang-Mills-Higgs energy functional, and show its gauge invariance. If time permits, we will then construct critical points of the energy functional using geometry and no analysis. Thu 28 Jan 2021 16:00-18:00Zoom meeting Discrepancy inequality   Speaker: Shengwen Wang (QMUL) Abstract:In this session we will present some L^\infty estimate of the discrepancy, which measures the difference of the 2nd term (Yang-Mills energy) and 3rd term (Potential energy) of the Yang-Mills-Higgs energy functional. The method is to apply Moser iteration to a sub-equation satisfied by the discrepancy, and the equation is deduced from some Bochner type identities for the Yang-Mills energy and potential energy. This L^\infty estimate will play a central role in the proof of (n-2)-monotonicity formula and the integrality later. Thu 4 Feb 2021 16:00-18:00Zoom meeting Monotonicity formula Speaker: Mario Schulz (QMUL) Abstract: We derive the inner-variation equation for critical points of the Yang–Mills–Higgs energy. This involves computing the differential of the energy density which we do in detail for the Euclidean case. The goal is to analyse the scaling properties of the Yang–Mills–Higgs energy over small balls. We show how the discrepancy inequality is used to improve the general codimension-four energy monotonicity of Yang–Mills problems and to obtain a codimension-two energy growth. Thu 11 Feb 2021 16:00-18:00 Zoom meeting Exponential decay away from the null set Speaker: Davide Parise (Cambridge) Abstract: Building on the previous talks we are going to refine the pointwise bounds on the curvature and gradient terms of the Yang-Mills-Higgs energy. We will then deduce the main result of the section, namely exponential decay of the Higgs field away from the null set. The sole technical tool needed is maximum principle for elliptic PDEs. Thu 25 Feb 2021 16:00-18:00Zoom meeting (gentle) Introduction to varifolds, GMT and Ambrosio-Soner varifolds + rectifiability argument Speaker: Fritz Hiesmayr (UCL) Abstract: The aim of this presentation is to give an introduction to the theory of varifolds, with a focus on the question of their rectifiability. This includes the theory of generalised varifolds developed by Ambrosio and Soner in their approach to the asymptotics of the Ginzburg-Landau equation. In the second part we will see how this theory, together with density bounds yields the rectifiability of the codimension two varifolds that arise in the abelian Higgs construction. This corresponds to Section 6.1 pp. 21-25. Thu 4 Mar 2021 16:00-18:00Zoom meeting Integrality argument + Jaffe-Taubes quantisation result I Speaker: Huy Nguyen (QMUL) Abstract: In this first of two talks we will show that the rectifiable varifold obtained from the the Abelian Yang-Mills-Higgs energy via the Ambrosio-Soner construction is an integral varifold. The key to this is a result due to Jaffe-Taubes on the quantisation of the two dimensional solutions. Thu 11 Mar 2021 16:00-18:00 Zoom meeting Integrality argument + Jaffe-Taubes quantisation result II Speaker: Huy Nguyen (QMUL) Abstract: In this second talk, we will go through the technical aspects of the proof of the integrality of limit varifolds obtained from the convergence of the Abelian Yang-Mills-Higgs energy. Thu 18 Mar 2021 16:00-18:00 Zoom meeting Min-Max construction on the trivial line bundle Speaker: Fritz Hiesmayr (UCL) Abstract: The objective of this talk is to present a min-max onstruction in the Yang-Mills-Higgs theory, in the special case where the line bundle in question is trivial. Roughly speaking two results will be explained. The first states that solutions produced from this mountain-pass-type argument must be non-trivial. This is justified by a topological argument, which in turn relies on elliptic estimates. The second result consists in estimates for the energies of said critical points, which must be uniformly bounded on either side. The upper bound is more difficult to prove, and requires picking a suitable connection in terms of a given section of the line bundle. This presentation covers Section 7.1, with an emphasis on pages 41-46.

### Term 1 2020 - 2021

Miscellaneous topics.

Minimal surfaces, geometric flows, manifolds with special holonomy.

 Thu 15 Oct 2020 14:00-16:00Zoom meeting Lagrangian mean curvature flow, exotic tori and mirror symmetry   Speaker: Chris Evans (UCL) Abstract: We investigate Lagrangian mean curvature flow in the context of Fano manifolds, specifically looking at the 2-sphere and complex projective plane CP2 as primary examples. We present Vianna's construction of an infinite family of monotone tori in CP2 and describe how mirror symmetry predicts a strong stability result for Lagrangian mean curvature flow. By restricting to symmetric versions of the first two members of this infinite family, we observe new and interesting topological consequences of singularities in mean curvature flow, and sketch a conjectural picture for all exotic tori. Thu 22 Oct 2020 14:00-16:00Zoom meeting 2-Convex Mean Curvature Flow: Weak solutions and Surgery  Speaker: Joshua Daniels-Holgate (Warwick) Abstract: The mean curvature flow of a mean convex hypersurface is very well understood, with a complete classification of singularities. Continuation of the flow through these singularities can be understood via the level set flow, however, one loses the ability to track how the topology changes. Alternatively, one could consider the mean curvature flow with surgery, which can be used to track topology, but is no longer a solution to the mean curvature flow in the classic or weak senses. Fortunately, these two flows are related in a fundamental way and can be used to complement each other. We will outline the basics of mean curvature flow, the level set flow and the surgical procedure. Finally, we will explore the convergence to the level set flow of surgical solutions, a result of both Lauer and Head. Thu 29 Oct 2020 14:00-16:00Zoom meeting Questions (and some answers) around Joyce's generalised Kummer construction Speaker: Daniel Platt (Imperial) Abstract: The first examples of compact manifolds with holonomy equal to G2 were constructed by Joyce in 1996. In the talk, I will define the Lie group G2, G2-structures, and explain Joyce's construction of a G2-structure with small torsion. He then perturbed this to a structure with 0 torsion using elliptic estimates for Sobolev norms, and I will explain an alternative proof of the perturbation result using weighted Hoelder norms. This will feature some ideas that also appear in a range of other gluing constructions. Thu 5 Nov 2020 14:00-16:00 Zoom meeting Smoothing out singular spaces via Ricci flow Speaker: Lucas Lavoyer (Warwick) Abstract: Let (M,g) be an n-dimensional compact Riemannian manifold. It is well-known that there exists a solution to the Ricci flow with (M,g) as initial data. When the initial manifold is non-compact but has bounded geometry, one also gets short-time existence. It is then natural to ask how rough the initial condition can be and if one could use the Ricci flow to smooth out singular spaces. In this talk, we will outline the basics of Ricci flow and discuss some recent work by Schulze and Gianniotis, where they consider smooth compact spaces with isolated singularities as initial conditions for the Ricci flow. Thu 12 Nov 2020 14:00-16:00Zoom meeting J-holomorphic curves in the nearly Kaehler CP^3 Speaker: Ben Aslan (UCL) Abstract: J-holomorphic curves are one of two classes of distinguished submanifolds in a nearly Kaehler manifold M. In the special case of M=CP^3 they are both related to minimal surfaces in S^4 and associative submanifolds in the Bryant-Salamon metric. A nearly Kaehler manifold is neither symplectic nor complex, meaning there are few general methods to study J-holomorphic curves in these spaces. In this talk, we will rely on methods from integrable systems and explain why J-holomorphic curves in CP^3 are described by 2D Toda equations and discuss special classes of examples. Thu 19 Nov 2020 14:00-16:00Zoom meeting Minimal Hypersurface Singularities Speaker: Paul Minter (Cambridge) Abstract: It is well-known that singularities can arise in minimal hypersurfaces. Indeed, since 1969 it has been known that even under the stronger assumption of an area-minimising hypersurface, singularities can still arise if the dimension is sufficiently large. Understanding the local structure of a minimal hypersurface about a singular point, namely the types of tangent object which can arise, is one of the biggest unsolved problems in geometric analysis. In this talk we shall discuss the current story of minimal hypersurface singularities, from the area-minimising case to the work of Schoen-Simon and finally that of Wickramasekera (who proved the most general dimension bound for the singular set known in the hypersurface case), whilst detailing the various problems of which little is still known. Thu 26 Nov 2020 14:00-16:00 Zoom meeting Application of the Kummer construction on complete 4-dimensional Hyperkaehler manifolds Speaker: Andries Salm (UCL) Abstract: In 1997 Sen conjectured a method to construct new hyperKaehler manifolds by glueing Multi-Taub-NUT to Atiyah-Hitchin manifolds. Several authors (Foscolo 2016, Hein et al. 2018, Schroers and Singer 2020) proved this conjecture and made several new families of these metrics. In this talk we explain their method, revisit the geometric building blocks needed for this construction and study the analysis needed to make the glueing happen. Thu 3 Dec 2020 14:00-16:00 Zoom meeting Area-minimising cones Speaker: Konstantinos Leskas (UCL) Abstract: It is known that area-minimisers are smooth, except on a set of Hausdorff dimension at most n-7. This result is optimal, since Simons' cone is an area-minimiser. In this talk, we prove this result using the method of sub-calibrations, developed by De Philippis and Paolini. This simplifies the original proof of Bombieri, De Giorgi and Giusti significantly. Thu 10 Dec 2020 14:00-16:00 Zoom meeting Phase transition functionals: existence of minimal surfaces and Gamma-convergence Speaker: Davide Parise (Cambridge) Abstract: We overview the recently developed level set approach to produce minimal submanifolds, and weak generalizations thereof. Special emphasis is given to the Gamma-convergence side of the above story. The talk is divided in two parts. The former, on the Allen-Cahn functional, develops the codimension one theory, while the latter focuses on the Ginzburg-Landau functional and the higher codimension case. Time permitting we mention recent works on the Yang-Mills-Higgs functional of Pigati and Stern that circumvent one of the major drawbacks of the Ginzburg-Landau theory, i.e. integrality of the limiting varifold.

### Term 3 2019 - 2020

New gauge theories.

Compactness for generalized Seiberg--Witten equations in dimension 3, ends of moduli spaces of Higgs bundles, Kapustin--Witten equations.

 Wed 13 May 2020 15:00-17:00Zoom meeting Compactness for generalized Seiberg--Witten equations in 3D, I   Speaker: Lorenzo Foscolo (UCL) Abstract: Last term we discussed Uhlenbeck's compactness results from the 1980's for connections with bounded curvature in L^p and compact structure group. In particular, Uhlenbeck's theorems yield strong compactness results for Yang--Mills connections in (subcritical) dimension 2 and 3. In these two lectures we will discuss new compactness results in dimension 3 for flat connections with structure group the complex Lie group PSL(2,C). These results, due to Taubes, have been extended to other gauge theoretic equations on 3-manifolds such as the Seiberg--Witten equations with multiple spinors. Our main reference is a recent paper by Walpuski--Zhang, which provides a general compactness result applying uniformly to a large class of generalised Seiberg--Witten equations in dimension 3. In this first lecture I will mostly focus on introducing the necessary background and on stating the main results. Wed 20 May 2020 15:00-17:00Zoom meeting Compactness for generalized Seiberg--Witten equations in 3D, II   Speaker: Lorenzo Foscolo (UCL) Abstract: In this second lecture I will discuss some of the analytic ingredients of the proof, in particular the crucial role of a version of Almgren's frequency function suitably adapted to the gauge theoretic context. Wed 27 May 2020 15:00-17:00Zoom meeting Ends of moduli spaces of Higgs bundles, I Speaker: Michael Singer (UCL) Abstract: In these talks, I'll give an account of the moduli space M of Higgs bundles/Hitchin's equations on a fixed bundle over a compact Riemann surface C. M is known to be smooth (in many cases), non-compact, and to have a natural hyperKaehler metric. I want to focus on the ends' of M, or at least the most friendly end, in particular describing the asymptotic behaviour of the metric. Wed 3 June 2020 (note different time) 13:30-15:30Zoom meeting Ends of moduli spaces of Higgs bundles, II Speaker: Michael Singer (UCL) Abstract: This talk will focus on some of the analytic issues for families of solutions of SU(2) Hitchin's equations (A_t, t Phi), for t going to infinity. It is assumed that (A_1,Phi) is a solution and the problem is to understand the behaviour of solutions in regions near and away from the zeros of det Phi. Wed 10 June 2020 (note different time) 15:30-17:30Zoom meeting The Kapustin--Witten equations and the Nahm pole boundary condition, I Speaker: Marco Usula (ULB) Abstract: In these three talks, we will discuss the Kapustin--Witten equations, mainly following the papers 1 and 2 by Rafe Mazzeo and Edward Witten. The Kapustin--Witten are first order gauge-theoretic equations on a bundle over a 4-manifold. We will study solutions in the interior of a 4-manifold with boundary, developing a "simple pole" near the boundary, and possibly an additional singularity along a knot in the boundary. It has been conjectured by Witten that an appropriate counting of such solutions should correspond to certain topological invariants of the knot. In this first talk, we will introduce the KW equations and the "Nahm pole boundary condition" when there are no knots on the boundary, and we will sketch a proof of the Fredholm theorem for the linearized gauge-fixed KW equations.   Before the talk we had a session on how to fight racism in academia, with discussion and sharing of resources. Wed 17 June 2020 (note different time) 15:30-17:30Zoom meeting The Kapustin--Witten equations and the Nahm pole boundary condition, II Speaker: Joel Fine (ULB) Abstract: I will discuss uniqueness of solutions to the Kapustin-Witten equation, following sections 2 and 3 of Mazzeo and Witten?s first paper. In the first part of the talk I will explain why on a closed manifold (with no boundary) the only solutions come from flat complex connections. In the second part of the talk I will explain why the only solution on the half-space which is asymptotic to the Nahm pole is the Nahm pole solution itself. Finally, I will show that the linearised KW equations at the Nahm pole solution are invertible. This is the key step to showing that the equations on a general four-manifold with boundary are Fredholm. Wed 24 June 2020 (note different time) 16:00-18:00 Zoom meeting The Kapustin--Witten equations and the Nahm pole boundary condition, III: the nightmare continues Speaker: Rafe Mazzeo (Stanford) Abstract: This talk will focus on the modification of the KW equations on manifolds with boundary when the boundary contains a knot or a link and the Nahm pole condition is modified one which detects the knot. Unlike the simpler Nahm pole condition in the absence of a knot, this problem exhibits a higher level of degeneracy, and its linearization is a depth two iterated edge problem''. The main types of results one can prove, e.g. polyhomogeneity of solutions, Fredholmness of the linearization, computation of the index, etc., are all direct analogues of results for the knot-free case, the computations and proofs are considerably different. I will focus on the main structural differences, including the computation of indicial roots, an introduction to the iterated edge calculus in this setting and the new techniques needed to prove regularity.

### Term 2 2019-2020

Uhlenbeck's Compactness and Removable Singularity Theorems for Yang-Mills.

In 1982 K. Uhlenbeck published two breakthrough papers, proving a weak compactness theorem for connections with L^p bounds on the curvature, a strong compactness theorem for Yang-Mills connections and a removable singularity theorem for 4D Yang-Mills connections. The original ideas developed in these papers are central to many problems in geometric analysis (as testified by the award of the 2019 Abel Prize).

 Wed 22 Jan 2020 13:00-15:00UCL Maths Department Room 707 Gauge Theory and Introduction to Uhlenbeck Compactness  Speaker: Daniel Platt (Imperial) Abstract: "Uhlenbeck Compactness", "Uhlenbeck Gauge", and "Removable Singularities of Yang-Mills Connections" are terms from the realm of gauge theory. I will explain the definitions of principal bundles, connections, gauge group, curvature, Yang-Mills energy and give some examples. Using this language, I will explain the three terms from the beginning of the abstract and will state the results that we will prove throughout this term. Wed 29 Jan 2020 13:00-15:00UCL Maths Department Room 707 Uhlenbeck Compactness Theorem I: Construction of the relative Coulomb gauge  Speaker: Fabian Lehmann (UCL) Abstract: This week we are going to discuss the existence of the relative Coulomb gauge. This gives a slice for the action of the gauge group on the space of connections and will be used later on in the proof of the Uhlenbeck compactness theorem. The gauge transformation and the preferred connection in the orbit of the gauge group will be found by a Newton iteration. It is important to keep track of various constants arising in the corresponding estimates to prevent circular reasoning. Wed 5 Feb 2020 13:00-15:00UCL Maths Department Room 707 Uhlenbeck Compactness Theorem II: Regularity of Yang Mills Equation in the relative Coulomb gauge Speaker: Konstantinos Leskas (UCL) Abstract: In this talk we are going to speak about regularity of weak Yang - Mills connections. We are going to start from where we left last time, making the extra assumption on our connection that it satisfies the relative Coulomb gauge condition. Under this assumption we are going to prove full regularity for the weak Yang-Mills connections . The interesting part will be on how to deal with the non-linearity of the problem. Wed 12 Feb 202013:00-15:00UCL Maths Department Room 707 Uhlenbeck Compactness Theorem III: Weak Compactness, Uhlenbeck gauge Speaker: Ektor Papoulias (Oxford) Abstract: Having studied the regularity of weak Yang-Mills connections we have half of the ingredients needed for strong compactness in the supercritical case (p>n/2). The next task is to obtain the weak compactness theorem of Uhlenbeck for connections with uniform L^p-bounds on curvature (Uhlenbeck, CMP, 1982). The proof relies on the construction of a suitable local slice of the gauge action near connections with small energy over a contractible neighbourhood of a point: the so called Uhlenbeck gauge. We will introduce and motivate this notion in the context of weak compactness and prove its existence when p>n/2. Our intermediate results will shed some light on the critical case p=n/2. Wed 26 Feb 2020 13:00-15:00UCL Maths Department Room 707 Uhlenbeck Compactness Theorem IV: Patching and Strong Uhlenbeck Compactness Speaker: Florian Litzinger (QMUL) Abstract: This week we shall conclude our study of Uhlenbeck's compactness theorems. In particular, we will complete the proof of the weak compactness theorem by patching together local Uhlenbeck gauges to obtain a global gauge transformation. Finally, we will demonstrate the strong compactness theorem for weak Yang-Mills connections with a uniform L^p-bound on the curvature. Wed 4 Mar 2020 13:00-15:00UCL Maths Department Room 707 Uhlenbeck's Removable Singularity Theorem for 4D Yang-Mills I: Canonical choice of gauge Speaker: Shengwen Wang (QMUL) Abstract: We will present the section constructing Coulomb gauge (in which d*A=0) when the sup norm of curvature F(A) of connection A is small in Uhlenbeck's removable singularity paper. One first construct a gauge in which the norm of F controls the sup norm of A, then one use implicit function theorem near flat connections (those with F(A)=0) to obtain such Coulomb gauge. The construction is done on spheres, disks and annuli regions. Wed 11 Mar 2020 13:00-15:00UCL Maths Department Room 707 Uhlenbeck's Removable Singularity Theorem for 4D Yang-Mills II: A priori estimates Speaker: Gianmichele Di Matteo (QMUL)   Abstract: In this talk, we will develop the a-priori estimates needed to remove the singularities. The argument is purely analytical, it uses standard elliptic regularity methods under a smallness assumption, necessary to be able to neglect the non-linear character of the equation. Wed 25 Mar 2020 13:00-15:00Microsoft Teams meeting (remote) Uhlenbeck's Removable Singularity Theorem for 4D Yang-Mills III: Conclusive argument Speaker: Benjamin Aslan (UCL) Abstract: In this talk, we will have pleasure to finish the proof of Uhlenbeck's removable singularities result. This is done by improving the given curvature bound to an L-infinity bound. To this end, we construct a broken Coloumb gauge on the unit ball which is divided into annuli centered around the origin. We then use curvature estimates on each of the annuli obtained in previous talks.

### Term 1 2019-2020

Recent classifications of ancient solutions in mean curvature flow and Ricci flow.

In recent years there has been fundamental progress in classifying ancient solutions in geometric flows such as mean curvature flow and Ricci flow, with important first applications to the structure of singularity formation. The techinques to achieve these results can be understood as an extension of asymptotic symmetries and first order behaviour to the entire solution and thus allowing for its classification. We thus expect that these ideas should be also of interest to a broader audience of people working in Differential Geometry/Geometric Analysis.

Thu 3 Oct 2019
13:00-15:00
UCL Maths Department Room 707
Recent classifications of ancient solutions in mean curvature flow and Ricci flow: Introduction and overview
Speaker: Felix Schulze (UCL)
Abstract: In recent years there has been fundamental progress in classifying ancient solutions in geometric flows such as mean curvature flow and Ricci flow, with important first applications to the structure of singularity formation. The techinques to achieve these results can be understood as an extension of asymptotic symmetries and first order behaviour to the entire solution and thus allowing for its classification. We will give an introduction and overview over the results covered this term, together with an outline of the general strategy.
Thu 17 Oct 2019
13:00-15:00
UCL Maths Department Room 707

Unique asymptotics of Ancient Solutions of the Mean Curvature Flow
Speaker: Thomas K�rber (Freiburg/UCL)
Abstract: In this talk, we present results obtained by Angenent, Daskalopolous and Sesum in 2014 regarding the classification of ancient, compact, non-collapsed, convex solutions of the mean curvature flow. If such solutions arise as symmetric surfaces of revolution, the corresponding profile function enjoys unique asymptotics as time approaches minus infinity. In order to prove these asymptotics, we study the PDE satisfied by the profile function on different scales and determine its long time behavior by comparing it with suitable barriers. These barriers are self-shrinking surfaces with boundaries and we will also discuss their construction.

See the paper Unique asymptotics of ancient convex mean curvature flow solutions', by S. Angenent, P. Daskalopoulos and N. Sesum, https://arxiv.org/abs/1503.01178

Thu 24 Oct 2019
13:00-15:00
UCL Maths Department Room 707

Uniqueness of convex ancient solutions to mean curvature flow
Speaker: Ben Lambert (Oxford/UCL)
Abstract: We will look at Brendle and Choi's beautiful proof of the uniqueness of convex, non-compact, ancient solutions to mean curvature flow in R^3. The key difficulty here is proving rotational symmetry, and we will focus on the main steps required to obtain this result. This includes the Neck Improvement Theorem, and an extension of many of the estimates from last week's talk to the non-rotationally symmetric case.

See the paper Uniqueness of convex ancient solutions to mean curvature flow', by S. Brendle and K. Choi, https://arxiv.org/abs/1711.00823

Thu 31 Oct 2019
13:00-15:00
UCL Maths Department Room 707

Uniqueness of two-convex ancient ovals in mean curvature flow
Speaker: Mario Schulz (QMUL)
Abstract: Angenent, Daskalopoulos and Sesum proved that closed, two-convex and noncollapsed ancient mean curvature flows are unique up to ambient isometries, parabolic scaling and translations in time. A first step towards uniqueness of solutions is to establish their rotational symmetry with similar arguments as in the noncompact case. In this presentation we take symmetry for granted and focus on the structure of the proof that two rotationally symmetric solutions coincide after suitable translations and scaling. We will state the key estimates and study how they are combined to prove uniqueness.

See the paper Uniqueness of two-convex closed ancient solutions to the mean curvature flow', by S. Angenent, P. Daskalopoulos ans N. Sesum, https://arxiv.org/abs/1804.07230

Thu 7 Nov 2019
13:00-15:00
UCL Maths Department Room 707

Ancient solutions to the Ricci flow in dimension 3, following Simon Brendle, part I
Speaker: Ovidu Munteanu (University of Connecticut)
Abstract: I will go over a recent result of S. Brendle, that classifies three dimensional noncompact ancient kappa-solutions. The classification hinges on proving that a three dimensional noncompact type II ancient kappa solution must be the Bryant soliton. This is achieved in two steps, first proving the result in the rotationally symmetric case, and then proving the rotational symmetry of such ancient solutions.

The classification assuming rotational symmetry is done by first understanding the asymptotic behavior of solutions, using barrier functions and applying the spectral decomposition of a certain linear operator. This will be the goal of the first talk.
The rotational symmetry of ancient kappa-solutions will be obtained in the second talk. First, by analyzing the Lichnerowicz equation on the cylinder, it will be proved that neck-like regions become more symmetric under the flow. By making use of this Neck Improvement theorem, it will also follow that large enough caps that are close to the Bryant soliton are also more symmetric under the evolution.

See the paper Ancient solutions to the Ricci flow in dimension 3', by S. Brendle, https://arxiv.org/abs/1811.02559

Thu 14 Nov 2019
13:00-15:00
UCL Maths Department Room 707

Ancient solutions to the Ricci flow in dimension 3, following Simon Brendle, part II
Speaker: Ovidu Munteanu (University of Connecticut)
Abstract: I will go over a recent result of S. Brendle, that classifies three dimensional noncompact ancient kappa-solutions. The classification hinges on proving that a three dimensional noncompact type II ancient kappa solution must be the Bryant soliton. This is achieved in two steps, first proving the result in the rotationally symmetric case, and then proving the rotational symmetry of such ancient solutions.

The classification assuming rotational symmetry is done by first understanding the asymptotic behavior of solutions, using barrier functions and applying the spectral decomposition of a certain linear operator. This will be the goal of the first talk.
The rotational symmetry of ancient kappa-solutions will be obtained in the second talk. First, by analyzing the Lichnerowicz equation on the cylinder, it will be proved that neck-like regions become more symmetric under the flow. By making use of this Neck Improvement theorem, it will also follow that large enough caps that are close to the Bryant soliton are also more symmetric under the evolution.

See the paper Ancient solutions to the Ricci flow in dimension 3', by S. Brendle, https://arxiv.org/abs/1811.02559

Thu 21 Nov 2019
13:00-15:00
UCL Maths Department Room 707

Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow
Speaker: Francesco Di Giovanni (UCL)
Abstract: In this talk we present a recent result by Angenent, Daskalopoulos, and Sesum on closed reflection invariant k-solutions to the 3D Ricci flow. If such solutions are not a family of contracting spheres, then they satisfy unique asymptotics. In the first part we will briefly review which (closed) k-solutions may appear as singularity models for the Ricci flow. We will then focus on describing the asymptotic shrinker of any k-solution satisfying the conditions above. The central part of the talk is dedicated to deriving the asymptotics in the Type-I region. We will then mainly skip the intermediate region (where one can essentially rely on Brendle's recent classification of 3D k-solutions) to dedicate time to the more interesting tip region.

See the paper Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow', by S. Angenent, P. Daskalopoulos and N. Sesum, https://arxiv.org/abs/1906.11967

Thu 28 Nov 2019
13:00-15:00
UCL Maths Department Room 707

Uniqueness of compact kappa-solutions to Ricci flow in 3D
Speaker: Andrew McLeod (UCL)
Abstract: Last week we saw that compact kappa-solutions to Ricci flow in 3D on S^3, whose asymptotic soliton is not a sphere, satisfy unique asymptotics backwards in time. We will now extend this to prove that, up to translations in time and parabolic rescalings, such solutions are unique. A particular consequence is that any such solution must coincide with a translated and parabolically rescaled copy of Perelman's solution. Obtaining the uniqueness relies upon estimates in suitable norms for the cylindrical and tip regions separately. Each norm must reflect the correct corresponding geometry of the region, whilst simultaneously 'agreeing' (in some sense) on the transitional overlap between the cylindrical and tip regions.

See the paper Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci Flow', by P. Daskalopoulos and N. Sesum, https://arxiv.org/abs/1907.01928

Thu 5 Dec 2019
13:00-15:00
UCL Maths Department Room 707

Embedded shrinkers of genus zero and wiggly planes
Speaker: Fritz Hiesmayr (UCL) and Konstantinos Leskas (UCL)
Abstract: We will talk about a short paper of Brendle, where he classifies self-similar shrinking solutions of MCF under topological assumptions. In the first theorem, presented by Kostas, we will show that embedded shrinkers of genus zero are automatically round spheres. In the second, presented by Fritz, we give a similar classification when the any two loops contained in the shrinker have vanishing intersection number mod two. This second result implies for example Ilmanen's Wiggly Plane Conjecture. Brendle proves both theorems using similar methods, inspired by Ros's work on minimal surfaces in S^3.

See the paper Embedded self-similar shrinkers of genus 0', by S. Brendle, https://arxiv.org/abs/1411.4640

Thu 12 Dec 2019
13:00-15:00
UCL Maths Department Room 707

A topological property of asymptotically conical self-shrinkers of small entropy
Speaker: Shengwen Wang (QMUL)
Abstract: We will present in this talk a result of Bernstein-Wang on topological property of asymptotic conical self shrinkers in R^{n+1}. When restricting to dimension n=2 and combining with Brendle?s classification of genus zero shrinking surfaces presented last time, one obtains a complete classification of low entropy (less than or equal to that of the round cylinder S^1\times R) shrinkers in R^3: it is either a flat plane, a shrinking round sphere 2S^2 or a shrinker round cylinder \sqrt{2}S^1\times R up to rotations. In the first half I will go over the backgrounds and strategy of proof, in the second half I will go over selected proofs of the propositions used.

See the paper A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy', by J. Bernstein and L. Wang, https://arxiv.org/abs/1504.01996

Wed 15 Jan 2020
13:00-15:00
UCL Gordon Sq 16-18 Room 101

Ancient low entropy flows, mean convex neighborhoods, and uniqueness
Speaker: Felix Schulze (UCL)
Abstract: In this talk we will discuss the recent proof of the positive mean curavture neighborhood conjecture of Choi-Haslhofer-Hershkovits for the mean curvature flow of embedded surfaces in R^3. The proof hinges on the complete classification of ancient solutions with low entropy. We will discuss the strategy and give an outline of the proof, but will also focus on the further implications of the result.

See the paper `Ancient low entropy flows, mean convex neighborhoods, and uniqueness', by K. Choi, R. Haslhofer and O. Hershkovits, https://arxiv.org/abs/1810.08467

### Term 3 2018-2019

The Seiberg Witten equations.

The Seiberg-Witten equations are a system of non-linear PDEs defined on any Riemannian 4-manifold. They have the remarkable property that the space of solutions (modulo gauge) is compact. When this space is zero dimensional, the number of solutions is independent of the Riemannian metric, giving an invariant of the underlying smooth 4-manifold. This invariant is intimately connected to the geometry and differential topology of themanifold. For example they enable one to detect different smooth structures on the same topological 4-manifold, the non-existence of a symplectic structure, the uniqueness of the symplectic structure on CP^2... The aim of the reading seminar is to give a "bird's-eye view" of the Seiberg-Witten equations and their applications, unfortunately only giving sketches of some of the more technical aspects of the subject (notably Taubes' work on the Seiberg-Witten invariants of symplectic 4-manifolds). Finally, time permitting, we will look briefly at variations of the Seiberg-Witten equations and their applications in G_2 geometry.

 Wed 24 April 2019 13:15-14:45UCL Christopher Ingold Building XLG1 Chemistry LT The Seiberg-Witten invariant of a compact 4-manifold Speaker: Joel Fine (ULB) Abstract: I will outline the definition of the Seiberg-Witten invariant of a compact 4-manifold. I will try and explain how the invariant fits into a framework of "Fredholm differential invariants," due to Donaldson. The specific case of the Seiberg-Witten equations is in some sense the most straightforward example, because compactness of the space of solutions is automatic, but to even state the equations requires quite a lot of geometric background. I will give a rapid overview of this (Spin-c structures and Dirac operators) before describing the Fredholm theory and compactness result. Finally, time permitting, I will explain why the number of solutions to the Seiberg-Witten equations is an invariant of the underlying smooth 4-manifold. Wed 1 May 2019 13:15-14:45UCL Drayton House B03 Ricardo LT Compactness, Seiberg-Witten invariants of K�hler 4-folds and the relation between Seiberg-Witten and Gromov-Witten invariants Speaker: Daniel Platt (UCL/LSGNT) Abstract: In the talk I will explain why the moduli space of solutions to the SW-equations is compact. Following this I will explain the canonical Spin^c structure and Dirac operator on K�hler 4-folds, and use this to compute their SW-invariants. The talk will end with a nod to the relation between SW-invariants and Gromov-Witten invariants discovered by Taubes. Wed 15 May 2019 13:30-14:45UCL Drayton House B03 Ricardo LT Defining the Seiberg-Witten invariant and applications to differential topology of 4-manifolds Speaker: Joel Fine (ULB) Abstract: I will complete the steps in the definition of the Seiberg?Witten invariant for compact 4-manifolds. I will then explain applications to 4-manifold topology, such as the existence of homeomorphic but not diffeomorphic 4-manifolds, as well as Donaldson?s diagonalisability theorem. Wed 22 May 201913:15-14:45UCL Drayton House B03 Ricardo LT Seiberg-Witten invariants of symplectic four-manifolds: Taubes' theorems Speaker: Ralph Klaasse (ULB) Abstract: In this talk we will take a look at the behavior of the Seiberg-Witten invariants on symplectic four-manifolds. We will show that the invariant of the canonical Spin^c structure has modulus 1, and sketch how this lead Taubes to be able to prove an equivalence between the Seiberg-Witten and (modified) Gromov-Witten invariants. In particular, we will discuss how from an appropriate sequence of Seiberg-Witten solutions, one can extract a pseudoholomorphic curve. Wed 29 May 2019 13:15-14:45UCL Cruciform Building B404 - LT2 Wall crossing for Seiberg-Witten invariants. And maybe if there's time, the uniqueness of the symplectic structure on CP^2 Speaker: Joel Fine (ULB) Abstract: When b_+>1, the Seiberg-Witten invariant does not depend on the choice of perturbation used in the equations. When b_+=1 however the space of perturbations is split by a codimension 1 wall. I will explain that when the perturbation crosses this wall the corresponding invariant jumps by 1. If there's time (which, frankly, given my ability to judge these things, is unlikely) I will explain a beautiful application of SW theory to symplectic topology: CP^2 admits a unique symplectic structure up to diffeomorphisms and overall scale. (This is a combination of arguments due to Gromov and Taubes.)

### Term 2 2018-2019

Recent progress in the theory of Poincar�-Einstein 4-manifolds.

A Poincar�-Einstein metric on a compact manifold with boundary determines a conformal structure on the boundary, called its ?conformal infinity?. (The paradigm is hyperbolic space itself, whose conformal infinity is the standard conformal structure on the sphere.) A long-standing goal in the field, first proposed by Michael Anderson, is to develop a degree theory for the map which sends a Poincar�-Einstein metric on a given 4-manifold to its conformal infinity. If one could define the degree of this map and prove it were non-zero it would follow that ALL conformal structures on the boundary could be filled by Poincar�-Einstein metric on the interior! The crux is to show the map is proper. There has been exciting recent progress on this problem, in recent papers of Chang-Ge and Chang-Ge-Qing, which prove compactness results for a sequence of Poincar�-Einstein metrics on a 4-manifold, assuming that their conformal infinities converge, together with some other supplementary conditions. After some talks setting the scene, we hope to go over the details of the following two papers: https://arxiv.org/abs/1809.05593https://arxiv.org/abs/1811.02112.

 Wed 16 Jan 2019 13:15-14:45UCL Maths Department Room 707 An introduction to the Dirichlet problem for Poincar�-Einstein 4-manifolds, Anderson?s programme and the papers of Chang-Ge and Chang-Ge-Qing Speaker: Joel Fine (ULB) Abstract: I will begin by introducing Poincar�-Einstein metrics: these are asymptotically hyperbolic metrics on the interior of a compact manifold with boundary. A Poincar�-Einstein metric determines a conformal structure on the boundary called its ?conformal infinity", the prototype being hyperbolic space itself with conformal infinity the standard structure on the sphere. Given a choice of conformal structure on the boundary, the Dirichlet problem asks for a Poincar�-Einstein metric on the interior with this structure as its conformal infinity. It is a deep problem. From the analytic point of view, one must solve a non-linear geometrically elliptic PDE whose symbol degenerates at infinity. I will describe an approach to this problem in 4-dimensions using Fredholm degree theory, which was suggested by Anderson. Whether this can be made to work is still an open question, but there has been recent exciting progress in the papers of Cheng-Ge and Cheng-Ge-Qing. I will state their results and try to explain the intuition behind their somewhat complicated hypotheses. Wed 23 Jan 2019 13:15-14:45UCL Maths Department Room 707 Linear analysis on asymptotically hyperbolic manifolds and the theorem of Graham-Lee Speaker: Marco Usula (ULB) Abstract: The theorem of Graham and Lee states that, for every small perturbation of the conformal infinity of the hyperbolic metric, there is a Poincare-Einstein metric on the ball whose conformal infinity is the perturbed conformal class. The proof of the theorem relies on Fredholm properties of the linearized Einstein equation modulo gauge. In this seminar, we will briefly introduce Mazzeo's elliptic theory of 0-differential operators, and we will use it to prove the theorem of Graham and Lee. Wed 30 Jan2019 13:30-14:45UCL Maths Department Room 706 Q-curvature, conformal powers of the Laplacian and the Fefferman-Graham compactification Speaker: Joel Fine (ULB) Abstract: Let g be a Poincar�-Einstein metric. Given a metric h representing the conformal infinity of g, Fefferman-Graham introduced a canonical choice of compactification x^2g, restricting to h on the boundary. This compactification has the special property that it has vanishing Q-curvature. These compcatifications are essential in the papers of Chang-Ge and Chang-Ge-Qing. I will firstly explain one way to define Q curvature together with the closely related sequence of differential operators which in conformal geometry play the role of powers of the Laplacian. I will then prove existence and uniqueness of the Fefferman-Graham compactification and the vanishing of its Q-curvature. This all follows the paper ?Q-curvature and Poincar� metrics? by Fefferman and Graham, available on the arxiv: https://arxiv.org/abs/math/0110271 Wed 6 Feb 201913:15-14:45UCL Maths Department Room 707 A first step in the proof of Chang-Ge's result: Blow-up analysis near the boundary Speaker: Bruno Premoselli (ULB) Abstract: In this talk we start to describe the proof of Change-Ge's compactness result for conformally compactified Poincar�-Einstein metrics. Assuming technical results that will be addressed in later talks (among which epsilon-regularity) we will briefly describe the strategy of proof and study in detail the first step of the blow-up analysis, namely when blow-up occurs near the boundary. Wed 13 Feb 2019 13:05-14:35UCL Maths Department Room 706 Interior Blow-up Analysis and completing the proof of Chang-Ge's Compactness Theorem Speaker: Andrew McLeod (UCL) Abstract: Following on from last week's talk, we rule our curvature blow up in the interior (I.e. far away from the boundary). It is within this step that the imposed topological restrictions on our sequence are vital. Once armed with uniform (in i) curvature bounds we may then complete the proof of the main compactness result, Theorem 1.1 in Chang-Ge. Wed 20 Feb 2019 13:15-14:45UCL Maths Department Room 706 Epsilon-regularity I Speaker: Daniel Platt (UCL) Abstract: To extract a converging subsequence of metrics we need high regularity of the metrics in the interior. epsilon-regularity can boost the a priori C^1-regularity to a certain version of C^{k+1}-regularity. It will be instructive to see how the assumptions from the main theorem (e.g. boundedness of Yamabe constant) and properties of the Fefferman-Graham compactification are used very explicitly in the proof of epsilon-regularity. This is the first out of two talks presenting epsilon-regularity. Wed 27 Feb 2019 13:15-14:45UCL Maths Department Room 706 Epsilon-regularity II Speaker: Konstantinos Leskas (UCL) Abstract: In this talk we shall see how to iterate Theorem 3.1 to obtain the L^\infty bounds on the derivatives of the curvature tensor up to order k-2. Then we finish the proof of Theorem 3.1, trying to clarify most of its steps. Wed 6 Mar 2019 13:15-14:45UCL Maths Department Room 707 Some useful Identities on the Boundary Speaker: Benjamin Aslan (UCL) Abstract: In this talk, several tensor identities which have been used in the talks on Epsilon Regularity will be proven. For example, we will see how the S-tensor transforms under conformal changes (Lemma 2.3) of the metric and find expressions for the k-th covariant derivative of the Weyltensor, Schoutentensor and the Scalar Curvature (Lemma 2.8 and 2.9). Wed 13 Mar 2019 13:15-14:45UCL Maths Department Room 706 Injectivity radius bounds via blow-up Speaker: Reto Buzano (QMUL) Abstract: In this talk, we show that for the Fefferman-Graham compactification of a conformally compact Einstein 4-manifold with positive Yamabe constant at conformal infinity, a bound on the intrinsic injectivity radius on the boundary implies a bound on the intrinsic injectivity radius in the interior. The proof relies on a blow-up argument and a Theorem of Li-Qing-Shi. Wed 20 Mar 2019 13:15-14:45UCL Maths Department Room 706 Sobolev inequalities and the compactness of Fefferman-Graham compactifications Speaker: Felix Schulze (UCL) Abstract: We show how the necessary Sobolev inequalities are obtained which replace the assumption of a positive lower bound on the first and second Yamabe constants in the proof in the paper of Chang-Ge of the compactness of Fefferman-Graham compactifications. We further demonstrate how these Sobolev inequalities can be used to obtain the compactness under suitable assumtions on the boundary Yamabe metrics, the nonconentration of the S-tensor, and the vanishing of the second homology. Wed 27 Mar 2019 13:15-14:45UCL Maths Department Room 706 Global uniqueness of the Graham-Lee metrics on the 4-ball Speaker: Jason D. Lotay (Oxford) Abstract: Recall that for every conformal class c on the 3-sphere,sufficiently near the conformal class of the standard round metric, atheorem of Graham-Lee gives a Poincare-Einstein metric g on the 4-ballwhich has c as its conformal infinity.  Moreover, g is locally uniquemodulo gauge.  It has been a long-standing open question whether g isglobally unique.  We will describe the proof of this uniqueness theorem,due to Chang-Ge-Qing, which is the key application thus far of theircompactness theory.

### Term 1 2018-2019

 Mon 1 October 2018 14:00-16:00Foster Court 233 A beginner?s guide to the Nash-Moser implicit function theorem  Speaker: Michael Singer (UCL) Abstract: The Nash-Moser theorem has a fearsome reputation in some quarters, but the basic ideas are quite simple.  In this talk I shall discuss a simple version of the theorem as presented in the short paper of X. Saint-Raymond, A simple Nash-Moser implicit function theorem, L?enseignement Mathematique, (2) 35 (1989), as well as some motivating examples. Mon 8 October 2018 14:00-16:00Foster Court 114 Min-Max and Geometry  Speaker: Huy Nguyen (QMUL) Abstract: In this talk, I will give a broad and not-that-technical overview of recent results in min-max and its applications to geometry. In particular, I will focus on the Allen-Cahn and Ginzburg-Landau equations and their relationship to minimal surfaces. Mon 15 October 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 Existence of critical points of Ginzburg?Landau functionals Speaker: Florian Litzinger (QMUL) Abstract: Critical points of Ginzburg?Landau functionals can be found by means of the min-max procedure. First we prove a fairly general version of the Saddle Point Theorem for C^1-functionals satisfying the Palais?Smale condition. Following Section 2 of D. Stern's paper, we then apply the construction to the family of Ginzburg?Landau functionals on a compact Riemannian manifold, establishing the existence of non-trivial critical points in the Sobolev space H^1. Mon 22 October 201814:00-16:00Birckbeck Gordon Sq (43) G01 Lower bounds on the energies Speaker: Gianmichele di Matteo (QMUL) Abstract: In this talk, we will prove the lower bounds on the energies of the minimax critical points constructed in the previous section, needed to ensure the nontriviality of the energy concentration set. See also section 3 in the paper by D. Stern. Mon 29 Ocotober 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 ?-ellipticity for the Ginzburg-Landau equation Speaker: Huy Nguyen (QMUL) Abstract: In this talk, we will consider the proof of a key estimate for the Ginzburg-Landau min-max procedure, the ?-ellipticity. This estimate was used to prove the upper bounds for the min-max values for the Ginzburg-Landau equation. They key ideas are the monotonicity formula for penalised harmonic maps and a Hodge-de Rham decomposition of the phase function. Mon 12 November 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 Upper bounds for the energies Speaker: Francesco di Giovanni (UCL) Abstract: In this talk we present Stern's construction of upper bounds for the min-max constants of the G-L Energy functional with logarithmic growth in epsilon. Along with the previously discussed existence of analogous lower bounds, that allows to complete the proof of Theorem 1.1 in Stern's paper. Mon 19 November 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 Varifolds and Rectifiability I Speaker: Costante Bellettini (UCL) Abstract: Integral varifolds are ?singular submanifolds?. The necessity for extending the class of submanifolds beyond smooth ones is evident whenever a limiting process is involved. The drawback of extending the class is that the the set where smoothness fails could be huge. We will address the notions of rectifiability, of first and second variation of their area, and of generalized mean curvature.Under suitably mild assumptions on the generalized mean curvature, Allard?s theorem provides the foundational steps in the regularity theory (giving a bit of regularity) and in the compactness theory. The latter requires the introduction of a more general notion of varifolds, which is also useful for other purposes (for example it naturally appears in the paper we are reading) and for which one can still talk of "area" and "mean curvature" in a suitable sense. Thu 29 November 2018 14:00-16:00Cruciform Building B404 - LT2 Varifolds and Rectifiability II Speaker: Fritz Hiesmayr (Cambridge) Abstract: This is the second talk of our introduction into the basics of varifolds and rectifiability. After briefly recalling the notions from the previous week (including general varifolds) we will state Allard's regularity and compactness theorems (without proof). We then move on to the monotonicity formula for stationary varifolds (and time permitting give a proof) and define the notion of density. We end the talk with a statement of Preiss' theorem, which gives conditions under which a Radon measure is rectifiable. Mon 3 December 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 Rectifiability of the Ginzburg Landau limit Speaker: Ben Lambert (UCL) Abstract: We will see that the Ginzburg Landau solutions constructed earlier in the reading group are "generalised varifolds" (to be defined). We aim to prove that the limit of these generalised varifolds is a rectifiable varifold. The key step in this is to prove suitable density estimates, which we will prove via a series of reductions. Mon 10 December 2018 14:00-16:00Birckbeck Gordon Sq (43) G01 Rectifiability of the Ginzburg Landau limit - part II Speaker: Felix Schulze (UCL) Abstract: I will complete the proof of the rectifiability of the limiting varifold arising form the Ginzburg-Landau approximation.

### Term 3 2017-2018

 Wed 25 April 2018 13:00-14:45UCL Room D103 An introduction to pseudoholomorphic curves Speaker: Kim Moore (UCL) Abstract: We will describe the basic theory of pseudoholomorphic curves that will be needed for future seminars, including the linearisation of the non-linear Cauchy-Riemann operator and positivity of intersections. Wed 2 May 2018 13:00-14:45UCL Room D103 The moduli space of pseudoholomorphic curves Speaker: Ben Lambert (UCL) Abstract: We will describe the moduli space theory of pseudoholomorphic curves as needed for future seminars, particularly focusing on the issue of transversality. Wed 9 May 2018 13:00-14:45UCL Room D103 Estimates for pseudoholomorphic curves Speaker: Thomas Koerber Abstract: We will describe the estimates for pseudoholomorphic disks and cylinders, and the removable singularities theorem for pseudoholomorphic maps, as required in applications in symplectic topology. Wed 16 May 2018 13:00-14:45UCL Room D103 Proof of Gromov's compactness theorem for pseudoholomorphic curves Speaker: Chris Evans (UCL) Abstract: We will describe the proof of the compactness theorem for pseudoholomorphic curves, building on the theory in previous seminars, which will include the bubbling analysis. Wed 23 May 2018 13:00-14:45UCL Room D103 Gromov's nonsqueezing theorem Speaker: Albert Wood (LSGNT/UCL) Abstract: We will show how the pseudoholomorphic curve theory developed so far in previous seminars allows us to prove the nonsqueezing theorem.

### Term 2 2017-2018

 Wed 17 January 2018 13:00-14:45UCL Maths Department Room 707 An introduction to harmonic maps Speaker: Costante Bellettini (UCL) Abstract: A harmonic map is, roughly speaking, a critical point for the Dirichlet energy. Depending on the precise requirement, we will distinguish the notions of minimizing harmonic/stationary harmonic/weakly harmonic maps. The weaker notion (the third one) leads to a second order non-linear elliptic PDE (in weak form). The second notion provides, additionally, a monotonicity formula for the Dirichlet energy. We will overview regularity aspects for these classes of maps and compactness issues, in particular the bubbling phenomenon that occurs when working with weak W1,2 convergence (the natural notion, due to the relevant energy). We will overview other related aspects: the criticality, for the exponent p=2, of the PDE from the point of view of elliptic bootstrapping; some special results valid in the case of 2-dimensional domains (mostly postponed to the next talk); the lack of a Palais-Smale property; the subtleties associated to the possible lack of approximability of W1,2 maps by means of smooth maps and the relation with the topology of the target. Wed 24 January 2018 13:00-14:45UCL Maths Department Room 707 An overview of the Sacks-Uhlenbeck construction and further developments Speaker: Huy Nguyen (QMUL) Abstract: In this seminar, we will discuss aspects of the two dimensional harmonic maps problem. We will discuss some of the regularity theory (Helein's theorem for weakly harmonic maps), the existence theory (Sacks-Uhlenbeck) and bubble tree convergence (Parker, Ding-Tian) . Wed 31 January 2018 13:00-14:45UCL Maths Department Room 707 Sacks-Uhlenbeck Chapter 1 Speaker: Udhav Fowdar (LSGNT/UCL) Abstract: I will start by the basic definitions and relate the critical points of the energy and volume functional. Then I will define branched minimal immersions and sketch the proofs of the main theorems of the first section. Wed 7 February 2018 13:00-14:45UCL Maths Department Room 707 Sacks-Uhlenbeck Chapter 2 Speaker: Fabian Lehmann (LSGNT/UCL) Abstract: Morse theory relates the topology of a compact finite dimensional manifold to the existence of critical points of a Morse function. I will explain that in a similar way a non-trivial topology of the target manifold N gives the existence of non-trivial critical maps for the perturbed energy functional for maps from S2 to N. For this it is crucial that the functional satisfies the Palais-Smale condition. Wed 21 February 2018 13:00-14:45UCL Maths Department Room 707 Estimates and Extensions for the Perturbed Problem (Sacks-Uhlenbeck Chapter 3) Speaker: Albert Wood (LSGNT/UCL) Abstract: In the last talk we looked at properties of the perturbed energy functional E?. In this talk we will cover section 3 of the paper, in which we derive important estimates for critical points of this functional, as well as an improved regularity result. Fri 2 March 2018 13:00-14:30UCL Maths Department Room 707 Convergence properties of critical maps of the perturbed problem (Sacks-Uhlenbeck Chapter 4) Speaker: Gianmichele di Matteo (QMUL) Abstract: We will show that the only obstacle to a regular (C1) convergence of the critical maps constructed in the previous chapters, is the presence of at least one minimal sphere near the point where the convergence fails. Fri 9 March 2018 13:00-15:00UCL Maths Department Room 707 Harmonic maps and topology (Sacks-Uhlenbeck Chapter 5) Speaker: Jason Lotay (UCL) Abstract: I will describe some applications of the analytic theory of harmonic maps developed by Sacks-Uhlenbeck. First, I will describe some results providing the existence of energy-minimizing harmonic maps representing homotopy classes of maps, when topological obstructions vanish. Second, I will describe how Morse theory can be used in the presence of non-trivial topology to produce harmonic spheres (and thus conformal branched minimal spheres) which may be higher index critical points for the energy (rather than minimizers). Wed 21 March 2018 13:00-15:00UCL Maths Department Room 707 An introduction to Gromov's compactness theorem for pseudoholomorphic curves Speaker: Jason Lotay (UCL) Abstract: I will aim to provide the background to and statement of Gromov's compactness theorem for pseudoholomorphic curves. I will also describe some important results in symplectic topology which are applications of this result.

### Term 1 2017-2018

 Wed 4 October 2017 13:00-14:45KCL Strand Building Room S4.23 Original proof of positive mass theorem up to dimension 7 Speaker: Mattia Miglioranza (UCL) Abstract: We give an overview of the strategy of the original proof of the positive mass theorem by Schoen and Yau up to dimension 7. Wed 11 October 2017 13:00-14:45KCL Strand Building Room S4.23 Introduction and motivation (Schoen notes Chapter 1) Speaker: Lothar Schiemanowski (Kiel/QMUL) Abstract: In this talk we will introduce three motivations to study scalar curvature and various phenomena unique to it. Wed 18 October 2017 12:00-13:45UCL Taviton 14 Room 129 Positive scalar curvature and the positive mass theorem (Schoen notes Chapter 2 Sections 1-2) Speaker: Udhav Fowdar (UCL) Abstract: In this talk we shall see how the nonexistence of positive scalar curvature on certain closed manifolds implies the (classical) positive mass theorem. Wed 25 October 2017 13:00-14:45KCL Strand Building Room S4.23 An introduction to minimal slicings (Schoen notes Chapter 2 Section 3 Part 1) Speaker: Albert Wood (UCL) Abstract: Last week, we saw that the Positive Mass Theorem is implied by the non-existence of a metric on Mn#Tn with positive scalar curvature. Our ultimate aim, then, is to prove this. The bulk of the remaining work is taken up with the theory of existence and regularity of minimal slicings, which are nested families of submanifolds minimal with respect to a certain weighted volume functional. In this talk, I will explain why we would like to study these objects, and I will prove a key theorem on the path to the PMT: 'Positive scalar curvature implies that an appropriately chosen minimal slicing is Yamabe positive'. Wed 1 November 2017 13:00-14:45KCL Strand Building Room S4.23 Regularity of minimal k-slicings: L^2 non-concentration of the first eigenfunction (Schoen notes Chapter 2 Section 3 Part 2) Speaker: Chris Evans (UCL) Abstract: Last week, we saw how minimal k-slicing could be used to prove the positive mass theorem. We now begin the work on regularity by discussing results that guarantee the first eigenfunction of the quadratic form Qj doesn't concentrate around singularities under certain partial regularity assumptions. Wed 15 November 2017 13:00-14:45KCL Strand Building Room S4.23 Homogeneous minimal slicings (Schoen notes Chapter 2 Section 4) Speaker: Ben Lambert and Kim Moore (UCL) Abstract: We will discuss the results of Chapter 4 in the notes and give an overview of how the rest of the proof will follow. Wed 22 November 2017 13:00-14:45KCL Strand Building Room S4.23 Top dimensional singularities (Schoen notes Chapter 2 Section 5) Speaker: Kim Moore (UCL) Abstract: We will discuss the material in Section 5 of the notes, in which it is proved that a positive minimiser for the quadratic form on a volume minimising (nonplanar) cone is homogeneous of strictly negative degree. Wed 29 November 2017 13:00-15:00KCL Strand Building Room S4.23 Cones and compactness of minimal slicings (Schoen notes Chapter 2 Sections 4-6) Speaker: Ben Lambert (UCL) Abstract: I will make some comments about how we obtain cones in section 4 of the notes before proving convergence of the quadratic forms using a capacity argument. Depending on time and interest, we can also talk about convergence in weighted measure. Wed 6 December 2017 13:00-14:45KCL Strand Building Room S4.23 Completing the compactness theorem (Schoen notes Chapter 2 Sections 6-7) Speaker: Ben Lambert (UCL) Abstract: We will finish going through the "convergence of u in H2" (that is, we will finish section 6 in the notes). If there is time, we will start proving partial regularity using Federer's dimension reduction technique (section 7 in the notes). Wed 13 December 2017 13:00-14:45KCL Strand Building Room S4.23 Partial regularity and existence of minimal slicings (Schoen notes Chapter 2 Sections 7-8) Speaker: Fritz Hiesmayr (Cambridge) Abstract: The talk will be in two parts: in the first part we prove that minimal slicings are partially regular. The dimension bound on the singular set of the bottom leaf of the slicing follows inductively via a dimension reduction argument, using the non-existence of homogeneous 2-slicings with an isolated singularity at the origin that was established last week. In the second part we are going to discuss the existence of minimal slicings in manifolds satisfying a topological restriction (this applies in particular for connected sums with a torus as in the compactification theorem). The proof is via a minimisation argument for the weighted volume, carried out in the class of rectifiable currents with integer multiplicity. This corresponds to Sections 7 and 8 in Chao Li's notes, and Section 4 of the Schoen-Yau paper.

### Term 3 2016-2017

 Thur 27 April 2017 13:00-15:00UCL Maths Department Room 500 Desingularizing Einstein metrics Speaker: Michael Singer (UCL) Abstract: The goal will be to present Biquard's paper on desingularizing 4-dimensional Einstein orbifolds via gluing, including discussions of the obstructions. Thurs 11 May 2017 15:30-17:30UCL Maths Department Room 505 An overview of Yang-Mills flow Speaker: Casey Kelleher (UC Irvine) Abstract: The aim is to provide a discussion of the basics of Yang-Mills 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 13:00-15:00UCL 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 Yang-Mills connection, given by work of Uhlenbeck. Wed 24 May 2017 13:00-15:00UCL Roberts Building Room 508 Yang-Mills flow in dimension 4: part 1 Speaker: Huy Nguyen (QMUL) Abstract: In this first of three talks discussing Waldron's preprint on Yang-Mills flow in dimension 4, the goal will be to discuss the optimal decay estimates in the elliptic case as background and an outline of part of the energy estimates. Thurs 1 June 2017 13:00-15:00UCL Maths Department Room 500 Yang-Mills flow in dimension 4: part 2 Speaker: Huy Nguyen (QMUL) Abstract: I will continue discussing Waldron's preprint on Yang-Mills flow in dimension 4. I'll talk about the optimal decay estimates, an overview of the paper, the monotonicity formula for Yang--Mills flow and epsilon-regularity. Tues 13 June 2017 13:00-15:00UCL Maths Department Room 500 Yang-Mills flow in dimension 4: part 3 Speaker: Kim Moore (Cambridge) Abstract: In this final talk discussing Waldron's preprint on Yang-Mills flow in dimension 4, the key steps from the previous talks will be brought together to summarize the proof.

### Term 2 2016-2017

 Wed 1 February 2017 13:00-15:00UCL Maths Department Room 707 Convergence of the Allen-Cahn 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 Allen-Cahn equation to Brakke�s motion by mean curvature. Wed 8 February 2017 13:00-15:00UCL Maths Department Room 707 The min-max construction of Allen--Cahn critical points Speaker: Fritz Hiesmayr (Cambridge) Abstract: In my talk I will discuss the paper by Guaraco entitled "Min-max 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 min-max construction, and then the energy upper and lower bounds. Time permitting, I might make a short comparison between the Allen--Cahn construction and the earlier Almgren--Pitts theory. Wed 22 February 2017 13:00-15:00UCL Maths Department Room 707 Stability and absence of classical singularities for the minimal hypersurface obtained as limit of Allen-Cahn stable solutions. Speaker: Costante Bellettini (UCL) Abstract: Two weeks ago Fritz described how to construct an index one critical point for the ?-Allen-Cahn functional: a suitable subsequence as ??0 delivers a minimal hypersurface (integral varifold) in the limit. I will describe the contents of Tonegawa and Tonegawa-Wickramasekera, which show respectively how the stability of Allen-Cahn 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 (codimension-7 singular set). Wed 1 March 2017 13:00-15:00UCL Maths Department Room 707 Calabi-Yau metrics on Kummer surfaces Speaker: Eleonora di Nezza (Imperial) Abstract: After Yau proved the Calabi conjecture, showing the existence of K�hler metrics with Ricci curvature identically zero on compact K�hler manifolds with vanishing first Chern class, there has been a lot of use of "gluing constructions" in order to give an almost-explicit 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 Calabi-Yau metrics on K3 surfaces. Wed 8 March 2017 13:00-15:00UCL Maths Department Room 707 Collapsing Calabi-Yau metrics on K3 surfaces via gluing Speaker: Fabian Lehmann (LSGNT) Abstract: After last week's gluing construction of a Calabi-Yau 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 T3/Z2. The geometry around points where the curvature blows up is modelled on rescaled ALF gravitational instantons. Wed 15 March 2017 13:00-15:00UCL Maths Department Room 707 Minimal surfaces in Poincar�-Einstein manifolds Speaker: Joel Fine (ULB) Abstract: The talk will be loosely based on the article "Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds" by Alexakis and Mazzeo. A Poincar�-Einstein 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 Poincar�-Einstein 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 Poincar�-Einstein 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 Poincar�-Einstein metric with given conformal structure at infinity. Wed 22 March 2017 13:00-15:00UCL Maths Department Room 707 Gluing Eguchi-Hanson 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 Ricci-flat metrics on a 4-manifold (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 Ricci-flat metric with full holonomy, which in particular would be the first compact non-Kaehler Ricci-flat 4-manifold. However, the end result is not a Ricci-flat metric, but instead an ancient solution to Ricci flow. I will explain the apparent obstructions to the Ricci-flat gluing problem and the modification required to obtain the solution to Ricci flow.

### Term 1 2015-2016

 Wed 9 December 2015 14:00-16:00UCL Maths Department Room 707 H�lder continuity of tangent cones of limit spaces Speaker: Reto Buzano (M�ller) (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 H�lder continuously along geodesics. This will follow from the H�lder 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 14:00-16:00UCL 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 14:00-16:00UCL Maths Department Room 707 Almost rigidity: volume convergence Speaker: Yong Wei (UCL) Abstract: This talk will focus on the volume convergence part of Cheeger-Colding theory. Wed 28 October 2015 14:00-16:00UCL 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 14:00-16:00UCL Maths Department Room 500 Introduction to Cheeger-Colding theory Speaker: Panagiotis Gianniotis (UCL) and Jason Lotay (UCL) Abstract: In this talk we will first discuss the Cheng-Yau gradient estimate, before starting on Cheeger-Colding theory, including quantitative maximum principles, rigidity and almost rigidity, and the structure of limit spaces. Wed 14 October 2015 14:00-16:00UCL 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.

### Term 3 2014-2015

 Wed 3 June 2015 14:00-16:00UCL Medical Sciences G46 HO Schild Pharmacology Lecture Theatre Monopole moduli spaces and metrics (Part 4) - The Atiyah-Hitchin 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 14:00-16:00UCL Medical Sciences G46 HO Schild Pharmacology Lecture Theatre The Kaehler-Ricci 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 Kaehler-Ricci 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 14:00-16:00UCL Medical Sciences G46 HO Schild Pharmacology Lecture Theatre Monopole moduli spaces and metrics (Part 3) Speaker: Michael Singer Wed 13 May 2015 14:00-16:00UCL Medical Sciences G46 HO Schild Pharmacology Lecture Theatre Monopole moduli spaces and metrics (Part 2) Speaker: Michael Singer Wed 6 May 2015 14:00-16:00UCL 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 non-compact 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.

### Term 2 2014-2015

 Wed 10 March 2015 11:00-1:00UCL Taviton (16) Room 431 An Introduction to the Kaehler--Ricci flow Speaker: Eleonora di Nezza (Imperial) Abstract: I will give an exposition of a number of well-known 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 Kaehler-Ricci flow. Finally, if time permits, I will show that the Kaehler-Ricci 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 11:00-1:00UCL Taviton (16) Room 431 Tangent cones to two-dimensional area-minimizing 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 two-dimensional area-minimizing currents, the inequality is proved by constructing explicit comparison surfaces using multiple-valued 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 11:00-1:00UCL 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 6-7 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 \epsilon-regularity theorem for Ricci flow. As an application, they derived a uniform L^2-bound for the Riemannian curvature in 4-dimensional Ricci flow with uniformly bounded scalar curvature and show that such flow converges to an orbifold at a singularity. Wed 4 February 2015 11:00-1:00UCL 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 3-5, in which they prove a distance distortion estimate (answering a question of Hamilton), construct good cut-off 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 1:00-3:00UCL 25 Gordon Street Room 707 Asymptotic Rigidity of Self-shrinkers in Mean Curvature Flow Speaker: Lu Wang (Imperial) Abstract: In this talk, we discuss uniqueness of self-shrinkers with prescribed asymptotic behavior at infinity. The main tool is the Carleman type estimates.

### Term 1 2014-2015

 Wed 3 December 2014 1:30-3:00UCL 25 Gordon Street Room 707 Hyperbolic Alexandrov-Fenchel inequality Speaker: Yong Wei Abstract: I will present one work in my Phd thesis. For any 2-convex and star-shaped hypersurface in hyperbolic space, we prove a sharp Alexandrov-Fenchel 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 Alexandrov-Fenchel 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 2-convex case, the proof relies on an application of Gerhardt's convergence result of inverse mean curvature flow for strictly mean-convex hypersurfaces in hyperbolic space, and sharp Sobolev inequalities on sphere; in the general 2-convex case, the proof involves an approximation argument. Wed 26 November 2014 1:00-3:00UCL 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 real-valued generalized inverse using different behavior of spherical eigenvalues. We prove that all the solutions near the Freund--Rubin solution are prescribed by three pairs of data on the boundary 6-sphere. Wed 19 November 2014 1:00-3:00UCL 25 Gordon Street Room 707 Uniqueness of Lawlor necks Speaker: Jason Lotay Abstract: I will discuss the paper by Imagi--Joyce--Oliveira dos Santos on uniqueness of special Lagrangians and Lagrangian self-expanders 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 1:00--3:00UCL 25 Gordon Street Room 707 Short-time existence of Lagrangian Mean Curvature Flow Speakers: Tom Begley and Kim Moore Abstract: We will present recent work on the short-time existence of Lagrangian mean curvature flow with non-smooth 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 self-expanders, a monotonicity formula for the self-expander equation and the local regularity theorem of Brian White. Wed 29 October 2014 12:30--2:00UCL 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 12:00--2:00UCL 25 Gordon Street Room 706 (12-1) Room 707 (1-2) Layer Potential Operators for Two Touching Domains in Rn 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 half-space in Rn 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 Rn. Using the Push-Forward 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 Push-Forward 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 Push-Forward Theorem, and the b- and ?-calculi 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 11:00--1:00UCL 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 Thomas-Yau 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.

### Term 3 2013-2014

 Wed 11 June 2014 11:00--1:00UCL 25 Gordon Street Room 505 Uniqueness of Lagrangian self-expanders (part 2) Speaker: Jason Lotay Abstract: I will continue to talk about my joint paper with A. Neves on the uniqueness of Lagrangian self-expanders with two planar ends. Wed 4 June 2014 1:00--2:30UCL Drayton House Room B.16 Uniqueness of Lagrangian self-expanders (part 1) Speaker: Jason Lotay Abstract: I will talk about my joint paper with A. Neves on the uniqueness of Lagrangian self-expanders with two planar ends and some related results on Lagrangian self-expanders, including results by Neves and Tian, Joyce--Lee--Tsui and Imagi--Joyce--Oliveira dos Santos. Wed 21 May 2014 1:00--2:30UCL Drayton House Room B.04 Finite-time singularities of Lagrangian mean curvature flow (part 4) Speaker: Felix Schulze Abstract: I will continue to speak on the paper of A. Neves: Finite-time singularities of Lagrangian mean curvature flow. Wed 14 May 2014 1:00--2:30 UCL 25 Gordon Street Room 500 Finite-time singularities of Lagrangian mean curvature flow (part 3) Speaker: Felix Schulze Abstract: I will continue to speak on the paper of A. Neves: Finite-time singularities of Lagrangian mean curvature flow. Wed 30 Apr 20141:00--2:30 UCL 25 Gordon Street Room 500 Finite-time singularities of Lagrangian mean curvature flow (part 2) Speaker: Felix Schulze Abstract: I will speak on the paper of A. Neves: Finite-time singularities of Lagrangian mean curvature flow.

### Term 2 2013-2014

 Tues 1 Apr 2014 11:00--1:00 UCL 25 Gordon Street Room 706 Finite-time singularities of Lagrangian mean curvature flow (part 1) Speaker: Felix Schulze Abstract: I will discuss some further details from the paper on the zero-Maslov class case for singularities in Lagrangian MCF by A. Neves. Wed 12 Mar 201411:30--1:30 KCL Strand Building Room S4.29 Singularities of Lagrangian mean curvature flow: zero-Maslov class case Speaker: Kim Moore Abstract: I will talk about the paper of the same name by A. Neves. Wed 5 Mar 2014 12:00--14:00 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 12:00--14:00 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 12:00--14:00 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 12:00--13:30 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 13:00--15:00 Birkbeck Torrington Square Room 254 Introduction to mean curvature flow (part 2) Speaker: Felix Schulze Wed 15 Jan 2014 13:00--15:00 Birkbeck Torrington Square Room 254 Introduction to mean curvature flow (part 1) Speaker: Felix Schulze

### Term 1 2013-2014

 Wed 11 Dec 2013 14:30--16:00 KCL Strand Building Room S4.36 Colding--Minicozzi's curvature estimate (part 2) Speaker: Giuseppe Tinaglia Abstract: I will be finishing the proof of the Colding--Minicozzi curvature estimate. We are going to assume the Choi--Schoen curvature estimate from last time and go from there. Wed 27 Nov 2013 14:30--16:00 UCL Drayton House Room B.06 Colding--Minicozzi's curvature estimate (part 1) Speaker: Giuseppe Tinaglia Abstract: Colding--Minicozzi's curvature estimate says that for an embedded minimal disk, the L2 norm of the second fundamental form bounds its L? norm. In this part, I will prove a curvature estimate of Choi--Schoen. Wed 20 Nov 2013 14:30--16:00 KCL Strand Building Room S4.36 Minimal surfaces and the Bernstein theorem Speaker: Francesca Tripaldi Abstract: I will discuss the basic theory of minimal surfaces and the proof of Bernstein's theorem.