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Author Archives: Victoria Hain
Toby Wiseman: Lectures
June 4, 2018
TITLE: Some applications of Ricci flow in physics
ABSTRACT: I will review two areas where Ricci flow makes contact with physics. Firstly I will review how Ricci flow arises from the renormalisation group equations of 2d `sigma models’ (I will try to explain what these words mean!). Secondly I will review a more recent link, where Ricci flow may be thought of as an algorithm to numerically find solutions to Einstein’s gravitational equations in exotic settings. In physics in both cases it is interesting to consider how black holes evolve under Ricci flow. Static black holes may be thought of as Riemannian geometries, while stationary black holes cannot, but still may be evolved using a ‘Lorentzian’ signature Ricci flow. I will also discuss the existence of Ricci solitons which are important to understand in the second context.
Lu Wang: Lectures
June 7, 2018
TITLE: Properties of selfsimilar solutions of mean curvature flow
ABSTRACT: I will survey some known results as well as some open problems about selfsimilar solutions of the mean curvature flow.
Felix Schulze: Lectures
June 4, 2018
TITLE: Singularity formation in Lagrangian mean curvature flow
ABSTRACT: We will survey results on singularity formation in mean curvature flow, both in codimension one and in higher codimension with a particular focus on Lagrangian mean curvature flow. We will also review different concepts of weak flows through singularities together with geometric applications.
Pranav Pandit: Lectures

 6/7/2018: Gradient flows, iterated logarithms, and semistability
 6/6/2018: Categorical Kähler Geometry
 6/7/2018: Gradient flows, iterated logarithms, and semistability
June 7, 2018
TITLE: Gradient flows, iterated logarithms, and semistability
ABSTRACT: The formalism of categorical Kähler geometry outlined in the previous lecture leads to the study of certain dynamical systems. A typical example is furnished by the YangMills flow on the space of hermitian metrics on a holomorphic bundle. It turns out that the asymptotic behaviour of these dynamical systems is governed by iterated logarithms. The talk will elaborate on this statement, and explain how it leads to the discovery of a canonical refinement of the HarderNarasimhan filtration in a variety of contexts. This is a report on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.
June 6, 2018
TITLE: Categorical Kähler Geometry
ABSTRACT: After introducing the paradigm of derived geometry, I will outline attempts to formalize and understand the mathematical structures underlying the physical notion of stability for Dbranes in string theory using the language of derived noncommutative geometry. These efforts build upon Bridgeland’s notion of stability conditions on triangulated categories, and are inspired by ideas from symplectic geometry, nonArchimedean geometry, dynamical systems, geometric invariant theory, and the DonaldsonUhlenbeckYau correspondence. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.
Rafe Mazzeo: Lectures
April 11, 2018
TITLE: Analysis of elliptic operators on complete spaces with asymptotically regular collapsing geometry
ABSTRACT: Elliptic theory for asymptotically cylindrical and asymptotically conical spaces is now classical and can be approached in many ways. When dealing with slightly more intricate geometries at infinity it is often helpful or even necessary to use more sophisticated tools. This talk will discuss a general and systematic theory which leads to sharp mapping results for ALF/ALG type metrics and prospects for a similar theory for singular fibrations over QAC spaces.
April 10, 2018
TITLE: The largescale structure of the Hitchin moduli space
ABSTRACT: The moduli space of solutions to the Hitchin equations on a Riemann surface carries a natural hyperKaehler metric, and questions and conjectures about its asymptotic structure have emerged out of the physics literature. There has been a lot of progress on this recently. I will discuss recent results, showing why this space might reasonably be called QALG.
Ruobing Zhang: Lectures
April 12, 2018
TITLE: Gravitational collapsing of K3 surfaces II
ABSTRACT: We will exhibit some new examples of collapsed hyperkähler metrics on a K3 surface. This is my recent joint work with HansJoachim Hein, Song Sun and Jeff Viaclovsky. We will construct a family of hyperkähler metrics on a K3 surface which are collapsing to a closed interval. Geometrically, each regular fiber is a Heisenberg manifold and each singular fiber is a singular circle fibration over a torus. In our example, each bubble limit is either the TaubNUT space or a complete hyperkähler space constructed by TianYau. The regularity estimates in this example in fact confirms a general picture given by the regularity theorem we present in Lecture 2. Technically, our examples are achieved by a new gluing construction. Continuing Jeff Viaclovsky’s introduction to the construction of this example, we will go through the details of the proof. We will also discuss some variations of the main gluing construction and some possible developments.
April 9, 2018
TITLE: Quantitative nilpotent structure and regularity theorems of collapsed Einstein manifolds
ABSTRACT: This talk is on the new developments of the structure theory for collapsed Einstein manifolds. We will start with some motivating examples of collapsed Ricciflat manifolds. Our main focus is the regularity and structure theorems for collapsed Einstein manifolds which is included in my joint work with Aaron Naber. First, in the context of manifolds with Ricci curvature uniformly bounded from below, we show that every point on such a manifold can be associated with a nilpotent rank which has a sharp upper bound. This follows from an effective version of the Generalized Margulis Lemma. The main part of the regularity theorem gives the following dichotomy: either the curvatures are uniformly bounded or the nilpotent rank drops.
April 9, 2018
TITLE: Introduction to Ricci curvature and the convergence theory
ABSTRACT: The first talk is an overview of the convergence and regularity theory of the manifolds with Ricci and sectional curvature bounds. Specifically, we will review some both classical and new structure theory such as the regularity theorems, the fibration theorems, and the structure of the limit spaces. The main part is to introduce the analytic tools in studying the noncollapsing manifolds and we will see why most tools legitimately fail in the collapsed context. Another emphasis is the development of the Generalized Margulis Lemma which gives the local collapsing geometry at the level of the fundamental group.
Jeff Viaclovsky: Lectures
April 11, 2018
TITLE: Gravitational collapsing of K3 surfaces I
ABSTRACT: I will discuss a construction of collapsing sequences of Ricciflat metrics on K3 surfaces with TianYau and TaubNUT metrics occurring as bubbles. This is joint work with HansJoachim Hein, Song Sun, and Ruobing Zhang. Lecture II will be given by Zhang.
Guofang Wei: Lectures
April 9, 2018
TITLE: Manifolds with integral curvature bounds
ABSTRACT: We begin with a review of early joint work with P. Petersen on the Laplacian and volume comparison for manifolds with only integral Ricci curvature bounds. We then present recent joint work with X. Dai and Z. Zhang producing a local Sobolev constant estimate for such manifolds without assuming a lower bound on volume. We close with applications of this theorem to produce a maximum principle, a gradient estimate, and to extend the Hessian estimate of CheegerColding and ColdingNaber to manifolds with only lower bounds on their integral Ricci curvature.
Xiaochun Rong: Lectures
April 11, 2018
TITLE: Collapsed manifolds with Ricci local bounded covering geometry
ABSTRACT: Collapsed manifolds with local bounded covering geometry (i.e., sectional curvature bounded in absolute value) has been wellstudied; the basic discovery by CheegerFukayaGromov is the existence of a compatible local nilpotent symmetry structures whose orbits point to all collapsed directions.
In this talk, we will report an ongoing work in generalizing the structural result to collapsed manifolds with (partially) local Ricci bounded covering geometry; which may contain a large class of collapsed CalabiYau manifolds and Ricci flat manifolds with special holonomy. Our construction of local nilpotent symmetry structures does not reply on the work of CheegerFukayaGromov; which gives alternative approach to the structural result.
Xuemiao Chen: Lectures
January 12, 2018
TITLE: Singularities of Hermitian Yang Mills connections and the HarderNarasimhanSeshadri filtration
ABSTRACT: I will talk about joint work with Song Sun on the tangent cones of Hermitian Yang Mills connections with point singularity.