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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 self-similar solutions of mean curvature flow

ABSTRACT: I will survey some known results as well as some open problems about self-similar 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

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 Yang-Mills 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 Harder-Narasimhan filtration in a variety of contexts. This is a report on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Slides of lecture

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 D-branes 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, non-Archimedean geometry, dynamical systems, geometric invariant theory, and the Donaldson-Uhlenbeck-Yau correspondence. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Slides of lecture

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 large-scale 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 Hans-Joachim 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 Taub-NUT space or a complete hyperkähler space constructed by Tian-Yau. The regularity estimates in this example in fact confirms a general picture given by the \epsilon-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 Ricci-flat manifolds. Our main focus is the \epsilon-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 \epsilon-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 \epsilon-regularity theorems, the fibration theorems, and the structure of the limit spaces. The main part is to introduce the analytic tools in studying the non-collapsing 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 Ricci-flat metrics on K3 surfaces with Tian-Yau and Taub-NUT metrics occurring as bubbles. This is joint work with Hans-Joachim 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 L_2 Hessian estimate of Cheeger-Colding and Colding-Naber to manifolds with only lower bounds on their integral Ricci curvature.

Slides of lecture

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 well-studied; the basic discovery by Cheeger-Fukaya-Gromov is the existence of a compatible local nilpotent symmetry structures whose orbits point to all collapsed directions.

In this talk, we will report an on-going work in generalizing the structural result to collapsed manifolds with (partially) local Ricci bounded covering geometry; which may contain a large class of collapsed Calabi-Yau manifolds and Ricci flat manifolds with special holonomy. Our construction of local nilpotent symmetry structures does not reply on the work of Cheeger-Fukaya-Gromov; which gives alternative approach to the structural result.

Slides of lecture

Xuemiao Chen: Lectures

January 12, 2018
TITLE: Singularities of Hermitian Yang Mills connections and the Harder-Narasimhan-Seshadri filtration

ABSTRACT: I will talk about joint work with Song Sun on the tangent cones of Hermitian Yang Mills connections with point singularity.