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Markus Upmeier: Lectures

September 11, 2018
TITLE: Canonical Orientations for the Moduli Space of G2-instantons

ABSTRACT: The moduli space of anti-self-dual connections for 4-manifolds has been generalized by Donaldson-Segal to special, higher-dimensional geometries. I will discuss a technique for fixing canonical orientations on these moduli spaces in dimension 7 and for the gauge group SU(n). These orientations depend on the choice of a flag structure, an additional piece of data on the underlying 7-manifold introduced by Joyce. After discussing the reconstruction of an SU(n)-bundle from its ‘Poincaré dual’ submanifold, the definition of canonical orientations will be presented, based on the excision principle from index theory.

Yuuji Tanaka: Lectures

September 10, 2018
TITLE: On the Vafa-Witten theory on closed four-manifolds

ABSTRACT: Vafa and Witten introduced a set of gauge-theoretic equations on closed four-manifolds around 1994 in the study of S-duality conjecture in N=4 super Yang-Mills theory in four dimensions. They predicted from supersymmetric reasoning that the partition function of the invariants defined through the moduli spaces of solutions to these equations would have modular properties. But little progress has been made other than their original work using results by Goettsche, Nakajima and Yoshioka.

However, it now looks worth trying to figure out some of their foreknowledge with more advanced technologies in analysis and algebraic geometry fascinatingly developed in these two decades. This talk discusses issues to construct the invariants out of the moduli spaces, and presents possible ways to sort them out by analytic and algebro-geometric methods; the latter is joint work with Richard Thomas.

Hans-Joachim Hein: Lectures

September 10, 2018
TITLE: Higher-order estimates for collapsing Calabi-Yau metrics

ABSTRACT: Consider a compact Calabi-Yau manifold X with a holomorphic fibration F: X to B over some base B, together with a “collapsing” path of Kahler classes of the form [F*(omega_B)] + t * [omega_X] for t in (0,1]. Understanding the limiting behavior as t to 0 of the Ricci-flat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of Gross-Wilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricci-flat metrics is still not well-understood in general even away from the singular fibers of F. A key difficulty arises from the fact that Yau’s higher-order estimates for the complex Monge-Ampere equation depend on bounds on the curvature tensor of a suitable background metric that are not available in this collapsing situation. I will explain recent joint work with Valentino Tosatti where we manage to bypass Yau’s method in some cases, proving higher-order estimates even though the background curvature blows up.

Donaldson & Sun lectures at 2018 ICM

Regarding the 2018 ICM:

Simon Donaldson presented the opening plenary lecture –
Some recent developments in Kähler geometry and exceptional holonomy. Written version:

Song Sun gave a sectional lecture in Geometry –
Degenerations and moduli spaces in Kähler geometry. Written version: https://math.berkeley.edu/~sosun/ICM.pdf

Thomas Walpuski awarded 2018 Sloan Fellowship

Thomas Walpuski (Michigan State University) has been awarded a 2018 Alfred P. Sloan Research Fellowship in mathematics.

Walpuski will receive a two-year, $65,000 stipend to advance his work on gauge theory of G2-manifolds and the analysis of generalized Seiberg-Witten equations.

Read more: https://msutoday.msu.edu/news/2018/msu-mathematics-professor-awarded-prestigious-fellowship/

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.

The audio is missing at the beginning of the video below. It starts up at about the 2 minute mark.

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.