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Markus Upmeier: Lectures
September 11, 2018
TITLE: Canonical Orientations for the Moduli Space of G_{2}instantons
ABSTRACT: The moduli space of antiselfdual connections for 4manifolds has been generalized by DonaldsonSegal to special, higherdimensional 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 7manifold 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 VafaWitten theory on closed fourmanifolds
ABSTRACT: Vafa and Witten introduced a set of gaugetheoretic equations on closed fourmanifolds around 1994 in the study of Sduality conjecture in N=4 super YangMills 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 algebrogeometric methods; the latter is joint work with Richard Thomas.
HansJoachim Hein: Lectures
September 10, 2018
TITLE: Higherorder estimates for collapsing CalabiYau metrics
ABSTRACT: Consider a compact CalabiYau 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 Ricciflat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of GrossWilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricciflat metrics is still not wellunderstood in general even away from the singular fibers of F. A key difficulty arises from the fact that Yau’s higherorder estimates for the complex MongeAmpere 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 higherorder 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:
https://arxiv.org/abs/1808.03995
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 twoyear, $65,000 stipend to advance his work on gauge theory of G2manifolds and the analysis of generalized SeibergWitten equations.
Read more: https://msutoday.msu.edu/news/2018/msumathematicsprofessorawardedprestigiousfellowship/
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.
The audio is missing at the beginning of the video below. It starts up at about the 2 minute mark.
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.