Page 354 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 354
GEOMETRIC ANALYSIS AND LOW-DIMENSIONAL TOPOLOGY (MS-59)
A Positive Mass Theorem for Fourth-Order Gravity
Paul Laurain, paul.laurain@imj-prg.fr
Université de Paris, France
Coauthors: Rodrigo Avalos, Jorge Lira
In classical Einstein gravity, the metric is a critical point of the Einstein-Hilbert functional,
i.e. g → M Rg dv. It has been proved that there exists a conserved quantity along space-
like hypersurface, called the ADM mass. In a celebrated work Schoen and Yau proved that if
the scalar curvature of the hypersurface is none-negative then the mass is none-negative, with
rigidity if the mass vanishes. After remembering some facts about this result, I will introduce
a new mass associated to a fourth order functional, namely g → M αRg2 + β|Ricg|2 dvg. Then
I will explain how we obtain an analogue of the positive mass theorem for this new mass by
replacing the none-negativity of the scalar curvature by the one of the Q-curvature.
Applications of virtual Morse–Bott theory to the moduli space of SO(3)
Monopoles
Thomas Leness, lenesst@fiu.edu
Florida International University, United States
Coauthor: Paul Feehan
We describe some recent joint work with P. Feehan on the Morse theory of a function on the
moduli space of SO(3) monopoles on a smooth four-manifold and sketch some applications to
the topology of smooth four-manifolds.
Recent progress in Lagrangian mean curvature flow of surfaces
Jason Lotay, lotay@maths.ox.ac.uk
University of Oxford, United Kingdom
Lagrangian mean curvature flow is potentially a powerful tool for tackling several important
open problems in symplectic topology. The first non-trivial and important case is the flow of
Lagrangian surfaces in 4-manifolds. I will describe some recent progress in understanding the
Lagrangian mean curvature flow of surfaces, including general results about ancient solutions
and the study of the Thomas-Yau conjecture in explicit settings.
Applications of Floer homology to clasp number vs genus
Tomasz Mrowka, mrowka@mit.edu
Massachusetts Institute of Technology, United States
A basic question in four dimensional topology is determining the smooth four-ball genus of a
knot in the the three sphere. A closely related question is determining the minimal number of
double points of a disk smoothly immersed bounding the knot. We’ll survey recent advances on
this problem using tools coming from gauge theory and related invariant. This circle of ideas is
related to interesting questions in algebraic geometry.
352
A Positive Mass Theorem for Fourth-Order Gravity
Paul Laurain, paul.laurain@imj-prg.fr
Université de Paris, France
Coauthors: Rodrigo Avalos, Jorge Lira
In classical Einstein gravity, the metric is a critical point of the Einstein-Hilbert functional,
i.e. g → M Rg dv. It has been proved that there exists a conserved quantity along space-
like hypersurface, called the ADM mass. In a celebrated work Schoen and Yau proved that if
the scalar curvature of the hypersurface is none-negative then the mass is none-negative, with
rigidity if the mass vanishes. After remembering some facts about this result, I will introduce
a new mass associated to a fourth order functional, namely g → M αRg2 + β|Ricg|2 dvg. Then
I will explain how we obtain an analogue of the positive mass theorem for this new mass by
replacing the none-negativity of the scalar curvature by the one of the Q-curvature.
Applications of virtual Morse–Bott theory to the moduli space of SO(3)
Monopoles
Thomas Leness, lenesst@fiu.edu
Florida International University, United States
Coauthor: Paul Feehan
We describe some recent joint work with P. Feehan on the Morse theory of a function on the
moduli space of SO(3) monopoles on a smooth four-manifold and sketch some applications to
the topology of smooth four-manifolds.
Recent progress in Lagrangian mean curvature flow of surfaces
Jason Lotay, lotay@maths.ox.ac.uk
University of Oxford, United Kingdom
Lagrangian mean curvature flow is potentially a powerful tool for tackling several important
open problems in symplectic topology. The first non-trivial and important case is the flow of
Lagrangian surfaces in 4-manifolds. I will describe some recent progress in understanding the
Lagrangian mean curvature flow of surfaces, including general results about ancient solutions
and the study of the Thomas-Yau conjecture in explicit settings.
Applications of Floer homology to clasp number vs genus
Tomasz Mrowka, mrowka@mit.edu
Massachusetts Institute of Technology, United States
A basic question in four dimensional topology is determining the smooth four-ball genus of a
knot in the the three sphere. A closely related question is determining the minimal number of
double points of a disk smoothly immersed bounding the knot. We’ll survey recent advances on
this problem using tools coming from gauge theory and related invariant. This circle of ideas is
related to interesting questions in algebraic geometry.
352
