AI generated image titled "A simple drawing of a calibrated submanifold"

Calibrated Geometry Fun Day

11 August 2023, King's College London

Calibrated submanifolds are widely studied objects in differential geometry. They are interesting from different points of view. For example: they are special cases of minimal submanifolds. In general, constructing minimal submanifolds is hard, because being minimal is a second order condition. Being calibrated is a first order condition and often an elliptic problem, so it can be easier to find these objects. Another application is to define a count of these submanifolds to define numerical invariants of manifolds, like Gromov-Witten invariants. Calibrated submanifolds also play an important role in mirror symmetry.

The three talks will cover connections between calibrated geometry and geometric flows, cohomogeneity one and two geometry, and with exceptional holonomy. The methods from these talks, i.e. parabolic, elliptic PDEs, and PDEs under symmetries are applicable in many other areas of differential geometry.

All talks take place in room K0.18 in the King's College London Strand building, Strand, London WC2R 2LS.

Schedule

9:30-10:30 - room K0.18 - Albert Wood (King's College London) - Special Lagrangians with symmetries
Abstract: In this talk, I will introduce the concept of calibrations and calibrated submanifolds, and focus on one of the more accessible and well-studied examples – special Lagrangians in Calabi-Yau manifolds. In particular, I will describe how one can construct symmetric examples in C^n using moment map techniques.

10:30-11:00 - Coffee break

11:00-12:00 - room K0.18 - Dominik Gutwein (Humboldt University Berlin) - Associative and coassociative submanifolds
Abstract: We discuss calibrated submanifolds in G_2-geometry which has been an active research area over the last 20 years. A G_2-manifold is a 7-dimensional Riemannian manifold whose holonomy is contained within the group G_2. We will not assume any prior knowledge about these objects but rather give a short introduction in the beginning of the talk. These manifolds contain two distinguished classes of (calibrated) submanifolds: associatives and coassociatives. We will focus on the moduli theory of these and the roles which they (are hoped to) play for their ambient manifold.

12:00-14:00 - Lunch at restaurant Wahaca Covent Garden (provided for registered participants)

14:00-15:00 - room K0.18 - Federico Trinca (University of Oxford) - Calibrated geometry, multi-moment maps and cohomogeneity-one
Abstract: A classical way to construct calibrated submanifolds is by supposing that they admit symmetries with generic orbits of codimension one. Such an assumption reduces the first-order linear PDE (the calibrated condition) into a system of ODEs, and it is called the cohomogeneity one method. Less classically, manifolds endowed with symmetries and a calibration come equipped with functions that generalize the moment maps in symplectic geometry. Moment maps have several interesting applications, even in calibrated geometry. For instance, they were used to construct Special Lagrangians in $C^n$ with symmetries. In this talk, we review the basic features of multi-moment maps, the cohomogeneity one method and how they are linked (mainly via examples).

Registration

If you want to join the free lunch, you must register by 28 July 2023. (Registration is now closed.) If you have no King's ID and want to attend any part of the event but do not want to attend the lunch, please email the organiser at daniel.platt.berlin@gmail.com.