Marlon Gomes

Research description

I work on a branch of Differential Geometry called Twistor Theory, which lies at the crossroads of Conformal Geometry (where angles are defined, but lenthgs are not), and Complex Geometry.

These two theories are intimately related in real dimension 2, in dimension 4, the relation is not so simple. At any point on a manifold, there is a plethora of linear complex structures compatible with a given conformal structure. On the other hand, globally defined complex structures are scarce (and may even be obstructed by topology), whereas global conformal structures are abundant.

In twistor theory, we make use of the many pointwise complex structures in an unexpected way: instead of trying to find a "canonical" family of such structures, varying smoothly from point to point, and compatible with the conformal structure (as in dimension 2), we study them all at once!

The gadget we use to parametrize linear complex structures is called twistor space. It has a tautological almost-complex structure of its own, which is a faithful complex structure when a suitable curvature condition, called anti-self-duality, is satisfied by the underlying conformal manifold. Thus, by restricting our attention to a anti-self-dual conformal structures, we can employ techniques of complex geometry in problems related to Riemannian and conformal structures.

Twistor theory bears its fruits in the flexibility that complex analysis brings to the table. The twistor construction is nearly reversible: there are certain conditions on a complex manifold which ensure it is a Twistor Space, from which we can reverse-engineer anti-self-dual 4-manifolds.

The challege of twistor theory is that the Complex structures arising from the Twistor construction are rather mysterious. For instance, they are almost never Kähler (let alone projective) compact complex manifolds.

While we have many results asserting the existence of anti-self-dual structures, the ones we understand explicitly are few and far between. My work exploits known examples of Twistor Spaces, and the correspondence between their complex geometry and the conformal geometry of the underlying 4-manifolds to shed light on anti-self-dual structures by means of complex-analytic techniques.

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Title description, Sep 2, 2017

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