Department of Mathematics

MIT

This series of lectures will aim to provide an overview of a relatively recent set of techniques which can be used to study the topology of symplectic manifolds (a class of smooth manifolds which naturally generalizes the better understood case of smooth projective varieties and plays a central role in mathematical physics). In particular, the main feature of this approach is that various classification questions can be reduced to algebraic considerations involving braid or mapping class groups.

Tuesday, November 1

4:00 - 5:00 PM Location TBA

Lecture 1.

In the first lecture, we will introduce symplectic manifolds, and show how they can be described by Lefschetz fibrations (i.e., fibrations over the 2-sphere with isolated nodal singular fibers), focusing particularly on the four-dimensional case. We will then discuss how the classification of Lefschetz fibrations reduces to that of "factorizations" in the mapping class group of an oriented surface, and give some partial classification results.

Wednesday, November 2

4:00 - 5:00 PM Location TBA

Lecture 2.

The second lecture will provide an alternative description of symplectic 4-manifolds, realizing them as branched covers of the complex projective plane. This establishes a bridge between the topology of symplectic manifolds and that of singular plane curves. We will introduce the "braid monodromy" approach to the study of plane curves, and give some examples.

Friday, November 4

4:00 - 5:00 PM Location TBA

Lecture 3.

The third lecture will continue in the same direction, discussing various results about the isotopy problem for symplectic plane curves (i.e. the question of determining whether a given curve is isotopic to a complex algebraic curve), outlining various types of behaviors (isotopy, stable isotopy, non-isotopy). We will also mention some results and conjectures about fundamental groups of complements of plane branch curves.

Monday, November 7

4:00 - 5:00 PM Location TBA

Lecture 4.

The fourth lecture will bring together the two strands of the discussion, showing how the monodromy data of a Lefschetz fibration can be determined explicitly (via the "lifting homomorphism") from that of a plane branch curve. In this context, we will formulate some questions and results about surgery operations on Lefschetz fibrations. We will also show how monodromy techniques can be used to study higher-dimensional symplectic manifolds, by considering singular fibrations over the complex projective plane. If time permits we will also briefly mention the case of "near- symplectic" structures on (almost arbitrary) smooth 4-manifolds and a natural way of generalizing Lefschetz fibrations to this setting.

Wednesday, November 9

4:00 - 5:00 PM Location TBA

Lecture 5.

The final lecture will discuss applications of Lefschetz fibrations to mirror symmetry, focusing particularly on Kontsevich's homological mirror symmetry conjecture for Fano varieties. We will first introduce and motivate the homological mirror symmetry conjecture in broad generality, and then focus specifically on the case of Fano surfaces, where it involves the category of Lagrangian vanishing cycles of a Lefschetz fibration. We will discuss in particular two concrete examples for which the conjecture can be checked explicitly: weighted projective planes, and Del Pezzo surfaces (blowups of the projective plane).

There will be a tea preceding each lecture in the Mathematics Commons Room, Herman Brown 438.

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