## Luís Diogo

I am a symplectic geometer working at Uppsala University. I am interested in pseudoholomorphic curves, Lagrangian submanifolds, mirror symmetry and knot contact homology.

**Email:** luis.diogo@math.uu.se** **

## Research

**Monotone Lagrangians in cotangent bundles of spheres**(joint with Mohammed Abouzaid)

We use the wrapped Fukaya category to study monotone Lagrangians L in cotangent bundles of spheres. If L has non-trivial Floer homology, then it is non-displaceable from either the zero section or from one element of a 1-parameter family of monotone Lagrangians diffeomorphic to S^1 x S^{n-1}. This follows from the fact that the zero section and this family of Lagrangians split-generate the Fukaya category of compact monotone Lagrangians.

**Augmentations, annuli, and Alexander polynomials**

We prove a formula for the Alexander polynomial of a knot K in terms of the augmentation polynomial of K (which is defined via the knot contact homology of K). The proof involves studying pseudoholomorphic curves in the cotangent bundle of Euclidean 3-space, with boundary components mapping to the zero section and to a Lagrangian that is diffeomorphic to the knot complement.

A survey of the paper *Augmentations, annuli, and Alexander polynomials*** **(joint with Tobias Ekholm), starting with an introduction to the Alexander polynomial and to symplectic geometry. Based on a seminar given at the conference Matemáticos Portugueses pelo Mundo, in Porto, in 2019.

*Boletim da Sociedade Portuguesa de Matemática* 77 (2019) 51-68.

**Symplectic homology of complements of smooth divisors**(joint with Samuel Lisi)

We explain how to compute the symplectic homology of a complement of a smooth Donaldson-type divisor in a closed symplectic manifold, in terms of (absolute and relative) Gromov-Witten invariants of the manifold and the divisor.

*Journal of Topology* 12 (2019) 966-1029.

Published version arXiv version video

**Morse-Bott split symplectic homology**(joint with Samuel Lisi)

We prove transversality for the moduli spaces used to compute symplectic homology in the paper above. We also show how monotonicity hypotheses in the manifold and the divisor reduce the Morse-Bott cascades that need to be computed to a small list.

*Journal of Fixed Point Theory and Applications *(2019) 21:77 *.*

Published version arXiv version

**Lifting Lagrangians from Donaldson-type divisors**(joint work in progress with Dmitry Tonkonog, Renato Vianna and Weiwei Wu)

We prove a formula relating the superpotentials of monotone Lagrangians in Donalson-type divisors with the superpotentials of their monotone lifts. This formula includes contributions from relative Gromov-Witten invariants of the pair (symplectic manifold, divisor). Applications include a proof that there are infinitely many non-symplectomorphic monotone Lagrangian tori in complex projective spaces, quadrics and cubics of arbitrary dimension, and a new symplectic proof of a quantum Lefschetz hyperplane theorem.

video of talk by Renato Vianna

Ph.D. Thesis:

**Filtered Floer and symplectic homology via Gromov-Witten theory**

Stanford University