# Geometria em Lisboa Seminar

## Past sessions

### On a special configuration of $12$ conics and a related $K3$ surface

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9{\mathbf A}_{2}$-configuration of $(-2)$-curves (ie smooth rational curves). Such a configuration plays the role of the $16$ disjoint $(-2)$-curves for the usual Kummer surfaces.

In this talk we will explain how construct $9$ other such $9{\mathbf A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve.

The new $9{\mathbf A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $12$ conics which are in special position with respect to the branch curve, plus some singular quartic curves. We will then explain how construct some automorphisms of the K3 surface sending one configuration to another.

(Joint work with David Kohel and Alessandra Sarti).

### A construction of $D_k$ asymptotically locally flat gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries

Complete hyperKaehler 4-manifolds with cubic volume growth (and suitable decay of the curvature), also known as ALF gravitational instantons, are known to come in two families, according to the fundamental group at infinity. This group must be a finite subgroup of $SU(2)$ and the only possibilities compatible with cubic volume growth are the cyclic groups ($A_k$) and binary dihedral groups ($D_k$).

This talk will be about the construction of $D_k$ ALF gravitational instantons by a gluing construction in which the ingredients are the moduli space of centred charge-2 monopoles ($D_0$) and a particularly symmetric, but singular, $A_k$ ALF gravitational instanton. This construction was suggested in a paper of Sen (1997). It is also closely related to a construction due to Foscolo, in which hyperKaehler metrics are constructed on the $K3$ manifold that are “nearly” collapsed to a 3-dimensional space.

This is joint work with Bernd Schroers.

### Non-commutative integrable systems and their singularities

The theory of singularities of non-commutative integrable systems (a.k.a isotropic fibrations), in contrast with the well-known theory for the commutative case (a.k.a. Lagrangian fibrations), is nonexistent. In this talk I will describe a few first steps toward such a theory.

Slides of the talk

### Invariance of symplectic cohomology under deformations

This is joint work with Gabriele Benedetti (Heidelberg).

I will describe how Floer cohomology changes as one deforms the symplectic form. I will then explain how these results are useful in applications in symplectic topology, e.g. finding generators for the Fukaya category of toric varieties or finding lower bounds on the number of magnetic geodesics.

Slides of the talk

### Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

(all joint work with Jason Lotay) A standing conjecture of Richard Thomas, motivated by mirror symmetry, gives a stability condition supposed to control the existence of a special Lagrangian submanifold in a given Hamiltonian isotopy class of Lagrangians. Later, Thomas and Yau conjectured a similar stability condition controls the long-time existence of the Lagrangian mean-curvature flow. In this talk I will explain how Jason Lotay and myself have recently proved versions of these conjectures on all circle symmetric hyperKahler 4-manifolds.

### Finite order period integrals in normal crossing K3 degenerations

I am presenting a new technique to compute period integrals of degenerating Calabi-Yau manifolds. In the K3 case, the formula can be used to study the Picard group of nearby fibres. In general, the technique leads to canonical parametrizations of degenerating families at the boundary of the moduli space. The result enters ongoing work by Gross-Hacking-Keel-Siebert on a modular compactification of the family of g-polarized K3 surfaces. This is joint with Bernd Siebert.

### Sharp inequalities relating volume and the min-max widths of Riemannian three-spheres.

Min-max theories for the area functional have undergone ground-break developments in recent years. One aspect of these theories is that they define many notions of "width" that can be understood as geometric invariants of compact Riemannian manifolds. As such, it is an interesting question to understand what sort of geometric information they encode. In this talk, we will focus on the Simon-Smith width of Riemannian three-dimensional spheres, discussing how large it can be among metrics normalised to have the same volume, and necessary conditions for maxima under further restrictions. This is a joint work with Rafael Montezuma (UM-Amherst).

### Lifting Lagrangians from symplectic divisors

We prove that there are infinitely many non-symplectomorphic monotone Lagrangian tori in complex projective spaces, quadrics and cubics of complex dimension at least $3$. This is a consequence of the following: if $Y$ is a codimension $2$ symplectic submanifold of a closed symplectic manifold $X$, then we can explicitly relate the superpotential of a monotone Lagrangian $L$ in $Y$ with the superpotential of a monotone Lagrangian lift of $L$ in $X$. This sometimes involves relative Gromov-Witten invariants of the pair $(X,Y)$. We will define the superpotential, which is a count of pseudoholomorphic disks with boundary on a Lagrangian, and which plays an important role in Floer theory and mirror symmetry. This is joint work with D. Tonkonog, R. Vianna and W. Wu.

### Construction of harmonic mappings

Geometric variational problems frequently lead to analytically extremely hard, non-linear partial differential equations, where the standard methods fail. Thus finding non-trivial solutions is challenging. The idea is to study solutions with a certain minimum level of symmetry (i.e. group actions with low cohomogeneity), and use the symmetry to reduce the original problem to systems of non-linear ordinary differential equations, typically with singular boundary values. In my talk I explain how to construct harmonic mappings between manifolds with a lot of symmetry (i.e. cohomogeneity one manifolds). If time permits, I will discuss applications of the developed methods.

### On the shape of a Riemannian manifold with large first nonzero eigenvalue for the Laplacian and the Dirichlet-to-Neumann operator

Classically, there is a strong relationship between the shape of a Riemannian manifold and the spectrum of its Laplace-operator, and, more generally, with the spectrum of Laplace type operators. In particular, for the spectrum of the Laplacian, the presence of a large first nonzero eigenvalue is related to some concentration phenomena. In the first part of the talk, I will recall these classical relations. In the second part of the talk, I will introduce the Dirichlet-to-Neumann operator on a manifold with boundary. I will survey the same types of questions in this context (existence of large first nonzero eigenvalue, concentration phenomena) without going into the technical details. This second part corresponds to work in progress with Alexandre Girouard.

### Projectively induced Ricci-flat Kaehler metrics

The aim of this talk is to discuss the problem of classifying Kaehler-Einstein manifolds which admit an isometric and holomorphic immersion into the complex projective space. We start giving an overview of the problem focusing in particular on the Ricci-flat case. Ricci-flat non-flat Kaehler manifolds are conjectured to be not projectively induced. Next, we give evidence to this conjecture for Calabi’s Ricci-flat metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds.

### Introduction to Mirror Symmetry on the Hitchin System

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

### On a conjecture about curve semistable Higgs bundles

We say that a Higgs bundle $E$ over a projective variety $X$ is curve semistable if for every morphism $f : C \to X$, where $C$ is a smooth irreducible projective curve, the pullback $f^\ast E$ is semistable. We study this class of Higgs bundles, reviewing the status of a conjecture about their Chern classes.

### Stability of symplectic leaves

In this talk I will give a gentle introduction to Poisson manifolds, which can be thought of as (singular) symplectic foliations. As an illustration of the kind of problems one deals in Poisson geometry, I will discuss and give some results on stability of symplectic leaves.

### Constructions of contact manifolds with controlled Reeb dynamics

The Reeb flow of a contact form is a generalisation of Hamiltonian flows on energy hypersurfaces in classical mechanics. In this talk I shall address the question of how "complicated" such flows can be. Among other things, I plan to discuss a construction of Reeb flows with a global surface of section on which the Poincaré return map is a pseudorotation. This is joint work with Peter Albers and Kai Zehmisch

### Geography of (bi)linearized Legendrian contact homology

The study of Legendrian submanifolds in contact geometry presents some similarities with knot theory. In particular, invariants are needed to distinguish Legendrian isotopy classes. Linearized Legendrian contact homology is one of these, and is based on the count of holomorphic curves. It is obtained after linearizing a differential graded algebra using an augmentation. A bilinearized version using two augmentations was introduced with Chantraine.

After a self-contained introduction to this context, the geography of these invariants will be described. In the linearized case, it was obtained with Sabloff and Traynor. The bilinearized case turned out to be far more general and was studied with Galant.

### Equivariant and nonequivariant contact homology

I will explain how to make use of geometric methods to obtain three related flavors of contact homology, a Floer theoretic contact invariant. In particular, I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over $\mathbb{Z}$ and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

### Spectrum of the Robin Laplacian: recent results, and open problems

I will discuss two recent theorems and one recent conjecture about maximizing or minimizing the first three eigenvalues of the Robin Laplacian of a simply connected planar domain. Conformal mappings and winding numbers play a key role in the geometric constructions. In physical terms, these eigenvalues represent decay rates for heat flow assuming a “partially insulating” boundary.

Joint with the CEAFEL seminar

### Hamiltonian $S^1$-spaces with large equivariant pseudo-index.

Let $$(M,\omega)$$ be a compact symplectic manifold of dimension $$2n$$ endowed with a Hamiltonian circle action with only isolated fixed points. Whenever $$M$$ admits a toric $$1$$-skeleton $$\mathcal{S}$$, which is a special collection of embedded $$2$$-spheres in $$M$$, we define the notion of equivariant pseudo-index of $$\mathcal{S}$$: this is the minimum of the evaluation of the first Chern class $$c_1$$ on the spheres of $$\mathcal{S}$$.

In this talk we will discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of $$M$$ are unimodal, we prove that it is at most $$n+1$$ . Moreover, when it is exactly $$n+1$$, $$M$$ must be homotopically equivalent to $$\mathbb{C}P^n$$.

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Current organizers: Rosa Sena Dias, José Mourão.