# Geometria em Lisboa Seminar

## Past

- 18/09/2017, 15:30 — 16:30 — Room P3.10, Mathematics Building

Ailsa Keating,*Cambridge University* -
### Examples of monotone Lagrangians

Joint work with Mohammed Abouzaid. We present some methods for constructing examples of compact monotone Lagrangians in families of $(n-1)$-dimensional affine hypersurfaces; they can be upgraded to Lagrangians in $C^n$. We will then explain different strategies for telling them apart, using Floer homology, and, time allowing, some counts of holomorphic annuli. The talk will only assume minimal knowledge of symplectic geometry.

- 26/07/2017, 16:30 — 17:30 — Room P4.35, Mathematics Building

Miguel Abreu,*Centro de Análise Matemática Geometria e Sistemas Dinâmicos, Instituto Superior Técnico* -
### Contact topology of Gorenstein toric isolated singularities

Links of Gorenstein toric isolated singularities are good toric contact manifolds with zero first Chern class. In this talk I will present some results relating contact and singularity invariants in this particular toric context. Namely,

- I will explain why the contact mean Euler characteristic is equal to the Euler characteristic of any crepant toric smooth resolution of the singularity (joint work with Leonardo Macarini).
- I will discuss applications of contact invariants of Lens spaces that arise as links of Gorenstein cyclic quotient singularities (joint work with Leonardo Macarini and Miguel Moreira).

- 26/07/2017, 15:00 — 16:00 — Room P4.35, Mathematics Building

Marta Batoréo,*Universidade Federal do Espírito Santo* -
### On periodic points of symplectomorphisms on closed manifolds

In this talk, we will discuss symplectomorphisms on closed manifolds with periodic orbits. We will present some results on the existence of (infinitely many) periodic orbits of certain symplectomorphisms on closed manifolds. Moreover, we will give a construction of a symplectic flow on a closed surface of genus $g$ greater than $1$ with exactly $2g-2$ fixed points and no other periodic orbits.

- 21/07/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

Ana Rita Pires,*Fordham University* -
### Symplectic embedding problems and infinite staircases, with some proofs

- 21/07/2017, 13:30 — 14:30 — Room P3.10, Mathematics Building

Nick Sheridan,*Princeton University* -
### Cubic fourfolds, K3 surfaces, and mirror symmetry

- 27/06/2017, 16:30 — 17:30 — Room P3.10, Mathematics Building

Gonçalo Oliveira,*Duke University* -
### $G_2$-instantons on noncompact $G_2$-manifolds

I will report on joint work with Jason Lotay on some existence and nonexistence results for $G_2$-instantons. I shall compare the behavior of $G_2$-instantons for two distinct $G_2$-holonomy metrics on $\mathbb{R}^4\times S^3$.

- 15/05/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

José Natário,*Instituto Superior Técnico* -
### A Minkowski-type inequality for convex surfaces in the hyperbolic 3-space

In this talk we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces arising from the normal flow and then applying the isoperimetric inequality. Using the same method, we also we give elementary proofs of the classical Minkowski inequalities for closed convex surfaces in the Euclidean 3-space and in the 3-sphere.

- 08/05/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

Thomas Baier,*Instituto Superior Tecnico* -
### Higher rank Prym varieties and Hitchin's connection

Prym varieties are abelian varieties similarly associated to a double covers of algebraic curves as Jacobians are to a curve. In this talk, we define a higher rank analogue of Prym varieties and investigate some of their geometric properties. In particular we are interested in deformation theoretic aspects that permit the construction of a generalized Hitchin's connection in this setting.

This talk is based on joint work in progress with Michele Bolognesi, Johan Martens and Christian Pauly. - 13/03/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

Cristiano Spotti,*Centre for Quantum Geometry of Moduli Spaces, Aarhus* -
### Kähler-Einstein Fano varieties and their moduli spaces.

Possibly singular Fano varieties which admit Kähler-Einstein metrics are of particular interest since, among other things, they form compact separated moduli spaces. In the seminar, I will talk about existence results for these canonical metrics and describe examples of compact moduli spaces of these special varieties, explaining how the existence and moduli problems are intimately related to each other when looking for explicit examples of such Kähler-Einstein Fano varieties.

- 21/02/2017, 16:30 — 17:30 — Room P3.10, Mathematics Building

André Gama Oliveira,*Centro de Matemática da Universidade do Porto* -
### Parabolic Higgs Bundles and Topological Mirror Symmetry

In 2003, T. Hausel and M. Thaddeus proved that the Hitchin systems on the moduli spaces of $\operatorname{SL}(n,\mathbb{C})$- and $\operatorname{PGL}(n,\mathbb{C})$-Higgs bundles on a curve, verify the requirements to be considered SYZ-mirror partners, in the mirror symmetry setting proposed by Strominger-Yau-Zaslow (SYZ). These were the first non-trivial known examples of SYZ-mirror partners of dimension greater than $2$.

According to the expectations coming from physicists, the generalized Hodge numbers of these moduli spaces should thus agree — this is the so-called topological mirror symmetry. Hausel and Thaddeus proved that this is the case for $n=2,3$ and gave strong indications that the same holds for any $n$ prime (and degree coprime to $n$). In joint work in progress with P. Gothen, we perform a similar study but for parabolic Higgs bundles. We will roughly explain this setting, our study and some questions which naturally arise from it.

- 10/01/2017, 16:30 — 17:30 — Room P3.10, Mathematics Building

Rui Albuquerque,*Universidade de Évora* -
### Riemannian $3$-manifolds and Conti-Salamon $\operatorname{SU}(2)$-structures

We present an $\operatorname{SO}(2)$-structure and the associated global exterior differential system existing on the contact Riemannian manifold $\cal S$, which is the total space of the tangent sphere bundle, with the canonical metric, of any given $3$-dimensional oriented Riemannian manifold $M$. This is part of a wider theory which can be studied in any dimension. In this seminar we focus on the first interesting dimension and show several new $\operatorname{SU}(2)$-structures on $\cal S$, following the recent ideas introduced by D. Conti and S. Salamon for the study of $5$-manifolds with special metrics.

- 06/12/2016, 16:30 — 17:30 — Room P3.10, Mathematics Building

Thibaut Delcroix,*Institut Fourier* -
### K-stability of Fano spherical varieties

The resolution of the Yau-Tian-Donaldson conjecture for Fano manifolds, that is, the equivalence of the existence of Kähler-Einstein metrics with K-stability, raises the question of determining when a given Fano manifold is K-stable.

I will present a combinatorial criterion of K-stability for Fano spherical manifolds. These form a very large class of almost-homogeneous manifolds, containing toric manifolds, homogeneous toric bundles, and classes of manifolds for which the Kähler-Einstein existence question was not solved yet, for example equivariant compactifications of (complex) symmetric spaces.

- 18/11/2016, 14:30 — 15:30 — Room P3.10, Mathematics Building

Martin Pinsonnault,*University of Western Ontario* -
### Finite Hamiltonian actions on 4-manifolds

- 10/11/2016, 15:00 — 16:00 — Room P4.35, Mathematics Building

Umberto Hryniewicz,*Universidade Federal do Rio de Janeiro, Brasil* -
### Negative and positive results in the intersection between systolic and symplectic geometry

How small is the smallest period of a closed trajectory of a Reeb flow? In this talk I will present recent answers to instances of this question in three-dimensions which reveal connections between systolic and symplectic geometry. I will present results both of a positive and of a negative nature. Namely, in some situations there are sharp bounds for the systolic ratio, which is defined as the ratio between the square of the smallest period and the contact volume, while in other situations the systolic ratio is unbounded. Our results confirm a conjecture of Babenko and Balacheff and disprove a conjecture of Hutchings. There are implications to middle-dimensional non-squeezing results which we hope to discuss if time permits. All this is joint work with Abbondandolo, Bramham and Salomão.

- 29/07/2016, 14:00 — 15:00 — Room P4.35, Mathematics Building

Bruno Oliveira,*University of Miami* -
### Twisted symmetric differentials and quadric envelopes III

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

- 28/07/2016, 11:00 — 12:00 — Room P3.10, Mathematics Building

Gonçalo Oliveira,*Duke University* -
### Random complexes in Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold and consider a Poisson process generating, in average, $n$ uniformly distributed points in $(M,g)$. In joint work with Omer Bobrowski we answer the following question: As the number, $n$, of points increases what is the smallest possible radius $r$, so that the union of the radius $r$ Riemannian balls centered at the randomly generated points has the same homology as that of the underlying Riemannian manifold $M$.

In terms of a data set lying in a Riemannian manifold, this is similar to asking: what is the minimum we must fatten the data points so that they recover the underlying topology being encoded.

- 27/07/2016, 15:00 — 16:00 — Room P4.35, Mathematics Building

Bruno Oliveira,*University of Miami* -
### Twisted symmetric differentials and quadric envelopes II

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

- 25/07/2016, 14:30 — 15:30 — Room P3.10, Mathematics Building

Bruno Oliveira,*University of Miami* -
### Twisted symmetric differentials and quadric envelopes

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

- 14/07/2016, 15:30 — 16:30 — Room P4.35, Mathematics Building

Nick Sheridan,*Princeton University* -
### Homological mirror symmetry for Greene-Plesser mirrors

I will start by explaining what mirror symmetry is about, paying special attention to the

*mirror map*which matches up the family of symplectic forms on one manifold with the family of complex structures on another. I will explain how this works for Batyrev's beautiful toric construction of mirror families from dual reflexive polytopes. Then I will give a template for proving cases of Kontsevich's homological mirror symmetry conjecture, based on a*versality*result for the Fukaya category, which roughly gives a criterion for the existence of a mirror map. The proof can be completed when the reflexive polytope in Batyrev's construction is a simplex: this special case of the construction is due to Greene and Plesser. The latter result is joint work with Ivan Smith. - 14/07/2016, 14:00 — 15:00 — Room P4.35, Mathematics Building

Ana Rita Pires,*Fordham University* -
### Symplectic embeddings and infinite staircases

McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a $4$-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid $E(2,3)$.

This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.

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