# Informal Geometry Seminar

## Past sessions

### Topics on moduli space of HIggs bundles

The work of Hausel proves that the Bialynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. These two stratifications do not coincide in general. Here, we give an approach for the rank three case of the classification of the Shatz stratification in terms of the Bialynicki-Birula stratification.

This talk will be a preliminary talk for the talk in the afternoon at "Geometria em Lisboa".

### Symplectic fillability of toric contact manifolds in higher dimensions

I will discuss strong and weak fillability of toric contact manifolds. In particular, I will show that every toric contact manifold in dimension greater than three is weakly symplectically fillable.

This result does not hold in dimension three since overtwisted toric contact 3-manifolds are known.

### Arnold Conjectures and introduction to the generating functions technique (II).

Symplectic diffeomorphisms possess certain rigidity properties (symplectic camel, non-squeezing theorem). An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism $h$ (i.e. symplectic diffeomorphism Hamiltonian isotopic to identity) of a compact symplectic manifold and a lower bound for the number of intersection points of a Lagrangian submanifold $L$ and $h(L)$. These lower bounds are greater than what topological arguments could predict.

The conjectures have been proved in many special cases, but not in full generality.

We will sketch the proof for $C^1$-small Hamiltonian diffeomorphisms. Then we introduce the technique of generating families which can be used to prove Arnold Conjectures for other Hamiltonian diffeomorphisms. If time permits we introduce contact version of Arnold Conjecture about the minimal number of translated points of a contactomorphism, and explain how to adjust the proof of Arnold Conjecture for $CP^n$ to obtain a proof of contact Arnold Conjecture for $RP^{2n-1}$.

Notice that time is shifted half an hour. we will start at 11:30.

### Arnold Conjectures and introduction to the generating functions technique.

Symplectic diffeomorphisms possess certain rigidity properties (symplectic camel, non-squeezing theorem). An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism $h$ (i.e. symplectic diffeomorphism Hamiltonian isotopic to identity) of a compact symplectic manifold and a lower bound for the number of intersection points of a Lagrangian submanifold $L$ and $h(L)$. These lower bounds are greater than what topological arguments could predict.

The conjectures have been proved in many special cases, but not in full generality.

We will sketch the proof for $C^1$-small Hamiltonian diffeomorphisms. Then we introduce the technique of generating families which can be used to prove Arnold Conjectures for other Hamiltonian diffeomorphisms. If time permits we introduce contact version of Arnold Conjecture about the minimal number of translated points of a contactomorphism, and explain how to adjust the proof of Arnold Conjecture for $CP^n$ to obtain a proof of contact Arnold Conjecture for $RP^{2n-1}$.

The date has been changed from 7 Oct. 7 to 14 Oct.

### On displaceability of pre-Lagrangian toric fibers in toric contact manifolds.

I will talk about some methods for showing displaceability and non-displaceability of pre-Lagrangian toric fibers. I will give some interesting examples, discuss possible generalisations and make comparison with toric symplectic case.

Room changed from 3.10 to 4.35.

### Einstein's equations in 4-dimensional Riemannian geometry

I will begin by reviewing some of the features of Riemannian geometry that are particular to dimension 4 as well as what is known about compact Riemannian solutions to Einstein’s equations in 4-dimensions. Then I will outline a new way to write Einstein’s equations in 4-dimensions as a gauge theory, somewhat analogous to Maxwell’s equations and the electromagnetic potential. If there is time I hope to mention some open questions suggested by this new formalism. This “gauge theoretic approach” is joint work with Kirill Krasnov and Dmitri Panov.

### GIT and symplectic stability(III)

Geometric Invariant Theory (GIT) is a powerful tool to study quotients of algebraic varieties by the action of Lie groups, related to symplectic quotients by the Kempf-Ness theorem. From both points of view a notion of stability for the orbits of the group action plays a prominent role.

In the lectures we will give the basic notions and ideas behind GIT stability, symplectic stability and the Kempf-Ness theorem.

GITsymplectic.pdf

### GIT and symplectic stability(II)

Geometric Invariant Theory (GIT) is a powerful tool to study quotients of algebraic varieties by the action of Lie groups, related to symplectic quotients by the Kempf-Ness theorem. From both points of view a notion of stability for the orbits of the group action plays a prominent role.

In the lectures we will give the basic notions and ideas behind GIT stability, symplectic stability and the Kempf-Ness theorem.

We will pay special attention to the unstable orbits for which one can finds a "maximal way to destabilize" them. This idea can be seen from both the algebraic and the symplectic point of view and we will show this coincidence. All the treatment will be done through three (basic but very illustrative) examples: the obtention of the projective space as a quotient, the moduli space of configurations of n points in the projective line (related to the moduli space of polygons) and the obtention of the grassmanian as a quotient.

### GIT and symplectic stability(I)

Geometric Invariant Theory (GIT) is a powerful tool to study quotients of algebraic varieties by the action of Lie groups, related to symplectic quotients by the Kempf-Ness theorem. From both points of view a notion of stability for the orbits of the group action plays a prominent role.

In the lectures we will give the basic notions and ideas behind GIT stability, symplectic stability and the Kempf-Ness theorem.

We will pay special attention to the unstable orbits for which one can finds a "maximal way to destabilize" them. This idea can be seen from both the algebraic and the symplectic point of view and we will show this coincidence. All the treatment will be done through three (basic but very illustrative) examples: the obtention of the projective space as a quotient, the moduli space of configurations of n points in the projective line (related to the moduli space of polygons) and the obtention of the grassmanian as a quotient.

The Informal Geometry Seminar, as its name says, is an informal seminar for graduate students and postdocs at IST to share their ideas with each other and a good place to ask simple questions without any pressure.