# Geometry

21 de Julho (sexta-feira), às 11:00, 6.2.33

G2-instantons with symmetry

Gonçalo Oliveira (Duke University - E.U.A.)

Abstract: G2-instantons, are 7-dimensional analogs of anti-self-dual connections in 4 dimensions, which however require the underlying manifold to have a so called G2-structure. The interest on these G2-instantons comes from the suggestion, of Donaldson and Segal, that it may be possible to use them to define enumerative invariants of G2-holonomy Riemannian metrics. In the talk, I will explain these ideas and report on joint work with Jason Lotay where we give existence, non-existence and classification results for these instantons. I will focus in the particular case of $\mathbb{R}^4 \times S^3$, with its two explicitly known distinct G2-holonomy metrics, and exhibit the different behavior of their instantons. I will also explain an explicit example of sequences of G2-instantons where “bubbling” and “removable singularity” phenomena occur in the limit. All this is joint work with Jason Lotay.

23 de Junho (sexta-feira), às 13:30, sala 6.2.33

Topological Lattice Models in Geometry and Physics

Björn Gohla (GFM - Universidade de Lisboa)

Abstract: Lattice models arise in physics as discrete approximations of quantum field theories (QFT). Topological quantum field theories (TQFT) on the other hand by definition are QFTs, that can be defined on smooth or topological space-times, as opposed to the usual pseudo-Riemannian space-times required by QFTs. Interesting TQFTs can be defined as lattice models, giving exactly solvable models in 2 dimensions for example. As an example in mathematics the Turaev-Viro TQFT gives a well-defined topological invariant of 3-manifolds. We will give an overview of these phenomena and present our work on a 4-dimensional gravitational lattice model.

16 de Junho (sexta-feira), às 14:00, sala 6.2.33

Classification of vector bundles over the Riemann Sphere

Javier Alcaide (Mestrado em Matemática - FCUL)

Abstract: We will give a brief introduction to the theory of compact Riemann surfaces, as an approach to algebraic geometry. Objects like sheaves and vector bundles play an important role in this subject, as they allow us to understand global properties of a surface looking only at local information. After giving some important results of sheaf cohomology, like Serre's duality or the Riemann-Roch theorem, we will use these tools to classify holomorphic vector bundles over the Riemann sphere.

19 de Maio (sexta-feira), às 15h00, sala 6.2.50

Compactification of Character Varieties

Sean Lawton (George Mason UNiversity - U.S.A.)

Abstract: In this talk we will first discuss a general procedure for compactifying G-character varieties of discrete groups, where G is a semisimple algebraic group of adjoint type over an algebraically closed field. We will then discuss various properties of this compactification in special cases of the discrete group. This work is in collaboration with Dan Ramras and Indranil Biswas.

19 de Maio (sexta-feira), às 14h00, sala 6.2.33

12, 24 and Beyond

Leonor Godinho (Dep. Matemática - Instituto Superior Técnico)

Abstract: Symplectic geometry and combinatorics are strongly intertwined due to the existence of Hamiltonian torus actions. These actions are associated with a special map (called the moment map) which “transforms” a compact symplectic manifold into a convex polytope. We will concentrate on the special class of reflexive polytopes which was introduced by Batyrev in the context of mirror symmetry and has attracted much attention recently. In particular, we will see how the famous “12 and 24” properties in dimension 2 and 3 can be generalized with the help of symplectic geometry.

21 de Abril (sexta-feira), às 14h00, sala 6.2.33

On the mean Euler characteristic of Gorenstein toric contact manifolds

Miguel Abreu (Dep. Matemática - Instituto Superior Técnico)

Abstract: In this talk I will prove that the mean Euler characteristic of a Gorenstein toric contact manifold is equal to half the normalized volume of the corresponding toric diagram. I will also give some immediate applications of this result. This is joint work with Leonardo Macarini.

31 de Março (sexta-feira), às 14h00, sala 6.2.38

Topological Lattice Models in Geometry and Physics

Björn Gohla (Group of Mathematical Physics - University of Lisbon)

Abstract: Lattice models arise in physics as discrete approximations of quantum field theories (QFT). Topological quantum field theories (TQFT) on the other hand by definition are QFTs, that can be defined on smooth or topological space-times, as opposed to the usual pseudo-Riemannian space-times required by QFTs. Interesting TQFTs can be defined as lattice models, giving exactly solvable models in 2 dimensions for example. As an example in mathematics the Turaev-Viro TQFT gives a well defined topological invariant of 3-manifolds.

We will give an overview of these phenomena and present our work on a 4-dimensional gravitational lattice model.

24 de Março (sexta-feira), às 14h00, sala 6.2.33

Normal complex spaces with near to quotient singularities

Daniel Barlet (Univ. Nancy)

Abstract: This is a joint work with J. Magnusson. We introduce an interesting class of normal complex spaces having only mild singularities (near to quotient singularities) in which we can generalize the notion of (analytic) fundamental class for complex cycles and also the notion of relative fundamental class for an analytic family of cycles. We also generalize to these spaces the intersection theory for cycles with rational positive coefficients. This also extends to the intersection of analytic families of cycles. We show that almost all the properties of these notions generalize to this context with the exception of the fact that the fundamental classes of the intersection of two cycles whose intersection has the expected co-dimension is not always given by the cup-product of their fundamental classes.

24 de Fevereiro (sexta-feira), às 14:00, sala 6.2.33

Quasi-abelian categories

Abstract: Bondal and Van den Bergh have constructed an equivalence between the notion of quasi-abelian category studied by Schneiders and that of tilting torsion pair on an abelian category. We will provide an overview of this result and hence we will extend this picture into a hierarchy of $n$-quasi-abelian categories and $n$-tilting torsion classes.

In particular $0$-quasi-abelian categories are abelian categories, $1$-quasi-abelian categories are Schneiders quasi-abelian categories, $2$-quasi-abelian categories are additive categories admitting kernels and cokernels. Any $n$-quasi-abelian category $E$ admits a “derived” category endowed with two canonical t-structures (the left and the right one) such that $E$ coincides with the intersection of their hearts.

20 de Janeiro (sexta-feira), às 14:00, sala 6.2.33

Rui Pacheco (CMAFCIO, Universidade da Beira Interior)

Surfaces in $\mathbb{R}^7$ obtained from harmonic maps in $S^6$

Abstract:

We will investigate the local geometry of surfaces in $\mathbb{R}^7$ associated to harmonic maps from a Riemann surface $\Sigma$ into the nearly Kähler $6$-sphere $S^6$. In this setting, the harmonicity of a smooth map $\varphi : \Sigma \to S^6$ amounts to the closeness of the differential $1$-form $\omega = \varphi \times {}^* \mathrm{d} \varphi$, where $\times$ stands for the $7$-dimensional cross product. This means that we can integrate on simply-connected domains in order to obtain a map $F : \Sigma \to \mathbb{R}^7$. By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions $\varphi : \Sigma \to S^6$ whose associated immersions $F : \Sigma \to \mathbb{R}^7$ belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces.

This is a joint work with Pedro Morais.

Acknowledgments. This work was supported by CMA-UBI through the project UID/MAT/00212/2013.

## 2016

16 de Dezembro (sexta-feira), às 14:00, sala 6.2.33

Pedro Morais (Departamento de Matemática, Universidade da Beira Interior)

Hypersurfaces of the sphere with a totally geodesic foliation of codimension one

Abstract:

Several authors have investigated whether on a given Riemannian manifold Mn there exists a totally geodesic foliation of codimension one, as well as the inverse problem of determining whether one can find a Riemannian metric on a manifold Mn with respect to which a given smooth foliation of codimension one on Mn becomes totally geodesic.

A related problem, but extrinsic in nature, may be formulated as follows: What are all Euclidean hypersurfaces f : Mn → Rn+1, n ≥ 3, that carry a foliation of codimension one with totally geodesic (complete or not) leaves?
This problem was solved by Dajczer, Rovenski and Tojeiro (2015).

In this talk we will study the spherical case.
This is a joint work with Susana Duarte Santos.

25 de Novembro , sala 6.2.38, 14:00

André Oliveira (Centro de Matemática, Univ. Porto)

Exotic components of Higgs bundle moduli spaces

Abstract:

25 de Novembro , sala 6.2.38, 15:00

Daniel Ramos (CMAFCIO, ULisboa)

Uniformization of cone surfaces using Ricci flow

Abstract:

Every smooth closed surface admits a Riemannian metric of constant curvature, determined by its Euler characteristic. Surfaces with cone-like singularities (such as certain orbifolds) may fail to admit such constant curvature metrics. We propose a Ricci soliton metic as the canonical metric on these cases, and we prove that Ricci flow converges to such soliton metrics for any initial metric on closed surfaces with cone angles less than or equal to pi. The Ricci flow is an evolution equation introduced by R. Hamilton in 1982 and used by G. Perelman in 2002 to prove the Thurston geometrization of closed 3-manifolds. We use Perelman's techniques for conesingular closed surfaces and we discuss some open problems of the flow in open surfaces.

28 de Outubro (sexta-feira)14h00, sala 6.2.33

Moduli spaces of Hitchin pairs for the indefinite unitary group U(p,q)

Abstract:

U(p,q)-Hitchin pairs on a Riemann surface consist of a a pair of holomorphic vector bundles together with a pair of twisted holomorphic maps between them, one in each direction. The natural stability condition for U(p,q)-Hitchin pairs depends on a real parameter. I will talk about wall crossing for the moduli spaces of polystable U(p,q)- Hitchin pairs as the stability parameter varies. The fact that the quiver associated to these objects contains an oriented cycle introduces new phenomena which was not present in the previously studied cases of triples and chains. From the obtained results we can deduce the birationality of the moduli spaces in a certain range of the parameter.

de Setembro, 15h00, sala 6.2.33

Marcos Dajczer (IMPA, Brasil)

A new class of minimal submanifolds

Abstract: A basic problem in submanifold theory is to decide if an isometric immersion of a given Riemannian manifold into a space form is unique, up to rigid motions. If this is the case, then the immersion is called rigid. If the immersion is not rigid, then it is a fundamental  problem  to determine all of its isometric deformations.

The local problem for hypersurfaces was mostly solved by Sbrana  and Cartan more than a century ago. A solution to the problem for compact hypersurfaces was given by Sacksteder and by Dajczer-Gromoll in the complete case.  I will discuss a contribution to the solution of the global problem in codimension two by means of new classes of minimal immersions.

This is joint work with Th. Vlachos.

24 de junho, 13:30, sala 2.2.33

Ana C. Silva (Universidade de Ghent)

Groups acting on CAT(0) polygonal complexes with prescribed local action

27 de Maio, 13h30, sala 6.2.33

João Faria Martins (CMA e FCT/UNL)

Crossed modules and 2-dimensional homotopy invariants of knotted surfaces in the 4-sphere

Abstract: Motivated by recently discovered connections to topological phases of matter of condensed matter Physics / topological quantum computing, I will describe work I have done some years ago oninvariants of knotted surfaces embedded in the 4-sphere derived from finite crossed modules. I will also report on oneiric (at this moment) extensions of this framework to Lie crossed modules invariants of knotted surfaces, to be addressed in the realm of gerbes and categorified BF-theories.

1 de Abril, 13h30, sala 6.2.33

André Neves (Imperial College London)

Min-max theory in Geometry

Abstract: Min-max theory was recently used by myself and Fernando Marques to prove the Willmore Conjecture, the Freedman-He-Wang Conjecture, and the Yau Conjecture for metrics with positive Ricci curvature. I will survey those results and talk about new directions in the area.

26 de Fevereiro, 13h30, sala 6.2.33

José Natário

CAMGSD, IST, ULisboa

A Minkowski-type inequality for convex surfaces in the hyperbolic 3-space

Abstract: 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 obtained from the normal flow and then applying the isoperimetric inequality. Using the same method, we also give elementary proofs of the classical Minkowski inequalities for closed convex surfaces in the Euclidean 3-space and in the 3-sphere.

22 de Janeiro, 13h30, sala 6.2.33

Luca Prelli

CMAFCIO, ULisboa

A Functorial approach to asymptotic expansions

Abstract: Asymptotically developable expansions of holomorphic functions are an important tool to study differential equations with irregular singularities. In this talk we will discuss their functorial nature using subanalytic sheaf theory and then we will construct several kind of asymptotics.

11 de Dezembro, 13h30sala 6.2.33

Gilles Lebeau

Université de Nice Sophia-Antipolis

On the holomorphic extension of the Poisson Kernel

20 de Novembro, 13h30, sala 6.2.33

João Pimentel Nunes

CAMGSD, IST, ULisboa

Complex time evolution in geometry and quantization

Abstract: We will describe the role of complexified symplectomorphisms in the geometry and quantization of Kahler manifolds. Examples including complex Lie groups and toric varieties will be mentioned. Based on joint work with W.Kirwin and J.Mourão.

16 de Outubro, 13h30, na sala 6.2.33

Margarida Mendes Lopes

CAMGSD/IST/UL

The (algebraic) fundamental group of surfaces with small invariants

Abstract:

The algebraic fundamental group of an algebraic variety is an important tool in Algebraic Geometry.
After an introduction to the main properties of invariants of algebraic surfaces, results about the fundamental group of surfaces with small invariants will be discussed, illustrating the fact that for algebraic surfaces with small invariants the structure of the algebraic fundamental group is relatively simple.

10 de Julho (sexta-feira), às 15:00h, na sala A2-25 do Instituto para a Investigação Interdisciplinar

Geometria das variedades de caracteres de grupos abelianos

Carlos Florentino
(IST, ULisboa)

Resumo: Dado um grupo de Lie G, a descrição do conjunto de n-tuplos dos seus elementos que comutam entre si, é um problema algébrico e geométrico com interesse em física-matemática, em particular nas teorias de Yang-Mills super-simétricas. Este espaço é também um espaço de "moduli" de fibrados de Higgs sobre uma variedade abeliana. Quando o grupo é redutivo complexo, este espaço designa-se a G-variedade dos caracteres de um grupo livre abeliano e é um quociente da forma Hom(Z^r,G)/G. Mostramos que Hom(Z^r,G)/G admite uma
deformação por retracção para o espaço Hom(Z^r,K)/K, onde K é umsubgrupo maximal de G, e obtemos condições necessárias e suficientes para que Hom(Z^r,G)/G seja uma variedade irredutível. Daremos alguns exemplos destes espaços e de alguns dos seus invariantes topológicos e algébricos, tais como os polinómios de Hodge-Deligne.

8 de Julho, 15h00, sala B3-01 do Instituto para a Investigação Interdisciplinar

Isometric immersions from a Kähler manifold into the quaternionic projective space

Bruno Simões (CMAFCIO, ULisboa)

Abstract:

In this talk we present a rigidity theorem for isometric immersions from Kähler manifolds into a quaternionic projective space (or its non compact real form), when its nullity index is everywhere positive.

8 de Maio 2015

15h00, no Instituto para a Investigação Interdisciplinar da Universidade de Lisboa (sala a anunciar).

César Rodrigo Fernández (CMAF-CIO, ULisboa)

Simplicial geodesic interpolation applied to the discrete variational formulation of field theories

Abstract:

In most cases Partial Differential Equations arising in Field theories
derive from a variational principle formulated on sections of a given
configuration bundle. Noether's principle associates then conservation laws
corresponding to infinitessimal symmetries of the Lagrangian density and
represents a great tool for the qualitative study of solutions. Numerical
schemes focused on the discretization of the PDE neglect these symmetries
and thus lead to approximations of the smooth solution without control of
its qualitative behaviour (namely, poor energy-conservation properties).
Since the beginning of the century, lots of work has covered this subject in
discrete mechanics (discrete time variable), with relevant results both for
the obtention of energy-conserving integration schemes or for the
translation of concepts from geometrical mechanics into the field of
numerical analysis of ODEs.

In this talk we shall present the variational formulation of discrete field
theories, seen as sections of some configuration bundle modelled over a
discrete n-D space. This discrete space has the structure of abstract
simplicial complex. We derive the main tools of variational
analysis: discrete Lagrangian densities lead to discrete action functionals;
a set of discrete Euler-Lagrange equations characterizes critical point for
some fixed-boundary variational principle formulated on this action
functional; a Noether theorem is proved showing how infinitessimal
symmetries of the Lagrangian lead to corresponding conservation laws valid
for any solution of the discrete Euler-Lagrange equations.

We then connect this formalism with the variational field theory derived
from some smooth Lagrangian density, particularly in the presence of
symmetries of the Lagrangian density that are also isometries of a given
Riemannian structure. We show how application of geodesic interpolation
techniques on simplices, together with a choice of quadrature on simplices
allows to determine a discrete Lagrangian density, keeping all the
symmetries given for the original one, and relating both formalisms, the
smooth and the discrete variational theories. An explicit computation of the
discrete objects depends only on the construction of geodesics joining two
neighboring points and is simple when the Riemannian structure has
well-knowm geodesics, as for example on the Lie-group SO(3). We show how to
discretize in this context the dynamics of a Cosserat rod, leading to
numerical integrators with work-energy conservation properties.