1999 seminars

Zuoqin Wang, University of Science and Technology of China Heifei
Equivariant Eigenvalues on Manifolds with Large Symmetry

Let $M$ be a compact Riemannian manifold on which a compact Lie group acts by isometries. In this talk I will explain how the symmetry induces extra structures in the spectrum of Laplace-type operators, and how to apply symplectic techniques to study the induced equivariant spectrum. In particular, I will discuss a) my joint works with V. Guillemin on inverse spectral results for Schrodinger operators on toric manifolds; b) my joint work with Y. Qin on the first equivariant eigenvalues of toric Kahler manifolds.

Pedro Boavida, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa
Spaces of smooth embeddings and the little disks operad

I will describe a homotopy theoretic approach, based on a method due to Goodwillie and Weiss, to study spaces of smooth embeddings of a manifold into another. This approach opened up interesting relations to operad theory and as such to fundamental objects in topology (e.g. configuration spaces) and algebra (e.g. graph complexes). I will survey some of these developments, focusing on the case of long knots and higher-dimensional variants, for which these relations are the sharpest.

Matias Del Hoyo, Universidade Federal Fluminense
Discrete dynamics and differentiable stacks

In a joint work with A. Cabrera (UFRJ) and E. Pujals (IMPA) we study actions of discrete groups over connected manifolds by means of their orbit stacks. Stacks are categorified spaces, they generalize manifolds and orbifolds, and they remember the isotropies of the actions that give rise to them. I will review the basics, show that for simply connected spaces the stacks recover the dynamics up to conjugacy, and discuss the general case. I will also discuss several examples, involving irrational rotations of the circle, hyperbolic toral automorphisms, and the lens spaces.

Pedro Freitas, Instituto Superior Técnico
The spectral determinant of the quantum harmonic oscillator in arbitrary dimensions

We show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity. We determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large.

As a consequence, we provide an alternative proof of a conjecture posed by Bar and Schopka concerning the convergence of the determinant of the Dirac operator on $S^{n}$, determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the proof by Moller.

Cyril Lecuire, Centre National de la recherche scientifique
Geometry in groups

The intent of Geometric Group Theory is to deduce algebraic properties of groups from their actions on metric spaces. A natural way to obtain such an action is to equip a group with an invariant distance.

First, to motivate the study of Geometric Group Theory, I will expose some of its achievements (solvability of the word problem, Tits alternative). Then I will define the word metric on a finitely generated group and explain the difficulties raised by the definition. Finally, as an example of geometric properties of interest, I will introduce hyperbolic groups.

Alessandro Ghigi, Università di Pavia
Compactifying automorphism groups of Kaehler manifolds

Around 1978 Akira Fujiki and David Lieberman independently introduced a natural compactification of the connected component of the identity in the automorphism group of a compact Kaehler manifold. In the talk I will recall the construction of this compactification using Barlet cycle space. Then I will describe some recent results obtained jointly with Leonardo Biliotti. The main result is the interpretation of boundary points in terms of non-dominant meromorphic inmaps of the manifold in itself.

Nitu Kitchloo, Johns Hopkins University
The Stable Symplectic Category and a Conjecture of Kontsevich

Motivated by his work on deformation quantization and his computations of Feynman integrals, Kontsevich conjectured that a certain group (related to the Grothendieck Teichmuller group) acts on the moduli space of quantum field theories. Even though this moduli space is not well-defined in general, we will show that a stable version of this space makes sense and can be identified as a space that represents an interesting cohomology theory. In addition, we will show that a solvable quotient of the Grothendieck-Teichmuller group acts on the stable moduli space, and as such, it can be identified with an algebraic functor of the underlying cohomology theory.

Lino Amorim, Kansas State University
Closed mirror symmetry for orbifold spheres

In this talk I will describe a closed mirror symmetry theorem for a sphere with three orbifold points. More precisely I will construct an isomorphism between the quantum cohomology ring of the orbifold and the Jacobian ring of a certain power series built from the Lagrangian Floer theory of an immersed circle. This is joint work with Cho, Hong and Lau.

José Mourão, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa
Imaginary time Hamiltonian flows and applications to Kahler geometry, Kahler reduction and representation theory

The formalism to complexify time in the flow of a nonholomorphic vector field on a complex manifold is reviewed. The complexified flow, besides acting on $M$, changes also the complex structure. We will describe the following applications:

  1. For a compact Kahler manifold the imaginary time Hamiltonian flows correspond to Mabuchi geodesics in the infinite dimensional space of Kahler metrics on $M$. These geodesics play a very important role in the study of stability of Kahler manifolds. A nontrivial nontoric example on the two-dimensional sphere will be described.
  2. Let the compact connected Lie group $G$ act in an Hamiltonian and Kahler way on a Kahler manifold $M$ and assume that its action extends to $G_C$. Then, by taking geodesics of Kahler structures generated by convex functions of the $G$-momentum to infinite geodesic time, one gets (conjecturally always, proved on several important examples) a concentration of holomorphic sections of holomorphic line bundles on inverse images of coadjoint orbits under the $G$-momentum map. A nontrivial toric example and the case of $M=G_C$ will be described.

On work with T Baier, J Hilgert, O Kaya, JP Nunes, M Pereira, P Silva.

Bruno Oliveira, University of Miami
Hyperbolicity of projective manifolds

In this talk we will discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold $X$ is said to be hyperbolic if there are no nonconstant holomorphic maps from the complex line to $X$. This is a subject that brings together methods of algebraic geometry, complex analysis and differential geometry.

We will discuss the key and well understood case of dimension $1$. We will have several distinct characterizations of hyperbolicity and see how that extend for projective manifolds of higher dimension. We will also discuss the related Green-Griffiths-Lang conjecture.

Jun Li, University of Minnesota, Minneapolis
The symplectomorphism groups of rational surfaces

This talk is on the topology of $\operatorname{Symp}(M, \omega)$, where $\operatorname{Symp}(M, \omega)$ is the symplectomorphism group of a symplectic rational surface $(M, \omega)$. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group (SMC) and describe the answer in terms of the Dynkin diagram of Lagrangian sphere classes. In particular, when deforming the symplectic form, the SMC of a rational surface behaves in the way of forgetting strand map of braid groups. We are also able to compute the fundamental group of $\operatorname{Symp}(M, \omega)$ for an open region of the symplectic cone. This is a joint work with Tian-Jun Li and Weiwei Wu.

Martin Pinsonnault, University of Western Ontario
Stability of Symplectomorphism Groups of Small Rational Surfaces

Let $(X_k,\omega_k)$ be the symplectic blow-up of the projective plane at $k$ balls, $1\leq k\leq 9$, of capacities $c_1,\ldots, c_k$. After reviewing some facts on Kahler cones and curve cones of tamed almost complex structures, we will give sufficient conditions on two sets of capacities $\{c_i\}$ and $\{c_i’\}$ for the associated symplectomorphism groups to be homotopy equivalent. In particular, we will explain when those groups are homotopy equivalent to stabilisers of points in $(X_{k-1},\omega_{k-1})$. We will discuss some corollaries for the spaces of symplectic balls.

Bruno Oliveira, University of Miami
Hyperbolicity of projective manifolds

We continue to discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold $X$ is said to be hyperbolic if there are no nonconstant holomorphic maps from the complex line to $X$. This is a subject that brings together methods of algebraic geometry, complex analysis and differential geometry.

We will discuss the key and well understood case of dimension $1$. We will have several distinct characterizations of hyperbolicity and see how that extend for projective manifolds of higher dimension. We will also discuss the related Green-Griffiths-Lang conjecture.

Levi Lima, Universidade Federal do Ceará
The mass of asymptotically hyperbolic manifolds with a noncompact boundary.

We discuss a positive mass inequality (and its consequences) for the class of manifolds in the title, under the spin assumption. This is a natural extension to this setting of a previous result by P. Chrusciel and M. Herzlich, who treated the boundaryless case. Joint work with S. Almaraz.

Dan Avritzer, Universidade Federal de Minas Gerais
Classical Geometry and the Moduli Space of Higgs bundles

One of the most beautiful objects of classical geometry is the Kummer Surface, that was studied by Kummer in the 19th century. In a celebrated paper of 1969 Narasimhan and Ramanan studied the moduli space of vector bundles of rank 2 and trivial determinant over a curve of genus 2, proving that this space is isomorphic to projective space of dimension 3. In this space the moduli space of non-stable bundles is parameterized by a Kummer Surface.

In this seminar, I will introduce the Kummer Surface in the classical setting and recall the main results of the paper of Narasimhan and Ramanan mentioned above. Then I will talk about joint work in progress with Peter Gothen, where we describe the moduli space of Higgs bundles over a curve of genus 2. We obtain a similar description as in the paper above of the moduli of Higgs bundles in the so called nilpotent cone. The aim is to study the geometry of this nilpotent cone as done in the Narasimhan-Ramanan paper.

Gonçalo Oliveira, Universidade Federal Fluminense
Yang-Mills flow and calibrated geometry

This is a report on joint work with Alex Waldron.

The Yang-Mills functional is the most studied functional on the space of connections on a vector bundle over an oriented Riemannian manifold. Its negative gradient flow leads to a semi-parabolic PDE known as the Yang-Mills flow.

I will introduce this flow and talk about its properties in the context of manifolds with special holonomy, particularly in Kahler, $G_2$, and $\operatorname{Spin}(7)$-manifolds. I intend to explain a blow-up criteria and talk about relationships with certain minimal “submanifolds” known as calibrated.