–
Room P3.10, Mathematics Building
Equivariant Eigenvalues on Manifolds with Large Symmetry
Let
Equivariant Eigenvalues on Manifolds with Large Symmetry
Let
Dynamics on locally symmetric spaces I
In this lecture series I will discuss the symplectic structure of the cotangent bundle of a (locally) symmetric space, natural Hamiltonian flows thereon and their (geometric) quantization.
Equivariant degenerations and applications I
In these lectures we shall explore the boundary region between algebraic and differential geometry for spaces possessing non-abelian symmetry. In algebraic geometry, the presence of a non-abelian reductive group acting on a variety gives rise to certain canonical equivariant degenerations of the variety. Moreover, the symmetry provides such control over the degenerations so as to make transitions between algebra-geometric and differential notions, which are often very difficult to handle, actually tractable. Motivating examples of this phenomenon that will be discussed include the switch between Kahler and real polarisations in geometric quantisation, or the link between K-stability and existence of Kahler-Einstein metrics.
Dynamics on locally symmetric spaces II
In this lecture series I will discuss the symplectic structure of the cotangent bundle of a (locally) symmetric space, natural Hamiltonian flows thereon and their (geometric) quantization.
Equivariant degenerations and applications II
In these lectures we shall explore the boundary region between algebraic and differential geometry for spaces possessing non-abelian symmetry. In algebraic geometry, the presence of a non-abelian reductive group acting on a variety gives rise to certain canonical equivariant degenerations of the variety. Moreover, the symmetry provides such control over the degenerations so as to make transitions between algebra-geometric and differential notions, which are often very difficult to handle, actually tractable. Motivating examples of this phenomenon that will be discussed include the switch between Kahler and real polarisations in geometric quantisation, or the link between K-stability and existence of Kahler-Einstein metrics.
Dynamics on locally symmetric spaces III
In this lecture series I will discuss the symplectic structure of the cotangent bundle of a (locally) symmetric space, natural Hamiltonian flows thereon and their (geometric) quantization.
Equivariant degenerations and applications III
In these lectures we shall explore the boundary region between algebraic and differential geometry for spaces possessing non-abelian symmetry. In algebraic geometry, the presence of a non-abelian reductive group acting on a variety gives rise to certain canonical equivariant degenerations of the variety. Moreover, the symmetry provides such control over the degenerations so as to make transitions between algebra-geometric and differential notions, which are often very difficult to handle, actually tractable. Motivating examples of this phenomenon that will be discussed include the switch between Kahler and real polarisations in geometric quantisation, or the link between K-stability and existence of Kahler-Einstein metrics.
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.
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.
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
The seminar is joint with CEAFEL seminar.
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.
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.
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.
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.
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
On work with T Baier, J Hilgert, O Kaya, JP Nunes, M Pereira, P Silva.
Hyperbolicity of projective manifolds I
In this talk we will discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold
We will discuss the key and well understood case of dimension
First session of a short course.
The symplectomorphism groups of rational surfaces
This talk is on the topology of
Stability of Symplectomorphism Groups of Small Rational Surfaces
Let
Hyperbolicity of projective manifolds II
We continue to discuss several ideas and methods used in studying the Kobayshi hyperbolicity of projective manifolds. A manifold
We will discuss the key and well understood case of dimension
This is the second part of a two seminar set, but can be followed independently of the first seminar.
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.
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.
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,