1999 seminars

Ana Margarida Melo, Univ. Coimbra
Compactified Picard stacks over the moduli space of curves with marked points

Let $X$ be a projective curve of genus $g$ and $\operatorname{Pic}^d(X)$ its degree $d$ Picard variety, parameterizing isomorphism classes of invertible sheaves of degree $d$ on $X$. $\operatorname{Pic}^d(X)$ is a very important invariant of the curve and it is well known that it is complete if and only if $X$ is a curve of compact type. The problem of compactifying $\operatorname{Pic}^d(X)$ has been widely studied in the last decades and in nowadays there exist several solutions, differing from one another in various aspects such as the geometrical interpretation or the functorial properties.

By the other hand, the introduction of (algebraic) stacks in the study of moduli problems in algebraic geometry allowed a deeper understanding of the objects one wants to parametrize exactly because of the good functorial properties of moduli stacks.

In the first part of my talk I will explain how to construct geometrically meaningful algebraic (Artin) stacks $\overline{\mathcal P}_{d,g}$ over the moduli stack of stable curves, $\overline{\mathcal M}_g$, giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves.

In the secont part, I will explain how to generalize this construction to curves with marked points using an inductive argument that consists, at each step, on giving a geometrical description of the universal family over the previous stack.

Ignasi Mundet i Riera, Univ. Barcelona
Jordan’s theorem for the diffeomorphism group of some manifolds

We will prove that if M is an n-dimensional smooth compact connected manifold, of cup length n, then there exists some constant c such that:

  1. any finite group of diffeomorphisms of M has an abelian subgroup of index at most c,
  2. if the Euler characteristic of M is nonzero,

then no finite group of diffeomorphisms of M has more than c elements. These statements can be seen as analogues of a classical theorem of Jordan for the diffeomorphism group of M (instead of the group GL(n,) as in the original theorem).

Sean Lawton, Maryland and CAMGSD
The topology of the moduli of free group representations

Let $G$ be a complex affine reductive group and let $K$ be a maximal compact subgroup. We have recently proved that the moduli space of representations $\operatorname{Hom}(F,G)/G$ deformation retracts to the quotient space $\operatorname{Hom}(F,K)/K$ for any rank $r$ free group $F$. If $F$ is replaced by other finitely generated groups the theorem may be false, but not always. In this talk we discuss this theorem and some examples.

Michèle Audin, IRMA Strasbourg
Lagrangian skeletons

I will discuss (but not solve) the following question: Given a closed manifold $L$, in which compact symplectic manifold can it be embedded as a Lagrangian submanifold?

Sérgio Mendes, ISCTE, Lisboa
Langlands Functoriality and $K$-Theory for $GL(n)$

We study the $K$-theory of the reduced $C^*$-algebra of $GL(n)$ over a local field $F$ with zero characteristic. For the archimedan case we obtain quite explicit formulas and we use automorphic induction to interpret a curious similarity for the $K$-theory of $GL(n,\mathbb{C})$ and $GL(2n,\mathbb{R})$. When $F$ is nonarchimedean we relate base-change with functoriality of affine buildings

Rahul Pandharipande, Princeton
Algebraic cobordism

I will give an introductory talk about a theory of cobordism for algebraic varieties defined by M. Levine and F. Morel from Quillen's axiomatic perspective. My point of view will be rather concrete with the goal of explaining a new geometric presentation of algebraic cobordism via the simplest normal crossings degenerations. Applications to computations in algebraic geometry will also be discussed. The talk is based on joint work with Levine.

Marina Logares, FC, Univ. Porto
Moduli of parabolic Higgs bundles and the Atiyah algebroid

We will discuss the holomorphic Poisson geometry of the moduli space of parabolic Higgs bundles over a Riemann surface with marked points in terms of Lie algebroids. Time permitting we will also discuss the Hitchin system in this setting and show how this provides a global analogue of the Grothendieck-Springer resolution. This is joint work with Johan Martens.

Olivier Brahic, Instituto Superior Técnico, Lisboa
Extensions of Lie brackets

Lie algebroids provide a rather flexible theory. They are relevant for describing many geometric structures such as Poisson, symplectic, or contact manifolds.

I will expose the notion of extension for Lie algebroids, and explain how they naturally get involved when studying fibrations with an extra structure on the fibers.

Ehresmann connections will be introduced in this context, for which the usual notions of curvature and parallel transport make sense. This approach allows a better understanding of the cohomology of an extension. It also provides a nice way to describe the topological groupoid integrating an extension. We will point out a lack of exactness of the “integration functor”, and explain the geometric counterpart.

Stavros Papadakis, Instituto Superior Técnico
Parallel unprojection of type Kustin-Miller

Unprojection theory, initiated by Miles Reid, aims to construct and analyze complicated commutative rings in terms of simpler ones. The unprojection of type Kustin-Miller is the simplest type of unprojection. It is specified by the data of a Gorenstein local ring $R$ and a codimension $1$ ideal $I$ with the quotient ring $R/I$ being Gorenstein, and constructs a new Gorenstein ring $S$, which geometrically corresponds to the birational contraction of the closed subscheme $V(I)$ of $\operatorname{Spec} R$. The talk will be about recent joint work with Jorge Neves (Coimbra) concerning the parallel unprojection of type Kustin-Miller, which is a generalization corresponding to the case where there are more than one Gorenstein subschemes of $\operatorname{Spec} R$ to be contracted.

Ana Rita Pires, MIT, USA
Symplectic Origami

An origami manifold is a manifold equipped with a closed $2$-form which is symplectic everywhere except on a hypersurface, where it is a folded form whose kernel defines a circle fibration. In this talk, I will explain how an origami manifold can be unfolded into a collection of symplectic pieces and conversely, how a collection of symplectic pieces can be folded (modulo compatibility conditions), into an origami manifold. Using equivariant versions of these operations, we will see how the classic symplectic results of convexity and classification of toric manifolds translate to the origami world. There will be pictures resembling paper origami, but no instructions on how to fold a paper crane. This is joint work with Victor Guillemin and Ana Cannas da Silva.

Thomas Ludsteck, Stuttgart, Germany
Homogeneous p-adic vector bundles on abelian varieties that are analytic tori

In the recent past, there were developed some analogues of parallel transport, a Riemann-Hilbert-correspondence or a Simpson correspondence in $p$-adic Geometry by several authors. We consider in more detail the special case of homogeneous vector bundles on abelian varieties, that are $p$-adic tori, and their relation to representations of $p$-adic fundamental groups.

Bart Van Steirteghem, Medgar Evers College, CUNY
Equivariant degenerations of spherical modules for groups of type $A$

Let $G$ be a complex connected reductive algebraic group. A normal affine $G$-variety $X$ is called spherical if its ring of regular functions $C[X]$ is a multiplicity free $G$-module. A natural question in the study of such varietes is the following: if we only know the $G$-module structure of $C[X]$, how well do we know $X$? Bringing geometry to this question, Alexeev and Brion introduced a certain moduli scheme. After recalling earlier examples of this scheme due to S. Jansou, P. Bravi and S. Cupit-Foutou, I will discuss joint work with S. Papadakis on the case where the module structure of $C[X]$ is that of $C[W]$, where $W$ is a spherical module for a group of type $A$.

Camilo Arias Abad, ETH, Zurich
Representations up to homotopy and cohomology of classifying spaces

Given a Lie group G, Bott proved the formula $H^p(\Omega^q(G_{\bullet}))\cong H^{p-q}(G,S^q({\cal g}^*))$, relating the cohomology of the classifying space BG to the representations in the polynomials in the Lie algebra. I the talk I intend to describe how the same formula holds true for general Lie groupoids, provided one works in the context of representations up to homotopy. If time permits, I will explain how this fomula relates to the models of Getzler and Cartan for equivariant cohomology and also describe some combinatorics which are involved in the construction of tensor products of representations up to homotopy. This is joint work with M. Crainic and B. Dherin.

Giorgio Trentinaglia, Goettingen, Germany
On the role of effective representations of proper Lie groupoids

In this talk, we discuss the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid $G$, the canonical homomorphism of $G$ into the reconstructed groupoid $T (G)$ is surjective, although -contrary to what happens in the case of groups- it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that $G$ may be isomorphic to $T (G)$ and, more generally, in order that $T (G)$ may be a Lie groupoid. We show that if $T (G)$ is a Lie groupoid, the canonical homomorphism $G \to T(G)$ is a submersion and the two groupoids have isomorphic categories of representations.

Florian Schaetz, Zurich
Chain-level Structures for coisotropic Submanifolds

We give an overview of two algebraic structures which can be associated to coisotropic submanifolds: the strong homotopy Lie algebroid on the one hand and the BFV-complex on the other. Both encode the Poisson bracket on the algebra of smooth functions on the quotient space (or a suitable replacement thereof in case the quotient is not smooth). We provide a Theorem which states that the two structures are "isomorphic up to homotopy". Nevertheless there are examples where the BFV-complex contains striclty more information about the Poisson geometry. If time permits connections to the deformation problem of coisotropic submanifolds are outlined.

Chenchang Zhu, Gottingen
Higher structures in geometry

In this talk, I will focus on some first examples of certain higher structures and their impact in geometry. My higher structures here are the $2$-group(oid) structures. They also have their infinitesimal versions, namely the higher Lie algebra(oid)s. In this view point, for example, a group (resp. a higher groupoid) is viewed as a space with a certain higher structure (resp. an even higher structure). The advantages of this new viewpoint will be explained in the second part of my talk.

Margherita Sandon, Instituto Superior Técnico
Contact Non-Squeezing via Generating Functions

In 2006 Eliashberg, Kim and Polterovich discovered an interesting non-squeezing phenomenon for a certain kind of domains in the contact manifold $\mathbb R^{2n} \times S^1$. I will present a new proof of their result, using generating functions instead of holomorphic curves techniques.

Leonardo Macarini, Universidade Federal do Rio de Janeiro
Symplectic homology of Morse-Bott stable hypersurfaces

We study the symplectic homology of compact subsets whose boundary is a stable Morse-Bott hypersurface. Under some conditions on this hypersurface, we conclude the non-triviality of the symplectic homology in certain degrees. Moreover, we address the question of the non-triviality of the monotonicity maps in these degrees.

Roberto Pignatelli, Univ. Trento
Compact complex surfaces of genus 0

There is only one compact complex curve of genus $0$, the Riemann sphere. In the XIXth century M. Noether conjectured a similar statement in dimension $2$: is every compact complex surface without nonzero holomorphic differential forms a rational surface? The first counterexample to this conjecture was given by F. Enriques in 1896. Enriques’ example led to new questions and conjectures about the existence of surfaces without nonzero holomorphic differential forms with special properties. I will report on the history of these questions, and construct some of these surfaces.

Victor Guillemin, Massachussets Institute of Technology
An equivariant "heat-type" trace formula for orbifolds plus some applications to inverse spectral theory

We will discuss some "heat-type" spectral invariants for a pair $(M,f)$ consisting of a compact Riemannian orbifold, $M$, and an isometry, $f$, of $M$, and will show how to obtain inverse results from these invariants involving the "equivariant" spectrum of $M$: the eigenvalues of the Laplace operator plus the representations of the isometry group of $M$ on the eigenspaces. We will also sketch some applications to the theory of toric orbifolds. (This is joint work with Emily Dryden and Rosa Sena-Dias.)

Mark Hamilton, Tokyo University
Toric degeneration and quantization of flag manifolds

Several authors over the past decade or so have developed a "toric degeneration" for flag manifolds, which is a family of projective varieties, parametrized by $C$, whose fibre over 1 is the flag variety and whose fibre over 0 is a toric variety. Nohara, Nishinou and Ueda extend this to a "toric degeneration of integrable systems," which is a degeneration in the above sense that also preserves the structure of an integrable system. They construct a degeneration on flag manifolds that takes the Gelfand-Cetlin integrable system on the flag manifold to the integrable system coming from the torus action on the toric variety.

In a recent paper Baier, Florentino, Mourão, and Nunes study a different kind of deformation: they deform the complex structure on a symplectic toric manifold to obtain a link between the real and Kähler quantization of the manifold. They show that, under this deformation, elements of a canonical basis for the space of holomorphic sections tend to distributional sections supported on Bohr-Sommerfeld fibres. We apply BFMN’s deformation of the complex structure to the toric degeneration considered by NNU, to obtain a direct relationship between the real and Kähler quantizations of the flag variety. This is joint work in progress with Hiroshi Konno.

Henrique Bursztyn, Instituto de Matemática Pura e Aplicada - IMPA
Morita equivalence and characteristic classes of star products

Morita equivalence is an important equivalence relation in algebra and geometry. The talk will discuss the classification of Morita equivalent star products on Poisson manifolds in terms of their characteristic classes. We will show that the correspondence between star products and formal Poisson structures (resulting from Kontsevich's formality theorem) has a suitable equivariance property which allows us to express Morita equivalence of star products in Poisson geometric terms (as a "B-field transform"). This is based on joint work with V. Dolgushev and S. Waldmann.

Carlos Grossi, Max-Planck-Institut, Bonn
Complex hyperbolic disc bundles

Riemannian geometry deals, roughly speaking, with manifolds covered by a model space whose geometric structure is important per se. In (real) dimension $4$, the complex hyperbolic plane is a basic and fundamental model space. We will describe a series of complex hyperbolic disc bundles over closed surfaces that arise from certain discrete groups generated by reflections in complex geodesics in the complex hyperbolic plane. The emphasis will be on the interplay between topology and geometry; for example, we will explain how the Euler number of a bundle can be obtained from certain fractional Euler numbers. Some of our results answer known problems in complex hyperbolic geometry. No previous knowledge of complex hyperbolic geometry is assumed. We will also give a brief and elementary introduction to the topology of disc bundles over closed surfaces. This is part of a joint work with Sasha Ananin (UNICAMP, Brazil) and Nikolay Gusevskii (UFMG, Brazil).

Laura DeMarco, University of Illinois at Chicago
The dynamical shape of a complex polynomial

Iteration of a complex polynomial induces a (singular) metric on the complex plane. I will describe this geometric structure and explain how it can be used to classify topological conjugacy classes in the moduli space of polynomials. I will begin with the basics of polynomial dynamics. This is joint work with Kevin Pilgrim.

Filippo Viviani, Universita degli Studi Roma Tre
Torelli theorem for stable curves

The classical Torelli theorem asserts that a smooth projective curve is determined by its Jacobian together with the principal polar- ization induced by the theta divisor. In modular terms, it asserts that the natural (Torelli) map from the moduli space of smooth projective curves of genus $g$ into the moduli space of principally polarized abelian varieties of dimension $g$ is injective on geometric points. Quite recently, Alexeev has extended the Torelli map, in a geometrically meaningfull way, to modular compactifications of the above moduli spaces, namely the moduli space of Deligne-Mumford stable curves and the moduli space of principally polarized stable semi-abelic pairs. I will report on a joint work with L. Caporaso, in which we study the geometric fibers of the above compactified Torelli map. If time permits, I will also outline some interesting connections with tropical geometry.

William Kirwin, MPI, Leipzig
Adapted Complex Structures and the Geodesic Flow

I will present a new construction of adapted complex structures. Adapted complex structures provide one way to understand the "complexification" of a compact, real-analytic Riemannian manifold $M$. I will explain how a tubular neighborhood of $M$ in its tangent bundle inherits a "canonical" complex structure, the so-called adapted complex structure, and furthermore that this complex structure can be constructed using the "imaginary time" geodesic flow. Time permitting, I will also discuss some applications. (joint w/ Brian Hall)

Vladimir Dragovic, GFMUL
Geometrization and generalization of the Kowalevski top

A new view on the Kowalevski top and the Kowalevski integration procedure is presented. The novelty of our approach is based on the following observations: First, the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski change of variables with the theory of two-valued Buchstaber-Novikov groups.

Carlo Rossi, Instituto Superior Técnico
Formality Theorems and Caldararu's conjecture

Caldararu's conjecture (now a Theorem) states that the Hochschild-Kostant-Rosenberg quasi-isomorphism on a complex manifold $X$, appropriately composed with actions of the Atiyah class of $X$, yields a quasi-isomorphism of dg-algebras between the Dolbeault cohomology of holomorphic polyvector fields/forms on $X$ and the Hochschild (co)homology of $X$: this result can be proved by using methods coming from deformation quantization, i.e. Kontsevich's and Tsygan's formality results.

Carlos Florentino, Instituto Superior Técnico
Topology and singularities of moduli spaces of free group representations

The moduli space $\operatorname{Hom}(F,G)/G$ of completely reducible representations of a finitely generated group $F$ into a Lie group $G$, known as the $G$-character variety of $F$, appears naturally in connection with knot theory, Higgs bundles and quantum field theories. We will discuss the geometry, topology and singularities of these varieties in the case when $G$ is a complex affine reductive Lie group with maximal compact subgroup $K$, and $F$ is a free group of rank $r$. In this situation, one can show that $\operatorname{Hom}(F,G)/G$ and $\operatorname{Hom}(F,K)/K$ have the same homotopy type. Moreover, if $G=SL(n,C)$, these character varieties admit a smooth structure only when $F$ or $G$ is abelian, or $r+n\leq 5$. Moreover, in the cases when $r+n=5$, these moduli spaces have the homotopy type of spheres. This is joint work with S. Lawton (arXiv:0807.3317 and arXiv:0907.4720).

Silvia Sabatini, École Polytechnique Fédérale de Lausanne
GKM spaces and GKM fiber bundles

The equivariant cohomology ring of a Hamiltonian manifold can be a rather difficult object to study. But for a particular category of manifolds, called GKM (Goresky-Kottwitz-MacPherson) spaces, the equivariant cohomology ring can be understood entirely in terms of a graph, the GKM graph. In this talk we introduce their main properties and describe how an equivariant map can be used to simplify the understanding of their cohomology rings. In particular we prove a graph theoretical version of the Serre-Leray theorem, and a generalization of the Chang-Skjelbred Theorem. As an example, we study the projection maps from complete flag varieties to partial flag varieties from this combinatorial perspective.

João Pedro Boavida, Instituto Superior Técnico
What is an Eisentein series

By the end of this talk, we will know how to evaluate the period of an Eisentein series of $O(n,1)$ along $O(n)$. That question will be the pretext to talk about $p$-adic numbers, adeles and ideles, cuspforms, Eisenstein series, automorphic spectral decompositions, and why anyone would want to know the period of an Eisenstein series.

Francisco Presas, Consejo Superior de Investigaciones Científicas, Madrid
Non-complete intersections

For a submanifold $N^k$ to be fiber of a submersion $f: \mathbb{R}^n \to \mathbb{R}^{n-k}$, it is necessary for $N$ to be parallelizable and with trivial normal bundle. If we request $N$ to be just one connected component of a (possibly disconnected) fiber, the $h$-principle applies to provide a complete answer. However, if we ask $N$ to be the whole fiber, then there is no $h$-principle argument. We construct examples of parallelizable, with trivial normal bundle, submanifolds such that they are not fibers of submersions. This is joint work with Daniel Peralta.

Rosa Sena Dias, Instituto Superior Técnico
Scalar flat Kahler metrics on non-compact toric surfaces

In this talk we will discuss a new construction of scalar-flat Kahler toric metrics on non-compact 4-manifolds. The construction actually gives a new perspective on Gibbons-Hawking\'s so called gravitational instantons. Perhaps more interestingly, it allows us to write down some new examples of complete scalar-flat Kahler metrics on some important non-compact toric varieties. We will give some background on such metrics and show how symplectic toric geometry is well suited to tackle them. This is joint work with Miguel Abreu.