In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case.

Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.

This is joint work with Vestislav Apostolov and Lars Sektnan.

In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.

The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.

References

We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:

In the first lecture, I will present the general problem of quantization. Starting from a classical phase space (symplectic manifold), how to define a quantum space (Hilbert space) and transform Hamiltonians into linear operators? I will give details about the left, right, and Weyl quantization on ${\mathbb R}^{2n}$.

In the second lecture, I will show how to develop a full microlocal theory: how to localize quantum mechanics in phase space? Using the notion of semiclassical wavefront set, we obtain a very flexible calculus that can be used to prove Bohr-Sommerfeld rules at any order for a very accurate description of the spectrum of semiclassical operators. I will also give a quick overview of what can be done for non selfadjoint operators using Sjöstrand's theory of complexified phase spaces.

In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.

The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.

References

We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:

In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.

The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.

References

We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:

In the third lecture, I will concentrate on the applications of microlocal analysis to integrable systems. What is the quantum analogue of the classical singular lagrangian foliation? How to compute the joint spectrum of commuting operators? In the case of semitoric sytems, I will present recent results concerning the inverse question: can you recover the symplectic geometry from the quantum spectrum?

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in $P^3$ with $A_n$ singularities and of low degree (low $c_1^2/c_2$). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree $8$ (and potentially $7$). The best known result till date was degree $13$.

Metrics of Poincaré type are Kähler metrics defined on the complement $X\setminus D$ of a smooth divisor $D$ in a compact Kähler manifold $X$ which near $D$ are modeled on the product of a smooth metric on $D$ with the standard cusp metric on a punctured disk in $\mathbb{C}$. In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. I will discuss that even in the toric case, this gives an obstruction to blowing up fixed points (which is unobstructed in the compact case). This additional condition is conjecturally related to ensuring the metrics remain of Poincaré type.

There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there are other Einstein metrics on them that are quite different from the currently-known ones. In this lecture, I will first present a characterization of the known Einstein metrics on Del Pezzo surfaces which I proved several years ago, and then describe an improved version which I obtained only quite recently.

Projecto FCT UID/MAT/04459/2019.

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Room P3.10, Mathematics Building

Adela Mihai, Technical University of Civil Engineering Bucharest

A Riemannian manifold $(M,g)$ of dimension $n \ge 3$ is called an Einstein space if $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, where trivially $\lambda = \kappa$, with $\kappa$ the (normalized) scalar curvature; in this case one easily proves that $\lambda = \kappa = \operatorname{constant}$.

We recall the fact that any $2$-dimensional Riemannian manifold satisfies the relation $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, but the function $\lambda= \kappa $ is not necessarily a constant. It is well known that any $3$-dimensional Einstein space has constant sectional curvature. Thus the interest in Einstein spaces starts with dimension $n=4$.

Singer and Thorpe discovered a symmetry of sectional curvatures which characterizes $4$-dimensional Einstein spaces. Later, this result was generalized to Einstein spaces of even dimensions $n = 2k \ge 4.$ We established curvature symmetries for Einstein spaces of arbitrary dimension $n \ge 4$.

This talk will be concerned with handling problems about embedding Lagrangians in symplectic four-manifolds where the target manifold is rational. In particular, we will determine those three-fold blow-ups of the symplectic ball which admit an embedded Lagrangian projective plane.

Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\).

In this talk we will discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of \(M\) are unimodal, we prove that it is at most \(n+1\) . Moreover, when it is exactly \(n+1\), \(M\) must be homotopically equivalent to \(\mathbb{C}P^n\).

Projecto FCT UID/MAT/04459/2019.

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Room P3.10, Mathematics Building

Richard Laugesen, University of Illinois at Urbana-Champaign

I will discuss two recent theorems and one recent conjecture about maximizing or minimizing the first three eigenvalues of the Robin Laplacian of a simply connected planar domain. Conformal mappings and winding numbers play a key role in the geometric constructions. In physical terms, these eigenvalues represent decay rates for heat flow assuming a “partially insulating” boundary.

I will explain how to make use of geometric methods to obtain three related flavors of contact homology, a Floer theoretic contact invariant. In particular, I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over $\mathbb{Z}$ and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

The study of Legendrian submanifolds in contact geometry presents some similarities with knot theory. In particular, invariants are needed to distinguish Legendrian isotopy classes. Linearized Legendrian contact homology is one of these, and is based on the count of holomorphic curves. It is obtained after linearizing a differential graded algebra using an augmentation. A bilinearized version using two augmentations was introduced with Chantraine.

After a self-contained introduction to this context, the geography of these invariants will be described. In the linearized case, it was obtained with Sabloff and Traynor. The bilinearized case turned out to be far more general and was studied with Galant.

The Reeb flow of a contact form is a generalisation of Hamiltonian flows on energy hypersurfaces in classical mechanics. In this talk I shall address the question of how "complicated" such flows can be. Among other things, I plan to discuss a construction of Reeb flows with a global surface of section on which the Poincaré return map is a pseudorotation. This is joint work with Peter Albers and Kai Zehmisch

Projecto FCT UID/MAT/04459/2019.

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Room P3.10, Mathematics Building

Rui Loja Fernandes, University of Illinois at Urbana-Champaign

In this talk I will give a gentle introduction to Poisson manifolds, which can be thought of as (singular) symplectic foliations. As an illustration of the kind of problems one deals in Poisson geometry, I will discuss and give some results on stability of symplectic leaves.

Projecto FCT UID/MAT/04459/2019.

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Room P3.10, Mathematics Building

Ugo Bruzzo, SISSA, Itália & Universidade Federal da Paraíba, Brazil.

We say that a Higgs bundle $E$ over a projective variety $X$ is curve semistable if for every morphism $f : C \to X$, where $C$ is a smooth irreducible projective curve, the pullback $f^\ast E$ is semistable. We study this class of Higgs bundles, reviewing the status of a conjecture about their Chern classes.

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

The aim of this talk is to discuss the problem of classifying Kaehler-Einstein manifolds which admit an isometric and holomorphic immersion into the complex projective space. We start giving an overview of the problem focusing in particular on the Ricci-flat case. Ricci-flat non-flat Kaehler manifolds are conjectured to be not projectively induced. Next, we give evidence to this conjecture for Calabi’s Ricci-flat metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds.

Classically, there is a strong relationship between the shape of a Riemannian manifold and the spectrum of its Laplace-operator, and, more generally, with the spectrum of Laplace type operators. In particular, for the spectrum of the Laplacian, the presence of a large first nonzero eigenvalue is related to some concentration phenomena. In the first part of the talk, I will recall these classical relations. In the second part of the talk, I will introduce the Dirichlet-to-Neumann operator on a manifold with boundary. I will survey the same types of questions in this context (existence of large first nonzero eigenvalue, concentration phenomena) without going into the technical details. This second part corresponds to work in progress with Alexandre Girouard.

Geometric variational problems frequently lead to analytically extremely hard, non-linear partial differential equations, where the standard methods fail. Thus finding non-trivial solutions is challenging. The idea is to study solutions with a certain minimum level of symmetry (i.e. group actions with low cohomogeneity), and use the symmetry to reduce the original problem to systems of non-linear ordinary differential equations, typically with singular boundary values. In my talk I explain how to construct harmonic mappings between manifolds with a lot of symmetry (i.e. cohomogeneity one manifolds). If time permits, I will discuss applications of the developed methods.