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12/11/2019, 16:30 — 17:30 — Room P3.10, Mathematics Building

Michela Zedda, *Università di Parma*

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Projectively induced Ricci-flat Kaehler metrics

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.

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06/11/2019, 15:00 — 16:00 — Room P4.35, Mathematics Building

Emilio Franco, *CAMGSD, Instituto Superior Técnico*

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Introduction to Mirror Symmetry on the Hitchin System

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.

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01/10/2019, 16:00 — 17:00 — Room P3.10, Mathematics Building

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

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On a conjecture about curve semistable Higgs bundles

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.

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19/07/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Rui Loja Fernandes, *University of Illinois at Urbana-Champaign*

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Stability of symplectic leaves

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.

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17/07/2019, 14:00 — 15:00 — Room P3.10, Mathematics Building

Hansjörg Geiges, *Universität zu Köln*

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Constructions of contact manifolds with controlled Reeb dynamics

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

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17/07/2019, 10:30 — 11:30 — Room P3.10, Mathematics Building

Frédéric Bourgeois, *Université Paris Sud*

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Geography of (bi)linearized Legendrian contact homology

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.

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21/06/2019, 11:00 — 12:00 — Room P3.10, Mathematics Building

Jo Nelson, *Rice University*

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Equivariant and nonequivariant contact homology

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.

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03/06/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Richard Laugesen, *University of Illinois at Urbana-Champaign*

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Spectrum of the Robin Laplacian: recent results, and open problems

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.

Joint with the CEAFEL seminar

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09/05/2019, 11:00 — 12:00 — Room P4.35, Mathematics Building

Isabelle Charton, *University of Cologne*

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Hamiltonian $S^1$-spaces with large equivariant pseudo-index.

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 \(\mathbb{C}P^n\).

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11/04/2019, 14:30 — 15:30 — Room P3.10, Mathematics Building

Gleb Smirnov, *ETH Zurich*

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Symplectic triangle inequality

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.

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14/03/2019, 14:30 — 15:30 — Room P3.10, Mathematics Building

Adela Mihai, *Technical University of Civil Engineering Bucharest*

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On Einstein spaces

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 $n \ge 4$.

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12/03/2019, 12:00 — 13:00 — Room P3.10, Mathematics Building

Claude LeBrun, *Stonybrook*

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Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry

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.

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07/03/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Lars Setktnan, *UQUAM Montréal*

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Blowing up extremal Poincaré type manifolds

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.

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26/02/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Bruno Oliveira, *University of Miami*

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Big jet-bundles on resolution of orbifold surfaces of general type.

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 $13$.

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07/01/2019, 14:00 — 15:00 — Room P3.10, Mathematics Building

Hugues Auvray, *Université Paris-Sud*

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Complete extremal metrics and stability of pairs on Hirzebruch surfaces

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.

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19/12/2018, 16:00 — 17:00 — Room P4.35, Mathematics Building

Gonçalo Oliveira, *Universidade Federal Fluminense*

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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.

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29/11/2018, 16:15 — 17:15 — Room P4.35, Mathematics Building

Dan Avritzer, *Universidade Federal de Minas Gerais*

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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.

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25/09/2018, 16:30 — 17:30 — Room P3.10, Mathematics Building

Levi Lima, *Universidade Federal do Ceará*

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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.

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26/07/2018, 16:30 — 17:30 — Room P3.10, Mathematics Building

Bruno Oliveira, *University of Miami*

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Hyperbolicity of projective manifolds (II)

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.

This is the second part of a two seminar set, but can be followed independently of the first seminar.

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24/07/2018, 16:30 — 17:30 — Room P3.10, Mathematics Building

Martin Pinsonnault, *University of Western Ontario*

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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.