19/01/2021, 17:00 — 18:00 — Online
Justin Sawon, University of North Carolina at Chapel Hill
Lagrangian fibrations by Prym varieties
Lagrangian fibrations on holomorphic symplectic manifolds and orbifolds are higher-dimensional generalizations of elliptic K3 surfaces. They are fibrations whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic form. Markushevich and Tikhomirov described the first example whose fibres are Prym varieties, and their construction was further developed by Arbarello, Ferretti, and Sacca and by Matteini to yield more examples. In this talk we describe the general framework, and consider a new example. We describe its singularities and show that it is a 'primitive' symplectic variety. We also construct the dual fibration, using ideas of Menet. This is joint work with Chen Shen.
12/01/2021, 17:00 — 18:00 — Online
Dusa McDuff, Columbia University
Counting curves and stabilized symplectic embedding conjecture
This is a report on joint work with Kyler Siegel that develops new ways to count $J$-holomorphic curves in $4$-dimensions, both in the projective plane with multi-branched tangency constraints, and in noncompact cobordisms between ellipsoids. These curves stabilize, i.e. if they exist in a given four dimensional target manifold $X$ they still exist in the product $X \times {\mathbb R}^{2k}$. This allows us to establish new cases of the stabilized embedding conjecture for symplectic embeddings of an ellipsoid into a ball (or ellipsoid).
See also
McDuff slides.pdf
05/01/2021, 17:00 — 18:00 — Online
Thibaut Delcroix, Université de Montpellier
On the Yau-Tian-Donaldson conjecture for spherical varieties
I will present how uniform K-stability translates into a convex geometric problem for polarized spherical varieties. From this, we will derive a combinatorial sufficient condition of existence of constant scalar curvature Kahler metrics on smooth spherical varieties, and a complete solution to the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.
See also
notes_Thibaut_Delcroix.pdf
15/12/2020, 17:00 — 18:00 — Online
Simon K. Donaldson, Simons Center for Geometry and Physics Stony Brook and Imperial College London
Co-associative fibrations of $G_{2}$-manifolds and deformations of singular sets
The first part of the talk will review background material on the differential geometry of $7$-dimensional manifolds with the exceptional holonomy group $G_{2}$. There are now many thousands of examples of deformation classes of such manifolds and there are good reasons for thinking that many of these have fibrations with general fibre diffeomorphic to a $K3$ surface and some singular fibres: higher dimensional analogues of Lefschetz fibrations in algebraic geometry. In the second part of the talk we will discuss some questions which arise in the analysis of these fibrations and their "adiabatic limits". The key difficulties involve the singular fibres. This brings up a PDE problem, analogous to a free boundary problem, and similar problems have arisen in a number of areas of differential geometry over the past few years, such as in Taubes' work on gauge theory. We will outline some techniques for handling these questions.
See also
Donaldson slides.pdf
Projecto FCT UIDB/04459/2020.
24/11/2020, 17:00 — 18:00 — Online
Leonardo Macarini, Instituto Superior Técnico and CAMGSD
Dynamical implications of convexity beyond dynamical convexity
We will show sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to furnish new examples of dynamically convex contact forms that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. Other applications are the multiplicity of symmetric non-hyperbolic closed Reeb orbits under suitable pinching conditions and the existence of symmetric elliptic periodic Reeb orbits.
This is ongoing joint work with Miguel Abreu.
See also
Macarini slides.pdf
Projecto FCT UIDB/04459/2020.
17/11/2020, 17:00 — 18:00 — Online
Silvia Anjos, Instituto Superior Técnico and CAMGSD
Loops in the fundamental group of $\mathrm{Symp}(M,\omega)$ which are not represented by circle actions
It was observed by J. Kędra that there are many symplectic $4$-manifolds $(M, \omega)$, where $M$ is neither rational nor ruled, that admit no circle action and $\pi_1 (\operatorname{Ham}( M))$ is nontrivial. In the case $M={\mathbb C\mathbb P}^2\#\,k\overline{\mathbb C\mathbb P}\,\!^2$, with $k \leq 4$, it follows from the work of several authors that the full rational homotopy of $\operatorname{Symp}(M,\omega)$, and in particular their fundamental group, is generated by circle actions on the manifold. In this talk we study loops in the fundamental group of $\operatorname{Symp}_h({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2) $ of symplectomorphisms that act trivially on homology, and show that, for some particular symplectic forms, there are loops which cannot be realized by circle actions. Our work depends on Delzant classification of toric symplectic manifolds and Karshon’s classification of Hamiltonian circle actions.
This talk is based in joint work with Miguel Barata, Martin Pinsonnault and Ana Alexandra Reis.
See also
Silvia slides.pdf
Projecto FCT UIDB/04459/2020.
10/11/2020, 17:00 — 18:00 — Online
Emilio Franco, Instituto Superior Técnico and CAMGSD
Torsion line bundles and branes on the Hitchin system
The locus of the Higgs moduli space fixed under tensorization by a torsion line bundle a key role in the work of Hausel and Thaddeus on topological mirror symmetry. We shall describe the behavior under mirror symmetry of this fixed locus.
See also
Slides of the talk
03/11/2020, 17:00 — 18:00 — Online
Andrew Lobb, Durham University
The rectangular peg problem
For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one.
Joint work with Josh Greene.
Projecto FCT UIDB/04459/2020.
27/10/2020, 17:00 — 18:00 — Online
Yaron Ostrover, Tel Aviv University
On symplectic inner and outer radii of some convex domains
Symplectic embedding problems are at the heart of the study of symplectic topology. In this talk we discuss how to use integrable systems to compute the symplectic inner and outer radii of certain convex domains.
The talk is based on a joint work with Vinicius Ramos.
See also
Ostrover slides.pdf
Projecto FCT UIDB/04459/2020.
20/10/2020, 17:00 — 18:00 — Online
Tristan C. Collins, MIT
SYZ mirror symmetry for del Pezzo surfaces and rational elliptic surfaces
I will discuss some aspects of SYZ mirror symmetry for pairs $(X,D)$ where $X$ is a del Pezzo surface or a rational elliptic surface and $D$ is an anti-canonical divisor. In particular I will explain the existence of special Lagrangian fibrations, mirror symmetry for (suitably interpreted) Hodge numbers and, if time permits, I will describe a proof of SYZ mirror symmetry conjecture for del Pezzo surfaces.
This is joint work with Adam Jacob and Yu-Shen Lin.
See also
Collins slides.pdf
Projecto FCT UIDB/04459/2020.
13/10/2020, 17:00 — 18:00 — Online
Xiuxiong Chen, Stony Brook University
On the space of Kähler metrics
Inspired by the celebrated $C^0, C^2$ and $C^3$ a priori estimate of Calabi, Yau and others on Kähler Einstein metrics, we will present an expository report of a priori estimates on the constant scalar curvature Kähler metrics. With this estimate, we prove the Donaldson conjecture on geodesic stability and the properness conjecture on Mabuchi energy functional.
This is a joint work with Cheng JingRui.
See also
Chen X slides.pdf
Projecto FCT UIDB/04459/2020.
06/10/2020, 17:00 — 18:00 — Online
Éveline Legendre, Université Paul Sabatier
Localizing the Donaldson-Futaki invariant
We will see how to represent the Donaldson-Futaki invariant as an intersection of equivariant closed forms. We will use it to express this invariant as the intersection on some specific subvarieties of the central fibre of the test configuration. As an application we provide a proof that for Kähler orbifolds the Donaldson-Futaki invariant is the Futaki invariant of the central fiber.
See also
Legendre slides.pdf
Projecto FCT UIDB/04459/2020.
29/09/2020, 17:00 — 18:00 — Online
Gonçalo Oliveira, Universidade Federal Fluminense, Brasil
$G_2$-monopoles (a summary)
This talk is aimed at reviewing what is known about $G_2$-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of $G_2$-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in $G_2$-geometry.
See also
Oliveira slides.pdf
Projecto FCT UIDB/04459/2020.
22/09/2020, 17:00 — 18:00 — Online
Yang Li, Institute for Advanced Study
Weak SYZ conjecture for hypersurfaces in the Fermat family
The SYZ conjecture predicts that for polarised Calabi-Yau manifolds undergoing the large complex structure limit, there should be a special Lagrangian torus fibration. A weak version asks if this fibration can be found in the generic region. I will discuss my recent work proving this weak SYZ conjecture for the degenerating hypersurfaces in the Fermat family. Although these examples are quite special, this is the first construction of generic SYZ fibrations that works uniformly in all complex dimensions.
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.
15/09/2020, 11:00 — 12:00 — Online
Robert Berman, Chalmers University of Technology
Kähler-Einstein metrics, Archimedean Zeta functions and phase transitions
While the existence of a unique Kähler-Einstein metrics on a canonically polarized manifold $X$ was established already in the seventies there are very few explicit formulas available (even in the case of complex curves!). In this talk I will give a non-technical introduction to a probabilistic approach to Kähler-Einstein metrics, which, in particular, yields canonical approximations of the Kähler-Einstein metric on $X$. The approximating metrics in question are expressed as explicit period integrals and the conjectural extension to the case of a Fano variety leads to some intriguing connections with Zeta functions and the theory of phase transitions in statistical mechanics.
See also
Notes of the talk
Projecto FCT UIDB/04459/2020.
08/09/2020, 17:00 — 18:00 — Online
Nick Sheridan, University of Edinburgh
Lagrangian cobordism and Chow groups
Homological mirror symmetry predicts an equivalence of categories, between the Fukaya category of one space and the derived category of another. We can “decategorify” by taking the Grothendieck group of these categories, to get an isomorphism of abelian groups. The first of these abelian groups is related, by work of Biran-Cornea, to the Lagrangian cobordism group; the second is related, via the Chern character, to the Chow group. I will define the Lagrangian cobordism and Chow groups (which is much easier than defining the categories). Then I will describe joint work with Ivan Smith in which we try to compare them directly, and find some interesting analogies.
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.
28/07/2020, 17:00 — 18:00 — Online
Mark Gross, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
Intrinsic Mirror Symmetry
I will talk about joint work with Bernd Siebert, proposing a general mirror construction for log Calabi-Yau pairs, i.e., a pair $(X,D)$ with $D$ a “maximally degenerate” boundary divisor and $K_X+D=0$, and for maximally unipotent degenerations of Calabi–Yau manifolds. We accomplish this by constructing the coordinate ring or homogeneous coordinate ring respectively in the two cases, using certain kinds of Gromov-Witten invariants we call “punctured invariants”, developed jointly with Abramovich and Chen.
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.
21/07/2020, 17:00 — 18:00 — Online
Colin Guillarmou, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud
On the marked length spectrum and geodesic stretch in negative curvature
I will review a couple of recent of results proved with T. Lefeuvre and G. Knieper on the local rigidity of the marked length spectrum of negatively curved metrics.
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.
14/07/2020, 17:00 — 18:00 — Online
Tian-Jun Li, University of Minnesota
Symplectic rational $G$-surfaces and the plane Cremona group
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to the one in algebraic geometry. The connection with the symplectic mapping class group will be mentioned.
This is a joint work with Weimin Chen and Weiwei Wu (and partly with Jun Li).
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.
07/07/2020, 17:00 — 18:00 — Online
Rahul Pandharipande, ETH Zürich
Abel-Jacobi maps and the moduli of differentials
The moduli of $(C,f)$ where $C$ is a curve and $f$ is a rational function leads to the well-developed theory of Hurwitz spaces. The study of the moduli of $(C,\omega)$ where $C$ is a curve and $\omega$ is a meromorphic differential is a younger subject. I will discuss recent developments in the study of the moduli spaces of holomorphic/meromorphic differentials on curves. Many of the basic questions about cycle classes and integrals have now been solved (through the work of many people) — but there are also several interesting open directions.
See also
Slides of the talk
Projecto FCT UIDB/04459/2020.