We present an $\operatorname{SO}(2)$-structure and the associated global exterior differential system existing on the contact Riemannian manifold $\cal S$, which is the total space of the tangent sphere bundle, with the canonical metric, of any given $3$-dimensional oriented Riemannian manifold $M$. This is part of a wider theory which can be studied in any dimension. In this seminar we focus on the first interesting dimension and show several new $\operatorname{SU}(2)$-structures on $\cal S$, following the recent ideas introduced by D. Conti and S. Salamon for the study of $5$-manifolds with special metrics.

In 2003, T. Hausel and M. Thaddeus proved that the Hitchin systems on the moduli spaces of $\operatorname{SL}(n,\mathbb{C})$- and $\operatorname{PGL}(n,\mathbb{C})$-Higgs bundles on a curve, verify the requirements to be considered SYZ-mirror partners, in the mirror symmetry setting proposed by Strominger-Yau-Zaslow (SYZ). These were the first non-trivial known examples of SYZ-mirror partners of dimension greater than $2$.

According to the expectations coming from physicists, the generalized Hodge numbers of these moduli spaces should thus agree — this is the so-called topological mirror symmetry. Hausel and Thaddeus proved that this is the case for $n=2,3$ and gave strong indications that the same holds for any $n$ prime (and degree coprime to $n$). In joint work in progress with P. Gothen, we perform a similar study but for parabolic Higgs bundles. We will roughly explain this setting, our study and some questions which naturally arise from it.

Possibly singular Fano varieties which admit Kähler-Einstein metrics are of particular interest since, among other things, they form compact separated moduli spaces. In the seminar, I will talk about existence results for these canonical metrics and describe examples of compact moduli spaces of these special varieties, explaining how the existence and moduli problems are intimately related to each other when looking for explicit examples of such Kähler-Einstein Fano varieties.

Prym varieties are abelian varieties similarly associated to a double covers of algebraic curves as Jacobians are to a curve. In this talk, we define a higher rank analogue of Prym varieties and investigate some of their geometric properties. In particular we are interested in deformation theoretic aspects that permit the construction of a generalized Hitchin's connection in this setting.

This talk is based on joint work in progress with Michele Bolognesi, Johan Martens and Christian Pauly.

In this talk we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces arising from the normal flow and then applying the isoperimetric inequality. Using the same method, we also we give elementary proofs of the classical Minkowski inequalities for closed convex surfaces in the Euclidean 3-space and in the 3-sphere.

I will report on joint work with Jason Lotay on some existence and nonexistence results for $G_2$-instantons. I shall compare the behavior of $G_2$-instantons for two distinct $G_2$-holonomy metrics on $\mathbb{R}^4\times S^3$.

In this talk, we will discuss symplectomorphisms on closed manifolds with periodic orbits. We will present some results on the existence of (infinitely many) periodic orbits of certain symplectomorphisms on closed manifolds. Moreover, we will give a construction of a symplectic flow on a closed surface of genus $g$ greater than $1$ with exactly $2g-2$ fixed points and no other periodic orbits.

Links of Gorenstein toric isolated singularities are good toric contact manifolds with zero first Chern class. In this talk I will present some results relating contact and singularity invariants in this particular toric context. Namely,

I will explain why the contact mean Euler characteristic is equal to the Euler characteristic of any crepant toric smooth resolution of the singularity (joint work with Leonardo Macarini).

I will discuss applications of contact invariants of Lens spaces that arise as links of Gorenstein cyclic quotient singularities (joint work with Leonardo Macarini and Miguel Moreira).

Joint work with Mohammed Abouzaid. We present some methods for constructing examples of compact monotone Lagrangians in families of $(n-1)$-dimensional affine hypersurfaces; they can be upgraded to Lagrangians in $C^n$. We will then explain different strategies for telling them apart, using Floer homology, and, time allowing, some counts of holomorphic annuli. The talk will only assume minimal knowledge of symplectic geometry.

In recent years, a surprising correspondence has been found between the spectral theory of certain trace class operators, and the enumerative geometry of certain Calabi-Yau threefolds. This correspondence leads to a new family of exactly solvable operators in spectral theory, as well as to a new point of view on Gromov-Witten theory. In this overview talk I will introduce the conjecture and some developments inspired by it.

A classical theorem by Mal'cev shows that the only obstruction to embedd a local Lie group to a global Lie group is the failure of (higher) associativity. A theorem of Olver characterizes local Lie groups in terms of Lie groups. We show that both results can be generalized to the setting of local Lie groupoids. More important, we show that (the lack of) associativity is intimately connected to (the lack of) integrability: we give a precise connection in the case of Lie algebroids and we conjecture that it holds also for reasonable infinite dimensional Lie algebras (e.g., Banach Lie algebras). This is joint work with my PhD student Daan Michiels.

Instantons are “finite energy” solutions to a geometric PDE for a connection on a vector bundle. These have their origin in Physics but have also been extensively studied by Mathematicians.

The first instanton on the Euclidean Schwarzschild manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold resisted many efforts for the past 40 years.

I shall explain, how using a duality between vortices and spherically symmetric instantons, Akos Nagy and I, recently gave a complete description of a connected component of the moduli space of unit energy instantons on the Euclidean Schwarzschild manifold. If time permits I will also explain how to use our techniques to:

Find new examples of instantons with non-integer energy;