This is a survey talk where we discuss how coadjoint orbits look symplectically, meaning in which cases they are known to coincide with well known/standard symplectic manifolds (e.g, orbits for nilpotent groups and hyperbolic and holomorphic orbits for non-compact semisimple Lie groups)

The importance of Lie algebras in geometry is well-known. However, the role of Jordan algebras in geometry seems less recognized. We will discuss the connections of Jordan algebras to Lie algebras and some recent applications to geometric analysis. This talk is intended for a general audience.

We study a secondary operation (Massey product) associated to a variation of Hodge structure of complex algebraic curves.

The Fujita decomposition of fibered surfaces is the splitting of the 1.0-relative Hodge bundle into a sum of a locally trivial unitary bundle and a positive vector bundle.

We show that the vanishing of the Massey product implies that the monodromy of the unitary part of the Fujita decomposition is finite.

On a projective complex manifold, the Abelian group of Divisors maps onto that of holomorphic line bundles (the Picard group). I shall explain a funny analogue of this for $G_2$-manifolds using coassociative submanifolds to define an analogue of Div, and a gauge theoretical equation for a connection on a gerbe to define an analogue of Pic.

Introduced by Deligne, mixed Hodge structures provide a generalization of the Kähler decomposition of cohomology. They were motivated by Deligne's attempt to assign “virtual” Hodge numbers to non-smooth varieties, a problem in turn related to the Weil conjectures.

I will give a short introduction to mixed Hodge structures, and calculate them in some simple illustrative examples. Afterwards, I will present a sketch of how to obtain these structures for varieties given by quotients of diagonal actions. Finally, I will finish by applying these results to the free abelian character varieties of $\operatorname{GL}(n,\mathbb{C})$ and $\operatorname{SL}(n,\mathbb{C})$.

Some contact manifolds have the property that the Reeb flow of every contact form supporting the corresponding contact structure has positive topological entropy. Examples of such manifolds are given by the unit sphere bundle of rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. In this talk I will discuss the construction of new examples using contact connected sums. If time allows, I will briefly explain how the idea of this construction leads us to the development of a Lagrangian Floer homology on the complement of certain codimension two invariantsubmanifolds in cotangent bundles. This is joint work with Marcelo Alves.

The notion of a free group action on a $C^\ast$-algebra provides a natural framework for noncommutative principal bundles which are not only of purely mathematical interest on their own. As a matter of fact, noncommutative principal bundles become also more and more prevalent in applications to geometry and physics. In this talk we discuss free group actions on $C^\ast$-algebras. In particular, we explain how to classify these actions, with an emphasis on the compact abelian case. If time allows, we further present some applications and open problems.

McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a $4$-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid $E(2,3)$.

This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.

I will start by explaining what mirror symmetry is about, paying special attention to the mirror map which matches up the family of symplectic forms on one manifold with the family of complex structures on another. I will explain how this works for Batyrev's beautiful toric construction of mirror families from dual reflexive polytopes. Then I will give a template for proving cases of Kontsevich's homological mirror symmetry conjecture, based on a versality result for the Fukaya category, which roughly gives a criterion for the existence of a mirror map. The proof can be completed when the reflexive polytope in Batyrev's construction is a simplex: this special case of the construction is due to Greene and Plesser. The latter result is joint work with Ivan Smith.

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

Let $(M,g)$ be a Riemannian manifold and consider a Poisson process generating, in average, $n$ uniformly distributed points in $(M,g)$. In joint work with Omer Bobrowski we answer the following question: As the number, $n$, of points increases what is the smallest possible radius $r$, so that the union of the radius $r$ Riemannian balls centered at the randomly generated points has the same homology as that of the underlying Riemannian manifold $M$.

In terms of a data set lying in a Riemannian manifold, this is similar to asking: what is the minimum we must fatten the data points so that they recover the underlying topology being encoded.

Subvarieties of projective space of low codimension have no symmetric differentials. If we twist the cotangent bundle by multiples of the polarization, \(\Omega^1_X\otimes O(a)\), one still has no sections of its symmetric powers if $a\lt 1$ (result of Schneider 92).

The case $a=1$ is the interesting border case, sections might exist and we call these sections twisted symmetric differentials. We will give a geometric description of the space of twisted symmetric differentials and show that if the codimension of the subvariety $X$ of $P^n$ is small relative to its dimension, $\operatorname{cod}(X)\lt \dim(X)/2$, then the algebra of twisted symmetric differentials of $X$ is the algebra generated by the quadrics containing $X$ (it will be shown for $\operatorname{cod}(X)=2$ and the general case will be discussed).

How small is the smallest period of a closed trajectory of a Reeb flow? In this talk I will present recent answers to instances of this question in three-dimensions which reveal connections between systolic and symplectic geometry. I will present results both of a positive and of a negative nature. Namely, in some situations there are sharp bounds for the systolic ratio, which is defined as the ratio between the square of the smallest period and the contact volume, while in other situations the systolic ratio is unbounded. Our results confirm a conjecture of Babenko and Balacheff and disprove a conjecture of Hutchings. There are implications to middle-dimensional non-squeezing results which we hope to discuss if time permits. All this is joint work with Abbondandolo, Bramham and Salomão.

The resolution of the Yau-Tian-Donaldson conjecture for Fano manifolds, that is, the equivalence of the existence of Kähler-Einstein metrics with K-stability, raises the question of determining when a given Fano manifold is K-stable.

I will present a combinatorial criterion of K-stability for Fano spherical manifolds. These form a very large class of almost-homogeneous manifolds, containing toric manifolds, homogeneous toric bundles, and classes of manifolds for which the Kähler-Einstein existence question was not solved yet, for example equivariant compactifications of (complex) symmetric spaces.