I will sketch some aspects of an interesting Gromov-Witten theory on weighted projective planes introduced by Gross, Pandharipande and Siebert. It admits a very special expansion in terms of tropical counts (called the tropical vertex), as well as a conjectural BPS structure. Then I will describe a refinement or "$q$-deformation" of the expansion using Block-Goettsche invariants, motivated by wall-crossing ideas. This leads naturally to a definition of a class of putative $q$-deformed BPS counts. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined (joint with S. A. Filippini).

I will discuss a generalization of the classical Clifford's theorem
to singular curves, reducible or non reduced. I will prove that for
2-connected curves a Clifford-type inequality holds for a vast set
of torsion free rank one sheaves. I intend to show that our
assumptions on the sheaves are the most natural when working with
this kind of results. I will moreover show that this result has
many applications to the study of the canonical morphism of a
singular curve, in particular that it implies a generalization of
the classical Noether's theorem to 3-connected curves. This is a
joint work with M. Franciosi.

Gromov width of a symplectic manifold $M$ is a supremum of
capacities of balls that can be symplectically embedded into $M$.
The definition was motivated by the Gromov's Non-Squeezing Theorem
which says that maps preserving symplectic structure form a proper
subset of volume preserving maps.

Let $G$ be a compact connected Lie group, $T$ its maximal torus,
and $\lambda $ be a point in the chosen positive Weyl chamber.

The group $G$ acts on the dual of its Lie algebra by coadjoint
action. The coadjoint orbit, $M$, through $\lambda $ is canonically
a symplectic manifold. Therefore we can ask the question of its
Gromov width.

In many known cases the width is exactly the minimum over the
set of positive results of pairing $\lambda $ with coroots of
$G$:

For example, this result holds if $G$ is the unitary group and
$M$ is a complex Grassmannian or a complete flag manifold
satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system
to construct symplectic embeddings of balls. In this way we prove
that the above formula gives the lower bound for Gromov width of
all $U(n)$ and most of $\mathrm{SO}(n)$ coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and
concentrate mostly on the case of regular \(U(n)\) orbits.

We prove that every contact form on the tight three-sphere has at least two geometrically distinct periodic orbits. This result was obtained recently by Cristofaro-Gardiner and Hutchings using embedded contact homology but our approach instead is based on cylindrical contact homology. An essential ingredient in the proof is the notion of a symplectically degenerate maximum for Reeb flows whose existence implies infinitely many prime periodic orbits (in any dimension). This is joint work with V. Ginzburg, D. Hein and U. Hryniewicz.

The linearization theorem for proper Lie groupoids, whose prove was
completed by Zung a few years ago, generalizes various results such
as Ehresmann theorem for submersions, Reeb stability for
foliations, and the Tube Theorem for proper actions. In a work in
progress with R. Fernandes we show that this linearization can be
achieved by means of the exponential flow of certain metrics,
providing both a stronger theorem and a simpler proof. In this talk
I will recall the classic linearization theorems, discuss its
groupoid formulation, and present our work on riemannian structures
for Lie groupoids.

The description of the space of commuting elements in a compact
Lie group is an interesting algebro-geometric problem with
applications in Mathematical Physics, notably in Supersymmetric
Yang Mills theories.

When the Lie group is complex reductive, this space is the
character variety of a free abelian group. Let \(K\) be a compact
Lie group (not necessarily connected) and \(G\) be its
complexiﬁcation. We consider, more generally, an arbitrary
finitely generated abelian group \(A\), and show that the
conjugation orbit space $\mathrm{Hom}(A,K)/K$ is a strong deformation
retract of the character variety $\mathrm{Hom}(A,G)/G$.

As a Corollary, in the case when \(G\) is connected and
semisimple, we obtain necessary and sufficient conditions for
$\mathrm{Hom}(A,G)/G$ to be irreducible. This is also related to an
interesting open problem about irreducibility of the variety of
\(k\) tuples of \(n\) by \(n\) commuting matrices.

Many geometric structures (such as Riemannian, conformal, CR, projective, systems of ODE, and various types of generic distributions) admit an equivalent description as Cartan geometries. For Cartan geometries of a given type, the maximal amount of symmetry is realized by the flat model. However, if the geometry is not (locally) flat, how much symmetry can it have? Understanding this "gap" between maximal and submaximal symmetry in the case of parabolic geometries is the subject of this talk. I'll describe how a combination of Tanaka theory, Kostant's version of the Bott-Borel-Weil theorem, and a new Dynkin diagram recipe led to a complete classification of the submaximal symmetry dimensions in all parabolic geometries of type $(G,P)$, where $G$ is a complex or split-real simple Lie group and $P$ is a parabolic subgroup. (Joint work with Boris Kruglikov.)

Let $(M,\omega)$ be a compact symplectic manifold with $[\omega]$ integral. It is a Theorem of Gromov (1970) and Tischler (1977) that $(M,\omega)$ symplectically embeds into $\mathbb{C}P^n$ with the Fubini-Study symplectic form, for $n$ large enough. Let $\beta_1(M)$ be the first Betti number of $M$. We refine this result of Gromov and Tischler by showing that the weak homotopy type of the space of symplectic embeddings of such a symplectic manifold into $\mathbb{C}P^\infty$ is $(S^1)^{\beta_1(M)}\times\mathbb{C}P^\infty$.

Mirror symmetry is a conjectural relationship between complex and symplectic geometry, and was first noticed by string theorists. Mathematicians became interested in it when string theorists used it to predict counts of curves on the quintic three-fold (just as there are famously 27 lines on a cubic surface, there are 2875 lines on a quintic three-fold, 609250 conics, and so on). Kontsevich conjectured that mirror symmetry should reflect a deeper equivalence of categories: his celebrated Homological Mirror Symmetry conjecture. Most of the talk will be an overview of mirror symmetry with a focus on the symplectic side, leading up to Kontsevich's conjecture. Finally I will describe a proof of Kontsevich's conjecture for the quintic three-fold, and more generally for a Calabi-Yau hypersurface in projective space of any dimension. If time permits I will draw lots of pictures in the one-dimensional case.

In this series of two lectures we will report about the
following Theorem: Let $Z$ be a subvariety of codimension $k>0$
of the moduli space ${\mathcal{M}}_{g}$ of curves of genus $g$ with
$g>3k+4$ and let $\chi :JC\prime \to JC$ an isogeny between two
Jacobians, where $C$ is generic in $Z$. Then $C$ and $C\prime $ are
isomorphic and $\chi $ is the multiplication by an integer. The
statement is also true for $k=1$ and $g\ge 5$. This extends a
result by Bardelli and Pirola for the case $Z={\mathcal{M}}_{g}$ and
$g\ge 4$.

There are two natural approaches to this problem. The first one,
which will be the content of the first talk, uses infinitesimal
variations of Hodge structures in order to translate the statement
into a problem concerning the quadrics containing a canonical
curve, probably of independent interest. The second one consists in
to degenerate to some special (mainly singular stable) curves. This
second approach, much more subtle, works for $g\ge 5$. However it
needs a good control of the intersection of the closure of $Z$ with
the boundary of $\overline{{\mathcal{M}}_{g}}$, and for this we have to
restrict to $k=1$. This procedure will be explained in the second
lecture.

This is a joint work of the speakers with V. Marcucci.

A compact connected semisimple Lie group G acts in a Hamiltonian fashion on its coadjoint orbits, i.e. G maps into the group of Hamiltonian transformations of the coadjoint orbit. McDuff and Tolman showed that the induced map on fundamental groups is injective, answering a question of A. Weinstein. In this talk we will show that this is not quite a symplectic phenomenon, but a topological one, since the (finite) fundamental group of G already injects in the fundamental group of the group of diffeomorphisms. This is joint work with I. Mundet.

Schubert varieties are certain closed subvarieties of the flag
variety of a complex algebraic group \(G\). They are indexed by
elements of the Weyl group \(W\) of \(G\). In general they are
singular and some of the structure of their singularities can be
understood via the resolutions known as the Bott-Samelson
resolutions and the resulting interplay between the intersection
cohomology of the Schubert variety and the ordinary cohomology of
the fibers of these resolutions.

This talk focuses on joint work with my student, Jennifer Koonz,
on some generalizations of the Bott-Samelson resolutions, which
provide some new information about the singularities of Schubert
varieties. Special cases of these resolutions have already appeared
in work by Polo, Wolper, Ryan, and Zelevinsky in the case where
\(G\) is the general linear group \(\operatorname{GL}(n)\) and
\(W\) is the symmetric group \(S_n\). I will explain the
Bott-Samelson resolutions and these new resolutions in some small
examples for the general linear group and perhaps finish with a
couple of propositions that are valid for arbitrary \(G\).

In this talk, we discuss a variant of the Conley conjecture which asserts the existence of infinitely many periodic orbits of a symplectomorphism if it has a fixed point which is unnecessary in some sense. More specifically, we discuss a result claiming that, for a certain class of closed monotone symplectic manifolds, any symplectomorphism isotopic to the identity with a hyperbolic fixed point must necessarily have infinitely many periodic orbits as long as the symplectomorphism satisfies some constraints on the flux.

Let \(V\) be an affine symplectic algebraic variety over \(\mathbb{C}\), and \(G\) a finite group of automorphisms of \(V\) (for example, \(V\) is a symplectic vector space, and \(G\) is a subgroup of \(\mathop{Sp}(V)\)). Let \(A\) be the algebra of regular functions on \(V/G\), and \(E\) be the space of linear functionals on \(A\) which are invariant under Hamiltonian vector fields on \(V/G\) (so called Poisson traces). It turns out that \(E\) is finite dimensional. I will explain how to prove and generalize this statement, using the theory of D-modules, and will also describe some applications to noncommutative algebra. This is joint work with Travis Schedler.

The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the topological generalizations of toric symplectic manifolds and projective toric varieties: quasitoric manifolds, topological toric manifolds and torus manifolds. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, and in the toric case its orbit space is an "origami polytope". In this talk we examine how the topology of a toric origami manifold can be read from its orbit space and how these results hold for the appropriate topological generalization of the class of toric origami manifolds, which includes quasitoric manifolds, and some torus manifolds. These results are from ongoing joint work with Tara Holm.

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3
dimensions. In this talk I shall introduce a generalization of the
monopole equations to both Calabi Yau and \(G_2\) manifolds. I will
motivate the possible relations of conjectural enumerative theories
arising from "counting" monopoles and calibrated cycles of
codimension 3. Then, I plan to state the existence of solutions and
sketch how these examples are constructed.

The Kadomtsev–Petviashvili (KP) equation is an important equation in fluid dynamics: it describes sea waves. It is an integrable model, meaning that we can find "all" solutions. In fact, the solutions are described by a purely geometric object, the grassmaniann. In the case of the equation D-KP (the orthogonal variation of KP) the solutions are described by isotropic grassmannians. In this talk, I'll explain how from an isotropic grassmannian we can build a solution for this equation.

In a moduli space, usually, we impose a notion of stability for
the objects and, when constructing the moduli space by using
Geometric Invariant Theory, another notion of GIT stability
appears, showing during the construction of the moduli that both
notions do coincide at the stable and semistable level. For an
object which is unstable (this is, contradicting the stability
condition) there exists a canonical unique filtration, called the
Harder-Narasimhan filtration. Onthe other habd, GIT stability is
checked by 1-parameter subgroups, by the classical Hilbert-Mumford
criterion, and it turns out that there exists a unique 1-parameter
subgroup giving some notion of maximal unstability in the GIT
sense. We show how to prove that this special 1-parameter subgroup
can be converted into a filtration of the object and coincides with
the Harder-Narasimhan filtration, hence both notions of maximal
unstability are the same, for the moduli problem of classifying
coherent sheaves on a smooth complex projective variety (cf.
[GSZ]). A similar treatment can be used to prove the analogous
correspondence for holomorphic pairs, Higgs sheaves, rank 2 tensors
and finite dimensional quiver representations.

[GSZ] T. Gómez, I. Sols, A. Zamora, A GIT characterization of
the Harder-Narasimhan filtration,
arXiv:1112.1886v2, (Preprint 2012)

We will describe an approach, inspired by geometric quantization, to the complexification of real analytic Hamiltonian flows on Kahler manifolds. The flows correspond to geodesics on the space of Kahler metrics, as suggested by Semmes and Donaldson. We will look at some examples, such as cotangent bundles of compact Lie groups. Based on joint work with J. Mourão.

I will describe joint work with M. Boekstedt and C. Wegner
aiming at uncovering fundamental features of \(N=(2,2)\)
supersymmetric quantum mechanics on moduli spaces of vortices on
compact Riemann surfaces, in analogy with the spectrum of quantum
dyon-monopole bound states that emerged in connection with Sen's
S-duality conjectures in the 1990s. My focus in this talk will be
on the (topological A-twisted) supersymmetric Abelian Higgs model
coupling to local systems, for both linear and nonlinear targets;
the corresponding ground states can be investigated by means of the
theory of \(L^2\)-invariants applied to the natural Kaehler metrics
on the moduli spaces. I shall explain why the quanta of such
Abelian gauge theories can nontrivially realize non-Abelian
statistics in particular examples, and motivate a precise
conjecture regarding the nonlinear superposition of ground
states.