The Arnold conjecture in symplectic topology says that for every Hamiltonian diffeomorphism on a compact symplectic manifold the number of fixed points is at least equal to the minimal number of critical points of a function on the manifold. In my talk I will present an analogue in contact topology of this conjecture, based on the notion of translated points of contactomorphisms, and a work in progress to prove it by constructing a Floer homology theory for translated points. I will also briefly discuss how this is related to some other contact rigidity phenomena, discovered after the work of Eliashberg and Polterovich in 2000, such as the existence of partial orders and biinvariant metrics on the group of contactomorphisms.

The Gromov-Eliashberg Theorem states that the \(C^0\)-limit of a sequence of symplectomorphisms is symplectic. This rigidity phenomenon motivated the study of \(C^0\) symplectic geometry which is concerned with continuous analogs of classical objects. In a joint work with V. Humilière and S. Seyfaddini, we showed that coisotropic submanifolds together with their characteristic foliations are also \(C^0\) rigid. I will discuss this result and in particular I will explain how it relies on continuous analogs of dynamical properties satisfied by coisotropics. Then I will discuss consequences of this rigidity phenomenon.

Floer homology is a powerful tool in symplectic geometry. It was developed by Andreas Floer at the end of the 1980's in order to prove Arnold's conjecture on the number of fixed points of so-called Hamiltonian diffeomorphisms.

In a joint work with Dietmar Salamon and Gregor Noetzel, we had generalized Floer homology to hyperkähler geometry. More precisely, we had defined and computed it on flat, compact hyperkähler manifolds. In contrast to the classical Floer setting where the critical points of the symplectic action functional are periodic Hamiltonian orbits, the critical points of the hypersymplectic action functional are 3-dimensional tori or spheres solving a certain triholomorphic equation. Since the whole setting still seems somewhat mysterious (at least to me), the idea is to come up with another way to describe the solutions.

In this talk, we will give an intuitive introduction to hyperkähler Floer homology. Then we will show, using an explicit construction, that the triholomorphic tori can be seen as periodic solutions of a Hamiltonian system on the iterated loop space.

I will talk about my joint work with Fernando Marques where we used Almgren-Pitts Morse Theory to solve some open problems in geometry.

I will introduce multi-matrix games which include Replicator equations and Bimatrix games . These are ordinary differential equations defined on cartesian products of simplexes. They govern a kind of "replicator" time evolution for groups of players which interact with themselves and other groups. If the loss of one player is the gain of the opponent, called zero-sum games, the system has an integral of motion. I will discuss the question:

Are zero-sum multi-matrix games Hamiltonian?

I will describe work in progress, joint with Dmitri Panov. A definite connection is an \(SO(3)\)-connection over a \(4\)-manifold, whose curvature is non-zero on every tangent \(2\)-plane. Given such a connection, the associated \(2\)-sphere bundle is naturally a symplectic manifold. In this talk I will be interested in definite connections invariant under a circle action, in which case the corresponding symplectic six-manifold is “Fano". I will explain how we hope to classify them, the only possibilities should be connections over \(S^4\) or \(\operatorname{CP}^2\) giving Fanos \(\operatorname{CP}^3\) and the complete flag on \(C^3\) respectively.

We are interested in understanding when a given Fano variety can be realised as a fibre of a Mori fibre space. We are able to provide two criteria, one sufficient and one necessary, which turn into a characterisation in the rigid case. In this talk we will also show how our criteria can be used to give a complete answer in the case of surfaces, an almost complete picture for 3folds and a combinatorial characterisation on the polytope in the toric case. This talk is based on a joint work with Giulio Codogni, Roberto Svaldi and Luca Tasin.

Our aim is to give uniqueness results for entire solutions of certain family of PDEs of divergence form on a parabolic Riemannian manifold of arbitrary dimension. Each equation is the minimal hypersurface equation on certain warped product ambient space. These equations appear from a natural variational problem of geometric interest. Combining geometrical and analytical tools it is presented a technical result from which it is given new Moser's weak Bernstein theorems on parabolic manifolds.

We present a class of isometric embeddings which are fully perturbable and therefore can encode all of the physical degrees of freedom of General Relativity. Assuming a 13+1 dimensional flat ambient space, we can formulate GR, in suitable gauge fixing, as a nonlinear wave equation system in the embedding coordinates. The approach is necessarily background dependent. Some applications and open problems will be discussed.

In 1999 Florian Deloup and I were attempting to find closed formulae for all abelian quantum invariants. These invariants can be expressed in terms of generalized Gauss sums, which depend on a quadratic form obtained from the linking form of the $3$-manifold. Toward this end  we formulated a conjecture that was intended to refine a theorem by Kawauchi and Kojma that demonstrated all linking pairings  on finite abelian groups (i.e. symmetric, non-degenerate bilinear forms into $\mathbb{Q}/\mathbb{Z}$)  arise as a "linking form" of some $3$-manifold. Their construction involves taking the connected sum of three different types of $3$-manifolds. The basis of this theorem was Wall's work and the subsequent work of Kawauchi and Kojma that classified all linking pairings on finite abelian groups.

Our  conjecture that was supposed to refine this theorem stated that any linking pairing on a finite abelian group arises from the linking form of a Seifert fibered rational homology sphere. We proved this result in the case when the abelian group has no $2$-torsion by 2004. In 2010 Jonathan Hillman gave counterexamples in the $2$-torsion case. The underlying reason for the failure of the linking form conjecture is that there are homology cobordism classes of $3$-manifolds that do not contain any Seifert manifolds.

It is possible to reformulate the linking form conjecture so it fulfils its original purpose. A corollary of this "new" linking form "theorem"  is that every homology cobordism class has a representative that arises from a "generalized Seifert presentation".

Furthermore, there are some interesting applications of these abelian quantum invariants to physics.

I will present the main objective of the series of lectures, namely the construction of deformation quantization of certain moduli spaces. In a second part I will present some of the basic notions of derived algebraic geometry, such as derived schemes and derived algebraic stacks.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

In this second lecture I will present more about derived algebraic geometry and will introduce the notion of shifted symplectic structures. I will state several existence theorems and deduce that many moduli spaces, when suitably considered as derived algebraic stacks, are endowed with natural shifted symplectic structures.

This last lecture is concerned with the construction of deformation quantization of moduli spaces endowed with shifted symplectic structures. More generally, I will present the notion of shifted Poisson structures as well as a shifted version of Kontsevich's formality theorem. I will explain how this implies the existence of quantizations. The lecture will end with examples, some recovering well known quantum objects (e.g. quantum groups), and some new.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

The theory of Higgs bundles has attracted a lot of interest since the moduli space of Higgs bundles has a very rich geometry. We will give an overview of the field and we will describe in detail this moduli space when the base variety is an elliptic curve. In this case, expliciteness can be achieved.

Typically, the first step in the quantization of a physical system is finding a Hilbert space whose vectors represent the quantum states of the system. Assuming we understand the classical configuration space, a Riemannian manifold $M$, geometric quantization provides a way to construct this Hilbert space. The Kähler version of geometric quantization constructs the quantum Hilbert space as the space of square integrable holomorphic sections of a certain line bundle over the tangent bundle $T_M$, which is often the same thing as holomorphic $L^2$ functions on $T_M$. For this to be meaningful, one needs to choose a complex structure on $T_M$ and a weight function (because $L^2$ refers to a weighted $L^2$ space).

The talk will discuss my joint results with Szöke on how one can make these choices and whether the quantum Hilbert spaces corresponding to different choices are canonically isomorphic.

After Gromov’s foundational work in in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold $(M,\omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in $(M,\omega)$.

I will discuss techniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in $\mathbb{R}^3$ with edges of lengths $(r_1, ..., r_n )$. Under some genericity assumptions on lengths $r_i$, the polygon space is a symplectic manifold. After introducing this family of manifolds, I will concentrate on the spaces of $5$-gons and calculate their Gromov width. I will also discuss higher dimensional polygon spaces, in particular the $6$-gons case.

This is joint work with Milena Pabiniak, IST Lisbon.

The work of Hausel proves that the Bialynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. These two stratifications do not coincide in general. Here, we give an approach for the rank three case of the classification of the Shatz stratification in terms of the Bialynicki-Birula stratification.