In this talk, we are concerned with the relation between two kinds of canonical Kähler metrics on Fano manifolds, the Calabi's extremal Kähler metrics and the Mabuchi solitons (or generalized Kähler-Einstein metrics in the literature). These are both generalizations of the concept of Kähler-Einstein metrics.

Mabuchi showed that the existence of Mabuchi solitons implies that of extremal Kähler metrics representing the first Chern class. It is also known that the converse is true for Fano manifolds of dimension up to two.

Based on the above, we present examples of Fano manifolds in ALL dimensions greater than two which admit extremal Kähler metrics in every Kähler class, but do not admit Mabuchi solitons.

Abstract: We will discuss smooth embeddings of 3-manifolds in 4-manifolds from both constructive and obstructive perspectives. We will present new results for the pair of Brieskorn spheres and sums of complex projective planes. Our results are based on Kirby calculus and Floer and gauge theoretic cobordism invariants.

For Legendrian submanifolds whose Rabinowitz Floer complex are acyclic we establish a relative Calabi-Yau structure as defined by Brav-Dyckerhoff, that can be seen as a generalisation of Sabloff duality for linearised legendrian contact homology. More precisely, the relative Calabi-Yau structure holds for the DG-morphism given by the inclusion of the DGA of chains on the based loop space of the Legendrian into the Chekanov-Eliashberg algebra of the same, with coefficients in the same DGA. Under certain conditions this can be used to show that the augmentation variety is a holomorphic Lagrangian.

This is joint work with N. Legout.

I will explain the vanishing of contact homology arising from certain contact +1 surgeries, which include all overtwisted manifolds. Combining the surgery cobordism and certain strong cobordisms, we produce contact manifolds with any algebraic (planar) torsion in higher dimensions, which settles a conjecture of Latschev and Wendl.

Many partial differential equations are encoded by proper Fredholm maps between (infinite dimensional) Hilbert spaces. By the Pontryagin-Thom construction these maps correspond to finite dimensional framed submanifolds. This gives a connection between finite and infinite dimensional topology.

In this talk, I will use this relation to classify proper Fredholm maps (up to proper homotopy) between Hilbert spaces in terms of the stable homotopy groups of spheres. This is based on joint work with Thomas Rot.

In this talk I will discuss some results about long time behaviors of solutions to Hamiltonian PDEs (Schrödinger, Quantum Harmonic Oscillator and Schrödinger-Poisson). In particular I will focus on a recent result where we (with J. Bernier and B. Grébert) prove exponential stability of small typical solutions of Schrödinger-Poisson equation by the so-called Rational Normal Form. For these resonant Hamiltonian PDEs the linear frequencies are fully resonant and we have to use the nonlinearity to break the resonances, which leads to a kind of new small divisors compared to Birkhoff Normal Form.

In this talk we will discuss a natural hierarchical structure that governs a vast teritory within the classical harmonic analysis area: (1) non-zero curvature problems: this usually involves the study of objects that lack (generalized) modulation invariance; promi- nent examples within this class are the “curved” Carleson oper- ator and the linear and bilinear Hilbert transforms along “non- flat” curves. (2) zero-curvature problems: this focuses on objects that, on top of the standard dilation and translation symmetries, also ex- hibit a (generalized) modulation invariance; prominent exam- ples within this class are the classical Carleson operator and the Bilinear Hilbert transform. (3) hybrid problems: this refers to the study of objects that share both zero and non-zero curvature feat

Knots have a double nature: according to each one’s preference, they can be considered as topological or as combinatorial objects. Unsurprisingly, they can be studied with a large array of techniques, spanning from algebraic topology to quantum physics. In this talk we will introduce some basic notions of knot theory and explain how different tools can be used for constructing interesting invariants. The driving example will be the Levine-Tristram signature, a classical topological invariant for which a combinatorial construction has been recently conjectured.