About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \\u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project.
We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them.
From joint works with Patrick Gérard.[-]

A theorem of Barbot (building on work of Ghys, Haefliger and others) says that Anosov flows on 3-manifolds are classified up to orbit equivalence by the data of a pair of transverse foliations of the plane and an action of the fundamental group of the 3-manifold. In recent work with T. Barthelmé, as well as C. Bonatti, S. Fenley and S. Frankel, we have been developing an abstract theory of Anosov-like group actions of bifoliated planes, applicable both to the study of flows and as an interesting class of foliation-preserving dynamical systems in its own right. This minicourse will explain some of this theory and the connections between flows and group actions in dimensions 1, 2 and 3. [-]

Taut co-orientable foliations are associated with non-trivial elements of Heegard-Floer homology, hence, if a 3-manifold admits a taut, co-oriented foliation, it is not an L-space (Kronheimer-Mrowka-Ozsváth-Szabó). Conjecturally (Boyer-Gordon-Watson, Juhász), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots in S3 is consistent with this conjecture. We consider the L-space Knot Conjecture: if a knot has no reducible or L-space surgeries, then it is persistently foliar, meaning that for each boundary slope there is a taut, co-oriented foliation meeting the boundary of the knot complement in curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a taut, co-oriented foliation in every manifold obtained by Dehn surgery on that knot. I will describe an approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. Recalling the work of Ghiggini, Hedden, Ni, Ozsváth-Szabó (and more recently, Juhász and Baldwin-Sivek) revealed that Dehn surgery on a knot in S3 can yield an L-space only if the knot is fibered and strongly quasipositive, we note that this approach seems to apply more easily when the knot is far from being fibered. As applications of this approach, we find that among the alternating and Montesinos knots, all those without reducible or L-space surgeries are persistently foliar. In addition, we find that any connected sum of alternating knots, Montesinos knots, or fibered knots is persistently foliar. Furthermore, any composite knot with a persistently foliar summand is easily shown to be persistently foliar. This work is joint with Charles Delman.[-]

This two-part tutorial will introduce the framework of conformal prediction, and will provide an overview of both theoretical foundations and practical methodologies in this field. In the first part of the tutorial, we will cover methods including holdout set methods, full conformal prediction, cross-validation based methods, calibration procedures, and more, with emphasis on how these methods can be adapted to a range of settings to achieve robust uncertainty quantification without compromising on accuracy. In the second part, we will cover some recent extensions that allow the methodology to be applied in broader settings, such as weighted conformal prediction, localized methods, online conformal prediction, and outlier detection.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

The density theorem of Schlesinger ensures that the monodromy group of a differential system with regular singular points is Zariski-dense in its differential Galois group. We have analogs of this result for difference systems such as q-difference and Mahler systems, whose only assumption is the regular singular condition. Moreover, solutions of difference or differential systems with regular singularities have good analytical properties. For example, the solutions of differential systems which are regular singular at 0 have moderate growth at 0. We have general algorithms for recognizing regular singularities and they apply to many systems such as differential and q-difference systems. However, they do not apply to the Mahler case, systems that appear in many areas like automata theory. We will explain how to recognize regular singularities of Mahler systems. It is a joint work with Colin Faverjon.[-]