Analyse des EDP, géométrie et physique

5 juin 2023, 08:00 → 9 juin 2023, 18:00 Europe/Paris

Amphis Laurent Schwartz (Toulouse)

Yuxin Ge (Institut Mathématiques Toulouse)

Analyse des EDP, géométrie et physique

Description de la manifestation:

Récemment un grand nombre d’avancées importantes en géométrie différentielle ou en physique mathématique ont été obtenues grâce à des méthodes d’analyse des EDP (que ce soit par des méthodes du calcul des variations, de flots géométriques ou à l’aide de l’analyse micro-locale) ou l’introduction de techniques transverses comme la théorie des systèmes complètement intégrables ou de cadres géométriques (géométrie multisymplectique et formalismes BRST et BV inspirés par des concepts mis au point par les physiciens). Ces progrès concernent une grande variété de problèmes :
- en géométrie, des applications du flot de Ricci, les surfaces et hypersurfaces minimales dans les variétés riemanniennes, les surfaces de Willmore, les applications harmonique,
- en physique mathématique, les équations de la supraconductivité, la théorie quantique des champs sur des espace-temps courbes et sa renormalisation, la structure géométrique des théories de jauge, etc.
L’objectif de ce congrès est de rendre compte de ces avancées récentes et de rassembler des spécialistes d’horizons différents mais connectés à cette variété de techniques et de questions.

Cette conférence internationale sur l’analyse géométrique et de la physique mathématique en l’honneur du Professeur Frédéric Hélein pour ses 60 ans aura lieu du 5 au 9 juin 2023 à Toulouse.

 

Orateurs invités:

Bennequin, D.

Bethuel, F.

Brouder, Ch.

Burstall, F.

Carbou, G.

Coron, J.-M.

De Pauw, T.

Dorfmeister, J. 

Frabetti, A

Khemar, I.

Laurain, P.

Laurent-Gengoux, C.

Leschke, K.

Levy, T. 

Michelat, A.

Pierard de Maujouy, J.

Pacard, F.

Rejzner, K.

Riviere, T. 

Roubtsov, V.

Ryvkin, L.

Stiénon, M.

Strzelecki, P.

Topping, P.

Uhlenbeck, K.

Ye, D. 

 

Le comité scientifique: Alice Chang (Princeton), Frédéric Hélein (Paris), Sylvie Paycha (Potsdam) et  Tillmann Wurzbacher(Metz)

Le comité d'organisation: Yuxin Ge (IMT), Nguyen-Viet Dang (IMJ)

 

Practical information:

The workshop will start on Monday June 5, 2023  and end on Friday June 9, 2023. 

It will take place within the Institut de Mathématiques de Toulouse, in the Amphitheater Laurent Schwartz. A detailed map of campus is available here : Access to campus. The amphitheater is located in building no. 23 on the campus map (rue Sébastienne Guyot).

From downtown, take the Metro B line up to "Université Paul Sabatier" station. The 1R3 mathematics building is two hundred meters away from the Metro station.

All speakers (participants) will be accommodated in Cité internationale Université de Toulouse.

Reception phone number: 0033 (0)532741467

E-Mail:   toulouse.cite@montempo.fr

How to reach Cité internationale Université de Toulouse:

From the train station "Matabiau", take the Metro A line up to "Jean Jaures" (one stop) then take the Metro Line B up to "Palais de Justice"(3 stops);

From the airport "Blagnac", take Tramway T2 up to the last stop "Palais de Justice".  

(see https://www.tisseo.fr/)

There are also taxis at the exit of the airport. The price is around 25-35 euros.

 

Cité internationale Université de Toulouse is an apartment hotel. There is a 24 hour reception service. During the day, the entrance to grande rue Saint-Michel is open. At night and during the day, the main entrance to rue Sainte Catherine is open.

 

poster2023V4.pdf

lundi 5 juin

09:50 Welcome

10:25 Fran Burstall, Eigenfunctions and representation theory

Originally discussed by Strichartz in the ‘80’s, there has been some recent interest from the harmonic map community in eigenfunctions of the Laplacian on symmetric spaces whose squares are also eigenfunctions. The state of the art until recently has been the construction of local examples on classical symmetric spaces. In this talk, I shall show how some simple ideas from representation theory explain the known examples and provide new results.

11:15 Pawel Strzelecki, Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives

Over 30 years ago Frédéric Hélein proved that all harmonic maps from surfaces into compact Rieman- nian manifolds are smooth. Despite the existence of several partial results, for n > 2 the counterpart of this theorem is wide open. In a recent work with two coauthors, Michał Miśkiewicz and Bogdan Petraszczuk, we prove regularity of n-harmonic maps into compact Riemannian manifolds and weak solutions to H-systems in dimension n, under an extra assumption: that n/2-th derivatives of the solution are square integrable. The tools used in the proof involve, as one might guess, Hardy spaces and BMO, and the Rivière–Uhlenbeck decomposition (with estimates in Morrey spaces). A par- ticularly prominent role is played by the Coifman–Rochberg–Weiss commutator theorem.

12:05 Lunch

Esplanade

14:00 Thierry De Pauw, Undecidably semilocalizable metric measure spaces and Radon- Nikodymification of arbitrary measure spaces

The questions raised here grew from the desire to give an integral representation for members of the dual of BV , the Banach space of functions of bounded variation. This potentially has application to the calculus of variations since BV dual contains subgradients of energy functionals to be minimized. The question quickly links to that of identifying the dual of L1(Rm, A, Hm−1) where Hm−1 is the Hausdorff measure. Whether the corresponding canonical map Υ : L∞ → (L1)∗ is injective or not depends upon the σ-algebra A. For A being the σ-algebra of measurable sets in the sense of Caratheodory, the surjectivity of Υ is undecidable in ZFC. This calls for trying to associate with every measure space (X, Σ, μ), in a universal way, a new measure space (Xˆ , Σˆ , mˆu) with respect to which the Radon-Nikodym theorem holds – alternatively such that the corresponding Υˆ is an isometric isomorphism – and L1(X,Σ,μ) ∼= L1(Xˆ,Σˆ,mˆu). I will explain how this is better stated in a specific category whose objects are “measurable spaces with negligibles”. In that context, the existence of the universal “Radon-Nikodymification” is obtained via several applications of Zorn’s Lemma and, therefore, is not much of practical use in general. In a particular case that pertains to BV dual, specifically when μ is an integral geometric measure (instead of Hausdorff measure), I will show that Xˆ can be described explicitly as a fibered space of Rm whose fiber above x consists of germs of rectifiable sets through x. Part of these results have been obtained in collaboration with Philippe Bouafia.

14:55 Thierry Lévy, New variations on old themes

In a first part, I will report on an ongoing collaboration with Adrien Kassel (CNRS, ENS Lyon) where we elaborate on a classical theorem of G. Kirchhoff (1847). In a second part, I will present recent developments due to Frédéric Hélein on an early work of W.A. Mozart (1762).

15:45 Coffee Break

16:20 Fabrice Bethuel, Asymptotics for two-dimensional elliptic Allen-Cahn systems

The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extent related to the absence of known monotonicity formula. I will focus on the elliptic case in two dimensions, and presents some results which extend to the vectorial case in two dimensions most of the results obtained for the scalar case. I will also emphasize some specific features of the vectorial case.

mardi 6 juin

09:00 Frank Pacard, Bouncing Jacobi Fields and the Allen-Cahn equation on surfaces

There is a strong parallel between the theory of minimal hypersurfaces and the solutions of the double-well phase transition Allen-Cahn equation on a manifold. In this talk, I will report some re- cent result on the relations between geodesics on surfaces and solutions of the Allen-Cahn equation with uniform bounds on the Morse Index and energy, as the phase transition parameter tends to 0. Our results show that the situation in 2-dimensions is strikingly different from the situation in dimension 3 and higher. This is a joint work with Juncheng Wei and Yong Liu.

09:50 Coffee Break

10:20 Paul Laurain, Q-Curvature and the Positive Mass Theorem

After a brief review of the classical Positive Mass Theorem and a short introduction to Q-curvature, I will present a theorem on the positive mass for Q-curvature and discuss some applications.

11:15 Jean-Michel Coron, Stability and Boundary stabilization of 1-D hyperbolic systems

Hyperbolic systems in one-space dimension appear in various real-life applications (navigable rivers and irrigation channels, heat exchangers, chemical reactors, gas pipes, road traffic, chromatography, ...). This presentation will focus on the stabilization of these systems by means of boundary control. Stabilizing feedback laws will be constructed. This includes explicit feedback laws that have been im- plemented for the regulation of the rivers La Sambre and La Meuse. The presentation will also cover robustness issues, the case where source terms exist and the case where optimal time stabilisation is considered.

12:05 Lunch

14:00 Karen Uhlenbeck, (zoom) Tight Best-Lipschitz Maps from Surfaces to Surfaces

In honor of the contributions of Frederic Helein to analysis, I pose several new problems rather than present solutions. George Daskalopoulos and I have been studying best Lipschitz maps from surfaces to surfaces. While this is motivated by Thurston’s distance function in Teichmuller space, it has connections with older ideas. I will give a bit of the history about Lipschitz extensions, define the notion of tight, remind the listeners about infinity harmonic functions, and describe our approximation scheme for infinity harmonic mappings. The goal is to motivate several interesting new and I believe hard questions in analysis.

14:55 Katrin Leschke, Periodic Darboux transforms

In classical differential geometry, geometric transformations have been used to create new curves and surfaces from simple ones: the aim is to solve the underlying defining compatibility equations of curve or surface classes by finding solutions to a simpler system of differential equations arising from the transforms. Classically, the main concern was a local theory. In modern theory, global questions have led to a renewed interest in classical transformations. For example, in the case of a torus, the investigation of closing conditions for Darboux transforms naturally leads to the no- tion of the spectral curve of the torus. In this talk we discuss closing conditions for smooth and discrete polarised curves, isothermic surfaces and CMC surfaces. In particular, we obtain new ex- plicit periodic discrete polarised curves, new discrete isothermic tori and new explicit smooth CMC cylinder.

15:45 Coffee Break

16:15 Josef Dorfmeister, The loop group method for harmonic maps with applications to Willmore surfaces

The first part of the talk gives a review of the loop group method for harmonic maps from Riemann surfaces to (semisimple) symmetric spaces. Applications of this will be illustrated by pointing out the relation between certain types of harmonic maps and certain classes of surfaces. In the second part we will describe in some detail the discussion of Willmore surfaces in spheres along the lines of the first part. This will include the occurrence of some unusual features. Finally, time permitting, we will talk about a harmonic map for Willmore surfaces, found by Frederic Helein, which gives an additional avenue for the loop group approach.

17:10 Idrisse Khemar, A Weierstrass type representation of Constrained Willmore sur- faces

In a first part, we present a joint work with Josef Dorfmeister : a Weierstrass type representation for constrained Willmore surfaces in spheres. Using appropriates (moving) frames and a appropriate Lie algebra decomposition of so(1,n+3), we translate the PDE of constrained Willmore surfaces into the Lie algebra setting : namely we rewrite it as a Maurer-Cartan equation of an extended Maurer Cartan form of the frame associated to the surface. Then using some “Bruhat” decomposition of SO(1,3), we consider appropriate versions of the Iwasawa and the Birkhoff decompositions of the loop group. This allows us to construct a DPW algorithm for constrained Willmore surfaces and hence to obtain a Weierstrass type representation for these surfaces in terms of holomorphic (or meromorphic) potentials. In a second part, we characterize the conformal Gauss maps of constrained Willmore surfaces (personal work) : in particular, we generalize a theorem of Dorfmeister and Wang for Willmore surfaces to constrained Willmore surfaces.

18:05 Tristan Rivière, Area Variations under pointwize Lagrangian and Legendrian constraints

mercredi 7 juin

09:00 Camille Laurent-Gengoux, About Koszul-Tate resolutions

I will present several results and open problems about Koszul-Tate resolutions. Several topics will be addressed : how symmetries “closing” only on a singular subset naturally give a Z-graded Q- manifold. with Koszul-Tate negative part, how to get constructive Koszul-Tate resolutions. Then I will detail several open questions about affine varieties. Joints works with Hancharuk, Kotov, Sal- nikov, Strobl.

09:50 Coffee Break

10:25 Alessandra Frabetti, Direct connections on jet groupoids

Gauge fields are the local expression of a principal connection on a principal bundle, and therefore encode the infinitesimal data of a parallel transport between the fibres along curves on the base manifold. One may wonder if there is a field interpretation for parallel transport. Reading principal connections as infinitesimal connections on the associated Atiyah Lie algebroid, this question can be answered by usual integration and gives rise to direct connections on Lie groupoids [A. Kock 1989, N. Teleman 2004, J. Kubarski and N. Teleman 2006]. In this talk we review the basic facts about direct connections on Lie groupoids, together with some interesting examples due to Teleman, and study their jet prolongations on jet groupoids. The talk is based on a work in progress with S. Azzali, Y. Boutaib and S. Paycha.

11:15 Mathieu Stiénon, Formality theorem for differential graded manifolds

The Atiyah class of a dg manifold $(\mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $\mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold $(\mathcal{M},Q)$, there exists an $L_\infty$ quasi-isomorphism of differential graded Lie algebras from the space of polyvector fields on $\mathcal{M}$ endowed with the Schouten bracket $[-,-]$ and the differential $[Q,-]$ to the space of polydifferential operators on $\mathcal{M}$ endowed with the Gerstenhaber bracket $[-,- ]$ and the differential $[m+Q,- ]$, whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold $(\mathcal{M},Q)$ with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we proved the Kontsevich--Shoikhet conjecture: a Kontsevich--Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This last result shows that, when understood in the unifying framework of dg manifolds, the classical Duflo theorem of Lie theory and the Kontsevich--Duflo theorem for complex manifolds are really just one and the same phenomenon.

12:05 Lunch

13:50 Social activities: visit Albi

20:00 Social Dinner

Restaurant "La braisière"

jeudi 8 juin

09:00 Gilles Carbou, Domain wall dynamics in notched ferromagnetic nanowires

Ferromagnetic nanowires have promising applications in data storage. In such devices, the infor- mation is encoded by Domain Walls (DW) which are thin zones of magnetization reversal. The magnetization behavior is described by the non linearLandau-Lifschitz model. In this talk, we in- vestigate the stability of DW configurations. In particular we highlight DW pinning properties by notches patterned along the wire, and DW depinning effect of applied magnetic field.

09:50 Coffee Break

10:20 Daniel Bennequin, (zoom) Singularities of degenerate complex Monge-Ampère equation, ac- cording to Alireza Bahraini

Conjugating a variety of methods, analytic series, Schauder a priori estimates, global continuity, Alireza Barhaini obtained recently a precise description of the singular behavior of the solution of a complex Monge-Ampère equation, degenerating along a smooth divisor in a compact Kähler manifold. Applications are given to the Bogomolov-Miyaoka-Yau inequality and to the Kodaira- Hodge theory on a class of singular complex structures, in view of the proof of the existence of Lagrangian surfaces in projective complex surfaces of the general type.

11:15 Volodya Roubtsov, Multiplicative kernels: from Bessel to Kontsevich, Buchstaber and Calabi-Yau

We discuss few very recent results of a work in progress (in collaboration with I. Gaiur and D. Van Straten and with V. Buchstaber and I. Gaiur) about interesting properties of multiplication Bessel kernels, which includes well-known Clausen and Sonin-Gegenbauer formulae of XIX cen- tury, special examples of Kontsevich discriminant loci polynomials, raised as addition laws for spe- cial two-valued formal groups (Buchshtaber-Novikov-Veselov) and period functions for some CY and Landau–Ginzburg models.

12:05 Lunch

14:00 Christian Brouder, The physical origin of the Batalin-Vilkovisky approach

This talk describes a (hypothetical) historical path that lead Batalin and Vilkovisky to the discovery of their (wonderful) approach, starting from the quantization of scalar fields, to the quantization of electrodynamics and non-Abelian gauge fields.

14:55 Leonid Ryvkin, Darboux theorems for volume-valued symplectic forms

We will discuss a Darboux type theorem for a certain class of multisymplectic (n+2)-form in a (n+2m)- dimensional manifold, generalizing the work for 3-forms in 5-dimensional space. In addition to closedness of the form, the involutivity of a certain distribution is a necessary condiition for flat- ness. When the distribution is zero or everything we recover the classical Moser and Darboux the- orems for volume resp. symplectic forms. This is joint work with Aliaksandr Hancharuk and Laura Leski.

15:45 Coffee Break

16:15 Jérémie Pierard de Maujouy, Hilbert-Einstein Lagrangian on a Generalised Frame Bundle

Einstein’s equation can be obtained as the Euler-Lagrange equation associated to the Hilbert-Einstein Lagrangian, which is essentially the scalar curvature. The curvature tensor, and therefore Einstein’s equation, can be formulated and studied on the frame bundle of spacetime. We will introduce a La- grangian defined on a 10-manifold such that the solutions to the Euler-Lagrange equations equip the manifold with a structure which is almost that of the frame bundle of an Einstein manifold. This will lead us to introduce a structure which generalises that of a frame bundle provided with a principal connection.

17:10 Kasia Rejzner, Non-perturbative renormalisation group from algebraic quantum field theory

In this talk I will present recent results obtained in collaboration with Brunetti, Duetsch and Freden- hagen, concerning the construction of a net of C*-algebras for interacting quantum field theories. Using these construction one can discuss features like the time-slice axiom, action of the appropri- ate version of the renormalisation group and anomalous Nother theorem.

vendredi 9 juin

09:00 Dong Ye, On the understanding of the Jacobian determinant

I will talk about the problem to prescribe the Jacobian determinant or the volume form, I will present some old and more recent results on this problem, and mention the link with the density problem in Sobolev spaces.

09:50 Coffee Break

10:20 Alexis Michelat, Morse Index Stability of Willmore Immersions

The Morse index of a critical point of a Lagrangian L is the dimension of the maximal vector space on which the second derivative D2L is negative-definite. In the classical theory of Hilbert spaces, one shows that the Morse index is lower semi-continuous, while the sum of the Morse index and nullity (the dimension of the Kernel of the differential operator associated to the second derivative) is upper semi-continuous. In a recent work (arXiv:2212.03124), Francesca Da Lio, Matilde Gianocca, and Tristian Rivière (ETH Zürich) developed a new method to show upper semi-continuity results in geometric analysis—that they applied to conformally invariant Lagrangians in dimension 2 (which include harmonic maps). The proof relies on a fine analysis of the second derivative in neck regions— that link the macroscopic surface to its “bubbles”—and a pointwise estimate of the sequence of critical points in the neck regions. In this talk, we will explain how to apply this method to the Willmore en- ergy, a conformally invariant Lagrangian associated to immersions of a surface into Euclidean spaces. Critical points of the Willmore energy—or Willmore immersions—satisfy a non-linear fourth-order elliptic differential equation, and this extension will give rise to several new technical difficulties. If time allows, we will try to show the universal character of this method, that could address (amongst others) the Morse index stability for Ginzburg-Landau energies in dimension 2, bi-harmonic maps in dimension 4, the Yang-Mills functional in dimension 4, and also apply to min-max problems.

11:15 Peter Topping, Pinching theorems via Ricci flow

There is a long tradition in Differential Geometry of results that deduce topological consequences from pointwise positive curvature hypotheses. In this talk we consider the consequences of pinched curvature of various types and explain how recent developments in Ricci flow have proved to be decisive in establishing results along these lines.
Joint work with ManChun Lee (CUHK)

12:05 Lunch