BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Prof Semyon Dyatlov (MIT) DTSTART:20210129T150000Z DTEND:20210129T160000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/1 DESCRIPTION:Title: Control of eigenfunctions on negatively curved surfaces\nby Pr of Semyon Dyatlov (MIT) as part of Open PDE and analysis seminar and lectu res\n\n\nAbstract\nGiven an $L^2$-normalized eigenfunction with eigenvalue $\\lambda^2$ on a compact Riemannian manifold $(M\,g)$ and a non-empty op en subset $\\Omega$ of $M$\, what lower bound can we prove on the $L^2$-ma ss of the eigenfunction on $\\Omega$? The unique continuation principle gi ves a bound for any $\\Omega$ which is exponentially small as $\\lambda$ g oes to infinity. On the other hand\, microlocal analysis gives a $\\lambda $-independent lower bound if $\\Omega$ is large enough\, i.e. it satisfies the geometric control condition. This talk presents a $\\lambda$-independ ent lower bound for any set $\\Omega$ in the case when $M$ is a negatively curved surface\, or more generally a surface with Anosov geodesic flow. The proof uses microlocal analysis\, the chaotic behaviour of the geodesic flow\, and a new ingredient from harmonic analysis called the Fractal Unc ertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves. Joint work with Jean Bourgain\, L ong Jin\, and Stéphane Nonnenmacher.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Prof Eugenia Malinnikova (Stanford) DTSTART:20210212T150000Z DTEND:20210212T160000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/2 DESCRIPTION:Title: On Yau’s conjecture for the Dirichlet Laplacian in C^1 domains\nby Prof Eugenia Malinnikova (Stanford) as part of Open PDE and analysi s seminar and lectures\n\n\nAbstract\nLet D be a bounded domain in R^n wit h C^1 boundary and let u be a Dirichlet Laplace eigenfunction in D with ei genvalue λ. We show that the (n − 1)-dimensional Hausdorff measure of t he zero set of u does not exceed C√λ. The opposite estimate follows fro m the work of Donnelly and Fefferman. The talk is based on a joint work wi th A. Logunov\, N. Nadirashvili\, and F. Nazarov..\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Claude Zuily (Universite Paris-Saclay) DTSTART:20210205T140000Z DTEND:20210205T160000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/3 DESCRIPTION:Title: (LECTURE) Quantitative unique continuation: an introduction. After A.Logunov and E. Malinnikova.\nby Claude Zuily (Universite Paris-Sacl ay) as part of Open PDE and analysis seminar and lectures\n\n\nAbstract\nT he question of the unique continuation from open sets for solutions of ell iptic equations with Lipschitz coefficients as well as its quantitative ve rsion have been positively answered a long time ago mainly using the techn ique of Carleman estimates. The same question where the open set is replac ed by a set of positive measure is more recent. In 2017 A. Logunov and E. Malinnikova introduced new ideas to face this problem. The present lecture is an introduction to their techniques which appear to have applications to the study of the size of the nodal sets of eigenfunctions as well as to control theory.\n\nThis meeting starts at 3pm in France time (9pm ET). He re is the zoom link:\nhttps://univ-cotedazur.zoom.us/j/82084903423?pwd=UUJ IS0ZSUm5lR1FZcDFoWTd4Wis3dz09\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Tarek Elgindi (Duke University) DTSTART:20210423T130000Z DTEND:20210423T150000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/4 DESCRIPTION:Title: (LECTURE) Singularity formation in incompressible fluids\nby T arek Elgindi (Duke University) as part of Open PDE and analysis seminar an d lectures\n\n\nAbstract\nI will discuss various aspects of singularity fo rmation in the incompressible Euler equation in two and three dimensions. In two dimensions\, important questions relate to the infinite time growth of smooth solutions\, filamentation of the vorticity\, and the genericity of this phenomenon. In three dimensions\, we will discuss two methods tha t have been used to rigorously construct finite time singularities. In bot h contexts\, an important theme is the identification of stable growth mec hanisms.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Maciej Zworski (UC Berkeley) DTSTART:20210305T150000Z DTEND:20210305T170000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/5 DESCRIPTION:Title: (LECTURE) Introduction to non-self-adjoint operators: a case stud y using a model of twisted bilayer graphene.\nby Maciej Zworski (UC B erkeley) as part of Open PDE and analysis seminar and lectures\n\n\nAbstra ct\nI will use a simple model from physics \n(Tarnopolsky--Kruchkov--Vishw anath\, 2019) to illustrate the wealth of \nstrange phenomena possible for non-self-adjoint (or rather non-normal) \noperators. The model\, which is a simple operator on the torus\, explains \nthe origin of ``magic angles" in twisted bilayer graphene\, a hot topic \nin physics going by the name of twistronics: when two sheets of graphene \nare twisted at a special ang le\, the material becomes a superconductor. \nBut please do not be scared by the physics: the talk will be an \nelementary blend of spectral theory\ , semiclassical version of \nHörmander's commutator condition\, represent ation theory of the finite \nHeisenberg group\, and theta functions. Easy to state open problems will \nalso be presented and the results will be il lustrated by colorful \nnumerics. Based on joint work with S Becker\, J Wi ttsten and M Embree.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Theodore Drivas (Stony Brooks University) DTSTART:20210416T130000Z DTEND:20210416T140000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/6 DESCRIPTION:Title: (SEMINAR) Some remarks on the long-time dynamics of 2D Euler.\ nby Theodore Drivas (Stony Brooks University) as part of Open PDE and anal ysis seminar and lectures\n\n\nAbstract\nWe describe some known results an d open questions regarding properties of steady solutions of the two-dimen sional incompressible Euler equations\, as well as properties of nearby tr ajectories. Specifically\, we focus on whether steady states can be isolat ed\, whether\, for solutions starting nearby steady states\, recurrence ca n occur or whether singularities must form at long times\, and finally som e results on the infinite-time limit near and far from equilibrium.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Charles Collot (CY Universite) DTSTART:20210409T140000Z DTEND:20210409T150000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/7 DESCRIPTION:Title: Singularities\, separation\, and generic self-similar behaviour fo r the inviscid unsteady Prandtl boundary layer\nby Charles Collot (CY Universite) as part of Open PDE and analysis seminar and lectures\n\n\nAbs tract\nThe inviscid unsteady Prandtl system in two dimensions describes an incompressible non viscous fluid close to a boundary. First\, we will pro ve that the boundary layer separates off the wall if and only if the solut ion becomes singular away from it. Second\, we will present a method to fi nd explicitly backward self-similar solutions forming finite time singular ities. Finally\, we will show that one of such self-similar solution is th e attractor for singular solutions near blow-up time\, in a generic sense (for a dense an open set). This explains the generic appearance of the so- called Van Dommelen and Shen singularity\, and describes completely and ri gorously the associated separating structure. The talk will combine ideas for transport equations\, such as Lagrangian coordinates and incompressibi lity\, and for singularity formation\, such as self-similarity and renorma lisation. This is joint work with T.-E. Ghoul and N. Masmoudi.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Francois Golse (Ecole Polytechnique) DTSTART:20210528T130000Z DTEND:20210528T150000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/8 DESCRIPTION:Title: (Lecture) Optimal Transport Distances in Quantum Mechanics\nby Francois Golse (Ecole Polytechnique) as part of Open PDE and analysis sem inar and lectures\n\n\nAbstract\nThe first part of this talk is focussed o n the definition of \nan extension of the Monge-Kantorovich-Wasserstein di stance of exponent 2 \nto the set density operators\, which correspond to probability measures \nin quantum mechanics. We shall mostly explore the m etric properties of \nthis extension\, in particular compare it with the W asserstein metric \nitself\, and discuss variants of the triangle inequali ty.\n\nThe second part of the talk presents some applications of this noti on of \nquantum Wasserstein distances\, to the uniform convergence of \nti me-splitting schemes in the Planck constant for quantum dynamics\, to \nef fective observation inequalities for the Heisenberg or the Schrödinger \n equations\, and to the uniformity in the Planck constant of convergence \n rates for the mean-field limit in quantum mechanics.\n(Based on a series o f works with E. Caglioti\, C. Mouhot and T. Paul)\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Thierry Paul (CNRS & Ecole Polytechnique) DTSTART:20210611T140000Z DTEND:20210611T150000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/9 DESCRIPTION:Title: (Seminar) Optimal transport and quantum mechanics: more facts and applications\nby Thierry Paul (CNRS & Ecole Polytechnique) as part of Open PDE and analysis seminar and lectures\n\n\nAbstract\nAfter showing th at the extension of the Monge-Kantorovich-Wasserstein distance introduced in the talk by F. Golse is more convenient to separate density matrices th an the usual Schatten topologies usually used in quantum mechanics\, we sh all show how (and explain why) they produce a cost for the quantum biparti te matching problem which is cheapper than the corresponding classical one . We shall then show that a quantum version of the Kantorovich duality pro vides a form of Knott-Smith-Brenier theorem in quantum mechanics\, under t echnical conditions on the density matrices involved\, with a suitable qua ntum definition of the gradient of an observable\, naturally constructed o n the classical one. The finite rank case\, always tractable\, will give r ise itself to a non-gradient «flow » without classical counterpart. Fina lly\, we will study transport associated to a semiquantum analogue of the Wasserstein distances and show that they involve a generalization the Lege ndre transform between classical and quantum densities. (Based on a series of works with E. Caglioti\, F. Golse and C. Mouhot)\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Chiara Saffirio (U. Basel) DTSTART:20210618T130000Z DTEND:20210618T140000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/10 DESCRIPTION:Title: (Seminar) Mean-field evolution of fermionic mixed states with sin gular interaction potentials\nby Chiara Saffirio (U. Basel) as part o f Open PDE and analysis seminar and lectures\n\n\nAbstract\nWe will consid er the many-body evolution of initially \nconfined fermions interacting th rough a singular potential. In a joint \nmean-field and semiclassical scal ing and using second quantization \ntechniques\, we will show that\, for m ixed states enjoying a semiclassical \nstructure\, the many-body dynamics can be approximated in Schatten norms \nby the time-dependent Hartree-Fock equation. In particular\, we will \nhighlight the advantages and drawback s of considering such strong \ntopology instead of the quantum Wasserstein distance introduced in [F. \nGolse\, C. Mouhot and T. Paul\, Commun. Math . Phys. 343\, 165-205 (2016]).\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Cambyze Rouze (U. München) DTSTART:20210618T141500Z DTEND:20210618T150000Z DTSTAMP:20260404T110825Z UID:OpenPDEA/11 DESCRIPTION:Title: (Seminar) Quantum modified logarithmic Sobolev inequalities\n by Cambyze Rouze (U. München) as part of Open PDE and analysis seminar an d lectures\n\n\nAbstract\nFunctional inequalities constitute by now a well -established \ntheory with many connections to other fields of mathematics such as \nconcentration of measure\, mixing times of Markov processes or optimal \ntransport to name only a few. Among these inequalities\, the mo dified \nlogarithmic Sobolev inequality (MLSI) controls the exponential en tropic \nconvergence of a Markov semigroup towards its stationary measure. \nAlthough introduced almost simultaneously\, their quantum analogues hav e \nlong suffered from the loss of certain key properties inherent to the \npassage to the non-commutative realm. Perhaps the most important of \nth ese is the tensorization property\, which often allows one to prove a \nfu nctional inequality for a Markov process on an uncountable state space \nb y reduction to the two-points space.\nDue to the absence of generic tensor ization results for the MLSI in the \nquantum setting\, one is often force d to prove it case by case. However\, \nin the recent years\, a new approa ch to the problem emerged from the \ncommunities of operator algebras and quantum information theory. Here\, \ninstead of proving the tensorization of MLSI for a product of \nsemigroups\, one considers a stronger inequalit y which naturally \ntensorizes\, namely the complete modified logarithmic Sobolev inequality \n(CMLSI). The latter consists in proving the inequalit y for the semigroup \ntensorized with the identity semigroup over an arbit rarily large matrix \nalgebra. The existence of CMLSI for all quantum Mark ov semigroups on \nmatrix algebras was however left as an open conjecture. \nIn this talk\, I will provide a proof of the conjecture for the class of \nreversible quantum Markov semigroups. This talk is intended to be \nsel f-contained and does not require previous knowledge about quantum \nmechan ics or quantum information theory. It is based on a joint work \nwith Li G ao\, a preprint of which is available here: \nhttps://arxiv.org/abs/2102.0 4146.\n LOCATION:https://stable.researchseminars.org/talk/OpenPDEA/11/ END:VEVENT END:VCALENDAR