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Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 200

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We study invariant differential operators on representations of supergroups associated with simple Jordan superalgebras, in the classical case this problem goes back to Kostant. Eigenvalues of Capelli differential operators give interesting families of polynomials such as super Jack polynomials of Sergeev and Veselov and factorial Schur polynomials of Okounkov and Ivanov. We also discuss connection with deformed Calogero-Moser systems in the super case.
We study invariant differential operators on representations of supergroups associated with simple Jordan superalgebras, in the classical case this problem goes back to Kostant. Eigenvalues of Capelli differential operators give interesting families of polynomials such as super Jack polynomials of Sergeev and Veselov and factorial Schur polynomials of Okounkov and Ivanov. We also discuss connection with deformed Calogero-Moser systems in the ...

17B10 ; 17A70 ; 17B60 ; 81T60

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Multi angle  Introduction to quantum optics - Lecture 3
Zoller, Peter (Auteur de la Conférence) | CIRM (Editeur )

Quantum optical systems provides one of the best physical settings to engineer quantum many-body systems of atoms and photons, which can be controlled and measured on the level of single quanta. In this course we will provide an introduction to quantum optics from the perspective of control and measurement, and in light of possible applications including quantum computing and quantum communication.
The first part of the course will introduce the basic quantum optical systems and concepts as ’closed’ (i.e. isolated) quantum systems. We start with laser driven two-level atoms, the Jaynes-Cummings model of Cavity Quantum Electro-dynamics, and illustrate with the example of trapped ions control of the quantum motion of atoms via laser light. This will lead us to the model system of an ion trap quantum computer where we employ control ideas to design quantum gates.
In the second part of the course we will consider open quantum optical systems. Here the system of interest is coupled to a bosonic bath or environment (e.g. vacuum modes of the radiation field), providing damping and decoherence. We will develop the theory for the example of a radiatively damped two-level atom, and derive the corresponding master equation, and discuss its solution and physical interpretation. On a more advanced level, and as link to the mathematical literature, we establish briefly the connection to topics like continuous measurement theory (of photon counting), the Quantum Stochastic Schrödinger Equation, and quantum trajectories (here as as time evolution of a radiatively damped atom conditional to observing a given photon count trajectory). As an example of the application of the formalism we discuss quantum state transfer in a quantum optical network.
Parts of this video related to ongoing unpublished research have been cut off.
Quantum optical systems provides one of the best physical settings to engineer quantum many-body systems of atoms and photons, which can be controlled and measured on the level of single quanta. In this course we will provide an introduction to quantum optics from the perspective of control and measurement, and in light of possible applications including quantum computing and quantum communication.
The first part of the course will introduce the ...

81P68 ; 81V80

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Multi angle  Introduction to quantum optics - Lecture 2
Zoller, Peter (Auteur de la Conférence) | CIRM (Editeur )

Quantum optical systems provides one of the best physical settings to engineer quantum many-body systems of atoms and photons, which can be controlled and measured on the level of single quanta. In this course we will provide an introduction to quantum optics from the perspective of control and measurement, and in light of possible applications including quantum computing and quantum communication.
The first part of the course will introduce the basic quantum optical systems and concepts as ’closed’ (i.e. isolated) quantum systems. We start with laser driven two-level atoms, the Jaynes-Cummings model of Cavity Quantum Electro-dynamics, and illustrate with the example of trapped ions control of the quantum motion of atoms via laser light. This will lead us to the model system of an ion trap quantum computer where we employ control ideas to design quantum gates.
In the second part of the course we will consider open quantum optical systems. Here the system of interest is coupled to a bosonic bath or environment (e.g. vacuum modes of the radiation field), providing damping and decoherence. We will develop the theory for the example of a radiatively damped two-level atom, and derive the corresponding master equation, and discuss its solution and physical interpretation. On a more advanced level, and as link to the mathematical literature, we establish briefly the connection to topics like continuous measurement theory (of photon counting), the Quantum Stochastic Schrödinger Equation, and quantum trajectories (here as as time evolution of a radiatively damped atom conditional to observing a given photon count trajectory). As an example of the application of the formalism we discuss quantum state transfer in a quantum optical network.
Parts of this video related to ongoing unpublished research have been cut off.
Quantum optical systems provides one of the best physical settings to engineer quantum many-body systems of atoms and photons, which can be controlled and measured on the level of single quanta. In this course we will provide an introduction to quantum optics from the perspective of control and measurement, and in light of possible applications including quantum computing and quantum communication.
The first part of the course will introduce the ...

81P68 ; 81V80

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In this lecture I discuss joint work with Eric Delaygue on supercongruences for certain truncated hypergeometric functions. There will also be a discussion of the hypergeometric motives that underlie these congruences.

33C70 ; 11A07 ; 33C20 ; 33C05

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Multi angle  Around Jouanolou-type theorems
Moosa, Rahim (Auteur de la Conférence) | CIRM (Editeur )

In the mid-90’s, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric contexts where model theory plays a role: in complex analytic geometry where it is an old theorem of Krasnov, in algebraic dynamics where it is a theorem of Bell-Rogalski-Sierra, and in meromorphic dynamics where it is a theorem of Cantat. I will report on work-in-progress with Jason Bell and Adam Topaz toward generalising and unifying these statements.
In the mid-90’s, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric contexts where model theory plays a role: in complex analytic geometry where it is an old theorem of Krasnov, in algebraic dynamics where it is a theorem of Bell-R...

03C60 ; 12H05 ; 12L12

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This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to ...

49N45 ; 65C40 ; 65C60 ; 65J22 ; 68U10 ; 62C10 ; 62F15 ; 94A08

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Multi angle  RNA secondary structures
Hofacker, Ivo (Auteur de la Conférence) | CIRM (Editeur )

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The gonality of a variety is defined as the minimal gonality of curve sitting in the variety. We prove that the gonality of a very general abelian variety of dimension $g$ goes to infinity with $g$. We use for this a (straightforward) generalization of a method due to Pirola that we will describe. The method also leads to a number of other applications concerning $0$-cycles modulo rational equivalence on very general abelian varieties.

14C15 ; 14C25 ; 14J70 ; 14J28 ; 14H51 ; 14Kxx

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I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.
I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the ...

11G18 ; 11F11 ; 11G05 ; 14G40

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Given a line bundle $L$ over a real Riemann surface, we study the number of real zeros of a random section of $L$. We prove a rarefaction result for sections whose number of real zeros deviates from the expected one.

32A60 ; 60D05 ; 53C65

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Multi angle  A new Northcott property for Faltings height
Mocz, Lucia (Auteur de la Conférence) | CIRM (Editeur )

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.
The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are ...

14G40 ; 11G50 ; 11R04 ; 12F05

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The plane Cremona group is the group of birational transformations of the projective plane. I would like to discuss why over algebraically closed fields there are no homomorphisms from the plane Cremona group to a finite group, but for certain non-closed fields there are (in fact there are many). This is joint work with Stéphane Lamy.

14E07 ; 14E30

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Simpson’s classic nonabelian Hodge correspondence establishes an equivalence of categories between local systems on a projective manifold, and certain Higgs sheaves on that manifold. This talk surveys recent generalisations of Simpson’s correspondence to the context of projective varieties with klt singularities. Perhaps somewhat surprisingly, these spaces exhibit two correspondences: one pertaining to local systems on the whole space, and one to local systems on its smooth locus. As one application, we resolve the quasi-étale uniformisation problem for minimal varieties of general type, and to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one.
Simpson’s classic nonabelian Hodge correspondence establishes an equivalence of categories between local systems on a projective manifold, and certain Higgs sheaves on that manifold. This talk surveys recent generalisations of Simpson’s correspondence to the context of projective varieties with klt singularities. Perhaps somewhat surprisingly, these spaces exhibit two correspondences: one pertaining to local systems on the whole space, and one ...

14E30 ; 53C07 ; 32Q30 ; 14E20 ; 32Q26

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This talk will be a survey of recent results and methods used in the classification of torsion subgroups of elliptic curves over finite and infinite extensions of the rationals, and over function fields.

11G05 ; 11R21 ; 12F10 ; 14H52

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A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet bundles. Shortly afterwards, Ya Deng obtained effective degree bounds by means of a refined technique. Our goal here will be to explain a drastically simpler proof that yields an improved (though still non optimal) degree bound, e.g. $d_n=[(en)^{2n+2}/5]$. We will also present a more general approach that could possibly lead to optimal bounds.
A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet ...

32Q45 ; 32L10 ; 53C55 ; 14J40

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Multi angle  Geometric recursion
Andersen, Jorgen Ellegaard (Auteur de la Conférence) | CIRM (Editeur )

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form.
Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson ...

51P05 ; 81Q30

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We give a new, more conceptual proof of the Decomposition Theorem for semisimple perverse sheaves of rank-one origin, assuming it for those of constant-sheaf origin, that is, assuming the geometric case proven by Beilinson-Bernstein-Deligne-Gabber. Joint work with Botong Wang.

14C30 ; 14F05 ; 14F43 ; 14D07

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Multi angle  How to make good resolutions
Véber, Amandine (Auteur de la Conférence) | CIRM (Editeur )

In this presentation, we shall discuss the reconstruction of demographic parameters based on the genetic variability observed within a sample of individual DNA. In the family of models that we consider, the statistics describing this genetic diversity (number of mutations, distribution of the mutations amongst individuals in the sample) depend on a more or less coarse ‘resolution of (i.e., level of information on) the hidden genealogical tree that relates the sampled individuals. Considering the optimal resolution thus allows to greatly improve the exploration of the space of possible genealogies when computing the likelihood of demographic parameters, compared to classical methods based on full labelled trees such as Kingmans coalescent. We shall focus on two examples, based on works with Raazesh Sainudiin (Uppsala Univ.) and with Julia Palacios (Stanford Univ.), Sohini Ramachandran (Brown Univ.) and John Wakeley (Harvard Univ.).
In this presentation, we shall discuss the reconstruction of demographic parameters based on the genetic variability observed within a sample of individual DNA. In the family of models that we consider, the statistics describing this genetic diversity (number of mutations, distribution of the mutations amongst individuals in the sample) depend on a more or less coarse ‘resolution of (i.e., level of information on) the hidden genealogical tree ...

92D15 ; 92D20 ; 60J10 ; 60J27

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I shall talk about an old, but not always correctly understood, paper which we wrote with N. Reshetikhin.

82B23 ; 82B20 ; 81T40 ; 81R50

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