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13:24 Joe Leys, Étienne Ghys, Aurélien Alvarez English 2012

Chaos | Chapter 2 : Vector fields - The lego race

  • Published: 2012
  • Publisher: Joe Leys, Étienne Ghys, Aurélien Alvarez
  • Language: English
13:21 Joe Leys, Étienne Ghys, Aurélien Alvarez English 2012

Chaos | Chapter 7 : Strange Attractors - The butterfly effect

  • Published: 2012
  • Publisher: Joe Leys, Étienne Ghys, Aurélien Alvarez
  • Language: English
1:02:05 Institut des Hautes Études Scientifiques (IHÉS) English 2014

2/4 Automorphic forms in higher rank

  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:15:07 Institut des Hautes Études Scientifiques (IHÉS) English 2015

Quantum Geometry and Strings

  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
49:58 Institut des Hautes Études Scientifiques (IHÉS) English 2013

Counterterms in gravity and N = 8 Supergravity

I will discuss counterterms in gravity using the light-cone frame formulation and show that also in this fully gauge fixed formulation we do need a local symmetry to find the correct counter terms.
  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:41:20 Institut des Hautes Études Scientifiques (IHÉS) English 2014

1/3 Mathematical Physics of Hurwitz numbers

Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'. When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:17:54 Institut des Hautes Études Scientifiques (IHÉS) English 2015

Quantum Field Theory and Gravitation

The incorporation of gravity into quantum physics is still an essentially open problem. Quantum field theory under the influence of an external gravitational field, on the other side, is by now well understood. I is remarkable that, nevertheless, its consistent treatment required a careful revision of traditional quantum field theory in the spirit of algebraic quantum field theory. Moreover, it allows a background independent perturbative construction of quantum gravity as an effective theory.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
56:40 Institut des Hautes Études Scientifiques (IHÉS) English 2013

The correlation numbers in Minimal Liouville gravity

The correlation numbers in Minimal Liouville gravity from Douglas string equation We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. Generalizing the earlier results we demonstrate that there exist such coordinates
  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:25:34 Institut des Hautes Études Scientifiques (IHÉS) English 2015

3/3 Supersymmetric Vacua and Integrability

"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra - all acquire meaning in the supersymmetric gauge theory." 29 avril 2015
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:03:38 Institut des Hautes Études Scientifiques (IHÉS) English 2014

2/3 Mathematical Physics of Hurwitz numbers

Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'. When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:01:45 Institut des Hautes Études Scientifiques (IHÉS) English 2014

4/4 Analytic number theory around torsion homology

  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
45:19 Institut des Hautes Études Scientifiques (IHÉS) English 2013

Equations for stability­

  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:54:58 Institut des Hautes Études Scientifiques (IHÉS) English 2013

3/4 Mathematical Structures arising from Genetics and Molecular Biology

  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:50:24 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/4 Singular support of coherent sheaves

Singular support is an invariant that can be attached to a coherent sheaf on a derived scheme which is quasi-smooth (a.k.a. derived locally complete intersection). This invariant measures how far a given coherent sheaf is from being perfect. We will explain how the subtle difference between "coherent" and "perfect" is responsible for the appearance of Arthur parameters in the context of geometric Langlands correspondence.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:06:56 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/4 Motivic periods and the cosmic Galois group

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights. Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties. Many Feynman amplitudes in quantum field theories are of this type. P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'. One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power. Topics to be addressed will include: 1) A Galois theory of periods, multiple zeta values. 2) Parametric representation of Feyman integrals and their mixed Hodge structures. 3) Operads and the principle of small graphs. 4) The cosmic Galois group: results, counterexamples and conjectures.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:55:07 Institut des Hautes Études Scientifiques (IHÉS) English 2015

4/4 Exponential Integral

  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:54:26 Institut des Hautes Études Scientifiques (IHÉS) English 2015

1/4 Motivic periods and the cosmic Galois group

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights. Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties. Many Feynman amplitudes in quantum field theories are of this type. P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'. One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power. Topics to be addressed will include: 1) A Galois theory of periods, multiple zeta values. 2) Parametric representation of Feyman integrals and their mixed Hodge structures. 3) Operads and the principle of small graphs. 4) The cosmic Galois group: results, counterexamples and conjectures.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:45:37 Institut des Hautes Études Scientifiques (IHÉS) English 2015

1/4 Singular support of coherent sheaves

Singular support is an invariant that can be attached to a coherent sheaf on a derived scheme which is quasi-smooth (a.k.a. derived locally complete intersection). This invariant measures how far a given coherent sheaf is from being perfect. We will explain how the subtle difference between "coherent" and "perfect" is responsible for the appearance of Arthur parameters in the context of geometric Langlands correspondence.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:03:48 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/3 Classical transversality methods in SFT

In this talk I will discuss two transversality results that are standard but perhaps not so widely understood: (1) Dragnev's theorem that somewhere injective curves in symplectizations are regular for generic translation-invariant J, and (2) my theorem on automatic transversality in 4-dimensional symplectic cobordisms (which generalizes earlier results for closed curves by Gromov, Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin). The common feature of these two theorems is that both can be proved by considering the restriction of the usual linearized Cauchy-Riemann operator to the "generalized normal bundle" of a (not necessarily immersed) holomorphic curve.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
04:26 Institut des Hautes Études Scientifiques (IHÉS) English 2015

Le Monde Quantique - Colloque de clôture - Opening remarks in franz. und engl.

  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:20:05 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/3 Supersymmetric Vacua and Integrability

"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra - all acquire meaning in the supersymmetric gauge theory." 22 avril 2015
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:22:26 Institut des Hautes Études Scientifiques (IHÉS) English 2015

1/3 Supersymmetric Vacua and Integrability

"I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra - all acquire meaning in the supersymmetric gauge theory."
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:10:44 Institut des Hautes Études Scientifiques (IHÉS) English 2015

1/2 Introduction to Polyfold Regularization

This lecture will discuss the overall ideas and challenges in regularizing moduli spaces, and introduce the two basic ideas behind polyfold theory: Making reparametrization actions "smooth" and making pregluing a "chart map". [related literature: Sections 2.1 and 3.3 of Polyfolds: A First and Second Look. Related videos: Lecture 8 and Lecture 20 from Wehrheim's special topics course.]
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:10:54 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/3 The rise of sc-retracts

In this talk, we discuss the second of two fundamental analysis concepts polyfold theory is built on: sc-retracts. In particular, we discuss how they arise naturally as a means of using pre-gluing maps to parametrize a neighborhood of nodal and non-nodal (or broken and unbroken) maps near a nodal (or broken) map. Despite locally varying dimensions, such retracts support a version of the sc-calculus on which the chain rule holds, and we define M-polyfolds (manifold-like polyfolds) to be those topological spaces locally modeled on such retracts. [Related literature: Sections 2.3 and 5.1 of Polyfolds: A First and Second Look.]
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
2:20:00 Institut des Hautes Études Scientifiques (IHÉS) English 2015

4/4 Motivic periods and the cosmic Galois group

In the 1990's Broadhurst and Kreimer observed that many Feynman amplitudes in quantum field theory are expressible in terms of multiple zeta values. Out of this has grown a body of research seeking to apply methods from algebraic geometry and number theory to problems in high energy physics. This talk will be an introduction to this nascent area and survey some recent highlights. Most strikingly, ideas due to Grothendieck (developed by Y. André) suggest that there should be a Galois theory of certain transcendental numbers defined by the periods of algebraic varieties. Many Feynman amplitudes in quantum field theories are of this type. P. Cartier suggested several years ago applying these ideas to amplitudes in perturbative physics, and coined the term `cosmic Galois group'. One of my goals will be to describe how to set up such a theory rigorously, define a cosmic Galois group, and explore its consequences and unexpected predictive power. Topics to be addressed will include: 1) A Galois theory of periods, multiple zeta values. 2) Parametric representation of Feyman integrals and their mixed Hodge structures. 3) Operads and the principle of small graphs. 4) The cosmic Galois group: results, counterexamples and conjectures.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:07:38 Institut des Hautes Études Scientifiques (IHÉS) English 2015

2/4 Polyfolds and the construction of Symplectic Field Theory

Topics: 1 Polyfold structures. 2 Consequences of polyfold structures. 3 Weighted categories and their smooth versions. 4 The polyfold of stable maps. 5 Bundle category
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:05:21 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Applications of Equivariant Cohomology

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:16:48 Institut des Hautes Études Scientifiques (IHÉS) English 2013

Wall-crossing and geometry at infinity of Betti moduli spaces

Linear algebraic differential equation (in one variable) depending on a small parameter produces a spectral curve, which is a point in the base of a Hitchin integrable system. Gaiotto, Moore and Neitzke discovered a remarkable structure on the Hitchin base, consisting in certain integer numbers (BPS counting) associated with cycles on the spectral curves, and satisfying universal wall-crossing constraint at hypersurfaces of discontinuity. For a generic spectral curve the wall-crossing structure leads to a preferred coordinate system on the Betti moduli space (a.k.a. the character variety, or the moduli space of monodromy data). I'll speak about interpretation of BPS counting as trees on the Hitchin base, about generalized Strebel differentials and associated quivers, and how one can effectively calculate BPS counting using algebraic curves in Betti moduli spaces.
  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:05:43 Institut des Hautes Études Scientifiques (IHÉS) English 2014

1/2 The orbital circle method and applications, toral eigenfuctions and their nodal sets

  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:59:46 Institut des Hautes Études Scientifiques (IHÉS) English 2013

4/4 Mathematical Structures arising from Genetics and Molecular Biology

  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
52:27 Institut des Hautes Études Scientifiques (IHÉS) English 2013

N = 4 Super Yang-Mills Theory on the Coulomb Branch

I will present a conjecture relating the world-volume action of a D3-brane in an AdS5 X S5 background to the effective action of N = 4 Super Yang-Mills Theory on the Coulomb branch.
  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:07:16 Institut des Hautes Études Scientifiques (IHÉS) English 2014

1/3 Bounded gaps between primes Download

  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:21:11 Institut des Hautes Études Scientifiques (IHÉS) English 2014

3/6 Nilsequences

Classical Fourier analysis has found many uses in additive number theory. However, while it is well-adapted to some pro - blems, it is unable to handle others. For example, if one has a set A, and one wishes to know how many 3-term arithmetic progressions are contained in A, then Fourier analysis is useful, but if one wishes to count 4-term progressions then it is not. For this, and other, problems the more general notion of a nilsequence is required. NIlsequences are a kind of «higher order character» forming the basis of what is becoming known as «higher-order Fourier analysis». The talks will be about this theory.
  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:10:52 Institut des Hautes Études Scientifiques (IHÉS) English 2014

On the cycle class map for zero-cycles over local fields

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.
  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
1:29:19 Institut des Hautes Études Scientifiques (IHÉS) English 2014

4/6 Nilsequences

Classical Fourier analysis has found many uses in additive number theory. However, while it is well-adapted to some pro - blems, it is unable to handle others. For example, if one has a set A, and one wishes to know how many 3-term arithmetic progressions are contained in A, then Fourier analysis is useful, but if one wishes to count 4-term progressions then it is not. For this, and other, problems the more general notion of a nilsequence is required. NIlsequences are a kind of «higher order character» forming the basis of what is becoming known as «higher-order Fourier analysis». The talks will be about this theory.
  • Published: 2014
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
55:24 Institut des Hautes Études Scientifiques (IHÉS) English 2013

Real Time Imaging of Quantum and Thermal Fluctuations

Tremendous progresses have been achieved in the last decade in realising and manipulating stable and controllable quantum systems, and these made possible to experimentally study fundamental questions posed in the early days of quantum mechanics. We shall theoretical discuss recent cavity QED experiments on non- demolition quantum measurements. While they nicely illustrate postulates of quantum mechanics and the possibility to implement efficient quantum state manipulations, these experiments pose a few questions such as: What does it mean to observe a progressive wave function collapse in real time? How to describe it? What do we learn from them? Their analysis will allow us one hand to link these experiments to basics notions of probability or information theory, and on the other hand to touch upon notions of quantum noise. As an illustration, we shall look at quantum systems in contact with a heat bath subject to quantum transitions between energy levels upon absorption or emission of energy quanta. Isolating the two indispensable mechanisms in competition, we shall describe the main physical features of thermally activated quantum jumps.
  • Published: 2013
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
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