Sur deux constructions de la gravité quantique de Liouville
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Title 
Sur deux constructions de la gravité quantique de Liouville

Title of Series  
Part Number 
7

Number of Parts 
17

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

Content Metadata
Subject Area  
Abstract 
We will try to briefly review two recent mathematical constructions of some random measures defined on the Riemann sphere. These objects are motivated by a rigorous description of the Liouville Quantum Gravity (here on the sphere). We will try to compare these two constructions and relate several key elements that appear naturally in both approaches. If time permits, we can also discuss the advantage of each approach. Joint work with Juhan Aru and Xin Sun.

00:00
Classical physics
Surface
Random number
Sheaf (mathematics)
Volume (thermodynamics)
Insertion loss
Sphere
Random measure
Summation
Frequency
Latent heat
Crosscorrelation
Matrix (mathematics)
Insertion loss
Lecture/Conference
Wellformed formula
Gravitation
Compact space
Link (knot theory)
Weight
Sphere
Measurement
Functional (mathematics)
Wellformed formula
Crosscorrelation
Function (mathematics)
Gravitation
Object (grammar)
01:39
Geometry
Point (geometry)
Link (knot theory)
Transformation (genetics)
Volume (thermodynamics)
Automorphism
Theory
Local Group
Measurement
Frequency
Insertion loss
Fiber (mathematics)
Link (knot theory)
Theory of relativity
Automorphism
Moment (mathematics)
Interior (topology)
Infinity
Transformation (genetics)
Limit (category theory)
Cartesian coordinate system
Bilinear form
Automorphism
Plane (geometry)
Computer animation
Phase transition
Units of measurement
Family
Identical particles
Resultant
03:35
Logical constant
Spacetime
Randomization
Group action
Zeitdilatation
Correspondence (mathematics)
Quantum fluctuation
Thermal fluctuations
Embedding
Sheaf (mathematics)
Insertion loss
Sphere
Mereology
Volume
Plane (geometry)
Insertion loss
Dedekind cut
Hausdorff dimension
Physical law
Circle
Vertex (graph theory)
Noise
Bounded variation
Family
Injektivität
Fisher's exact test
Theory of relativity
Process (computing)
Point (geometry)
Moment (mathematics)
Sampling (statistics)
Iterated function system
Thermal expansion
Mereology
Measurement
Bilinear form
Functional (mathematics)
Modulo (jargon)
Sample (statistics)
Bessel function
Quadratic equation
Mathematical singularity
Bounded variation
Point (geometry)
Slide rule
Random number
Free group
Conformal map
Process (computing)
Distribution (mathematics)
Algebraic structure
Modulform
Volume (thermodynamics)
Translation (relic)
Limit (category theory)
Average
Subgroup
Equivalence relation
Axonometric projection
Rule of inference
Random measure
Local Group
Measurement
Frequency
Lecture/Conference
Average
Quotient
Reduction of order
Lie group
Units of measurement
Focus (optics)
Twin prime
Distribution (mathematics)
Surface
Volume (thermodynamics)
Weight
Group action
Limit (category theory)
Sphere
Estimator
Planar graph
Plane (geometry)
Formal power series
Voting
Computer animation
Circle
Field (mathematics)
Dependent and independent variables
Social class
Object (grammar)
Units of measurement
Family
09:56
Spacetime
Group action
Randomization
Zeitdilatation
Sphere
Mereology
Cartesian coordinate system
Explosion
Summation
Casting (performing arts)
Physical law
Noise
Arc (geometry)
Point (geometry)
Infinity
Mereology
Measurement
Flow separation
Modulo (jargon)
Proof theory
Sample (statistics)
Bessel function
Theorem
Point (geometry)
Random number
Free group
Conformal map
Volume (thermodynamics)
Equivalence relation
Axonometric projection
Metric tensor
Measurement
Frequency
Operator (mathematics)
Boundary value problem
Normal (geometry)
Shift operator
Distribution (mathematics)
State of matter
Weight
Sphere
Approximation
Plane (geometry)
Calculation
Summation
Computer animation
Circle
Social class
Units of measurement
00:06
and in the back of of the head is unknown to talk to you about that to some sections of the hearings on this fear resurgence .period nary a soul in his previous start when you have a measure of new Newton you can have insurgents without explaining what is on and knowledge of this In the map well this to constructions of random measure 1 by 1 inspired the decay Oravida interruptions soul they gave explicit formulas with correlation functions they we must have more than 3 insertion .period and suitable for comfort services so you have seen the previous thought that we can construct this kind of fury for a lot of services now there's another construction by DuPont the Meron scheduled for example sphere they have less than 2 insurgents were same weights but in the case of the pure gravity in case where for the specific value of gonna death and matrix instead of measures so have something that is more of more powerful in some sense and have couplings was a classical probability objects is just as and there suitable for noncovered services I hope that the reason why 1 is suitable for Compaq Services and . 1 is suitable for not hapless services will be clear so the goal of our work I'm sorry that presentment too
01:40
often this is a joint where was the 1 who wins and so the goal of our work is to find a
01:47
link between its construction was not clear in the beginning that for example on this feared it would give the same kind of women To begin with wouldn't do something an algebra of geometry of simple geometry so you 1st talk about mortgage transformation while we talk about motivations for missions because all the theory that some I this intuition from the large random and that this should be the limits when you in bed over enlargement the map onto the so in battle Annapolis fear you knew it was a conformal moment so will start with the cultural automorphisms of the despair of the country's supplying all the above motivations forms with fibers on the country .period sir forgery fission somewhere relations between them yes reading reasonable but the 1st exercise defined all transformations that fixed 3 points on so there's trading result liberty you must restrict .period there's only 1 transformation is that solutions the 1st exercise you only have 1 man is the identity solutions the 2nd exercise now if you want to be true .period you have a family of applications for example all applications of this fall act all applications was full of phase 2 . 0 infinity act so I have a large family of applications and so but there were
03:38
no useful version so we can talk about the injection of random act and that the measure on the sphere this conjecture is that if you take a very large random act and issues 3 words as uniformly on the Net In map is reverses 2 0 0 1 Infinity on this then in the limits should get around the measures so this is a simulation truly on if you think about everyone is trying to if you if you take everyone enshrined was to have a voting 1 a urinal lies at the end you would give us some kind random measure on the the point is because of what we have said before if the 3 .period you have 1 random measuring and render wedges welldefined because the embedding is welldefined there's a unique waiting that dramatic honest get so what happens if you want to fix 2 points now the Mannings welldefined because Everett's you can actually pass for embedding has from 1 embedding to amending but not it and push forward the measured by 1 of this amount of attention again adding that so if you want to describe the limits of what is the the limits of the random acts Ashley you must define simultaneously a family of measures and old tune into measures in his family linked by some push forward relations by this action so in the construction if you want to construct this kind of the construction must be invariant under this action if you want to get it assessing and the construction must be influenced us in the beginning so it's not about to constructions that the 1st impression we just so In end the other construction dad response to the tape was 2 .period so this is the part of the little technical just give you some definitions of formalism not playing with this so this was the new refuting the previous talks and the way of adding insertions adding if those wastes are done on a slide as officials free points winches uniformly appointed corresponds to insertion of Wickham so this our .period of insertions and this lot singularities is the picture that Universal with singularity at 3 points so this is still a movie feel with insertions and you can define a volume associated to it by taking the expansion Our lots of focus on the case of unit volume measure so somewhere have to read lies the measure it you have to do some shifting of 2 probability but I'm not entering the tell the point is you have something that well defined here and everything can be calculated for example moments as certain regions for example every estimation it in calculating the explicit form where have 3 points now it's not about a
07:01
school .period construction this is a very interesting construction in the sense that they use sometimes of encoding of of your service you can construct surface by using a basso process somewhere so the idea behind this is that if you take a function that is the final complex plane you can decompose into 2 parts the 1st I would tell you just the average of dysfunction on initially circles the valley averaged over the function of a functional each 1 in circles you don't know much about its function if you want to know entirely the function you must ask legislation but dysfunction on everyone is surfaced so if it combines 2 parts you will get your function or distribution now the point of this construction is that you can sample actually reduce Greece parts in the independent way and we can give explicit constructions explicit ways to sample the now I would not talk about the fluctuation Papas has given by some actress matter but is not very the difficulty and outslugged about this part the average office and with the wife of the construction is invariant under section so what you do well this is a little too hard to read perhaps the 1st meal but when you do we use the latter Basso's 1st and it's taken a lot of these specimens version it looked like something like this is so are you process like that you like something that we call them to sign a drifting running motion it was my brother motion that was once year and wanted now didn't Didier the rule is that we were Paris twice this so that it will look like a brand but if you think about it I can translate my picture that constant and it does not change the fact that has put variations so actually we do this you're defining a process that is invariant underneath on translation so if you if you take this fisher back to sphere the whole plane you're defining an object that is invariant under this because the longer does not depend on the special so
09:57
out of Europe is the following weekend pass from the truth .period definition to frequent definition or with the hospital's reply intervention to support the finish how we do it will use the Web to explain this use our intuition from the random so I recorded 2 points of the case should correspond to the case where you paid too virtues is on a random act in Badakhshan Street and it's free .period case should correspond to wear shoes free assists in Balochistan so what we do is that we take the 2 .period I would take the measure in province has 4 points it will take a 3rd point according this measure it's like as if you're picking a 3rd .period uniformly among all over but this will give you something that is doing the cruelest cast if you want to pass through the measure we must use this kind of push for operation twofaced history .period will fix it to 0 1 infinity you push forward measure by this action and you should get the measure was 4 .period so this is intuition behind what we do but the difficulty here is that this sum for all map it's a little early surrender map actually so the little bit of a nontrivial to characterize role to to strong the consequences of this furor Indonesia from passing to 2 . 3 points is that it takes history .period say with insertion point 0 1 infinity take away 1 of of those points so you forget about the fact that have 1 point as 0 you just say that I have 2 . 0 and a high of 1 . 1 in each of at 2 points as the infinite the Fed forget about .period you're applying this kind of led shifts or you played this action to measure so that in the past Truman's fast and then you get influenced measure up to 2 . construction so
12:12
that's about it's hope this number along and this was a there was a lot of Parliament's work was done and the Newton insisted and I want to thank them for that particular for listening b next to the information and he has said he will result from the database has construction and hold the elections some something about 2 years yes OK if you wish In the beginning or what is year it is this very simple we want to do you take this view you do this the composition and see if you have to say in part for the for the rich it is that the only thing is that this thing is welldefined when you only have to search while should become clear that have 1 measure but several measures so we can't do this for 2 points but if you knew this calculation actually we find something so how about the proof that we do we actually use another kind another destruction Indian mascot they approaches this kind of viewed by approximations so imagine you sphere if you have 3 assertion .period so did take away a little reason you can take away reasonable age and considers kind of G affected finalists the as a it's if the boundary condition here is wellchosen newcomers to so this is the land crucial Antonino proved that we used to identified reports the instead with simple theft comes since