An overview of the topological recursion

Video in TIB AV-Portal: An overview of the topological recursion

Formal Metadata

An overview of the topological recursion
Title of Series
Number of Parts
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.
Release Date

Content Metadata

Subject Area
The "topological recursion" defines a double family of "invariants" W associated to a "spectral curve" (which we shall define). The invariants W are meromorphic n-forms defined by a universal recursion relation on |\chi|=2g-2+n, the initial terms W and W being the canonical 1-form and 2-form on the spectral curve. Those invariants have fascinating mathematical properties, they are "symplectic invariants" (invariants under some symplectic transformations of the spectral curve), they are almost modular forms, they satisfy Hirota-like equations, they satisfy some form-cycle duality deformation relations (generalization of Seiberg-Witten), they are stable under many singular limits, and enjoy many other fascinating properties... Moreover, specializations of those invariants recover many known invariants, including Hurwitz numbers to which this conference is dedicated (see M. Kazarian' lecture), intersection numbers, Gromov-Witten invariants, numbers of maps (Tutte's enumeration of maps), or asymptotics of random matrices expectation values. And since very recently, it is conjectured that they also include knot polynomials (Jones, HOMFLY, super polynomials...), which provides an extension of the volume conjecture. We shall present a few examples and mention how these invariants were first discovered in random matrix theory, and then observed or conjectured in many other areas of maths and physics.
Category of being Multiplication sign Prisoner's dilemma Network topology Nichtlineares Gleichungssystem Family Recursion
Spectrum (functional analysis) Topology INTEGRAL State of matter Modulform Algebraic structure Curve Student's t-test Number Sequence Invariant (mathematics) Oval Modulform Series (mathematics) Nichtlineares Gleichungssystem Normal (geometry) Associative property Condition number Addition Surface Moment (mathematics) Algebraic structure Variance Incidence algebra Complete metric space Sequence Meromorphe Funktion Fisher information Plane (geometry) Category of being Number Arithmetic mean Symmetry (physics) Oval Spectrum (functional analysis) Resultant Recursion
Area Polynomial Modulform Knot Algebraic structure Volume (thermodynamics) Curve Enumerated type Number Force Partition (number theory) Number Hyperbolischer Raum Invariant (mathematics) Order (biology) Spacetime Nichtlineares Gleichungssystem Extension (kinesiology) Volume Combinatorics Singuläres Integral
Geometry Functional (mathematics) Principal ideal domain Uniqueness quantification Coordinate system Barrelled space Volume (thermodynamics) Curve Number Sign (mathematics) Boundary value problem Modulform Spacetime Right angle Summierbarkeit Musical ensemble Volume Resultant Spacetime
Functional (mathematics) INTEGRAL Moment (mathematics) Algebraic structure Volume (thermodynamics) Curve Power (physics) Oval Modulform Spacetime Table (information) Volume Singuläres Integral
Statistical hypothesis testing Topology Group action Correspondence (mathematics) Direction (geometry) Barrelled space Curve Open set Special unitary group Fraction (mathematics) Term (mathematics) Boundary value problem Modulform Spacetime Nichtlineares Gleichungssystem Dean number Dependent and independent variables Geometry Sine Volume (thermodynamics) Hyperbolischer Raum Theorem Right angle Volume Resultant Recursion Singuläres Integral
Category of being Hyperbolischer Raum Topology Geometry Sine Term (mathematics) Indirekter Beweis Theorem Boundary value problem Nichtlineares Gleichungssystem Series (mathematics) Recursion
Point (geometry) Computer programming Functional (mathematics) Divisor Multiplication sign Modulform 1 (number) 3 (number) Mereology Grothendieck topology Number Product (business) Explosion Sign (mathematics) Mathematics Plane (geometry) Different (Kate Ryan album) Natural number Complex number Square number Modulform Noise (electronics) Process (computing) Differential (mechanical device) Sine Military base Forcing (mathematics) Projective plane Physical law Thermal expansion Riemann sphere Residual (numerical analysis) Arithmetic mean Symmetric matrix Symmetry (physics) Tower Hill differential equation Right angle Recursion Maß <Mathematik>
Covering space Functional (mathematics) Differential (mechanical device) Point (geometry) Modulform Riemann sphere Cartesian coordinate system Wave Symmetric matrix Many-sorted logic Function (mathematics) Right angle Diagonal Nichtlineares Gleichungssystem Pole (complex analysis) Recursion
Point (geometry) Surface Spectrum (functional analysis) Topology Functional (mathematics) Modulform Curve Open set Analytic set Event horizon Number Frequency Plane (geometry) Square number Pole (complex analysis) Local ring Wave Symmetric matrix Symmetry (physics) Function (mathematics) Time evolution Order (biology) Diagonal Fundamental theorem of algebra Recursion
Axiom of choice Surface Spectrum (functional analysis) Topology Functional (mathematics) Point (geometry) Modulform Curve Analytic set Equivalence relation Symmetric matrix Many-sorted logic Function (mathematics) Manifold Modulform Right angle Diagonal Local ring Pole (complex analysis) Family Recursion Condition number
Resolvent formalism Model theory Set (mathematics) Curve Analytic set Perspective (visual) Cartesian coordinate system Measurement Expected value Fraction (mathematics) Many-sorted logic Dedekind cut Different (Kate Ryan album) Matrix (mathematics) Pole (complex analysis) Polynomial Constraint (mathematics) Interior (topology) Thermal expansion Complete metric space Asymptote Sequence Symmetric matrix Symmetry (physics) Oval Right angle Summierbarkeit Figurate number Resultant Recursion Probability distribution Spectrum (functional analysis) Topology Random number Functional (mathematics) Constraint (mathematics) Connectivity (graph theory) Modulform Partition function (statistical mechanics) Product (business) Cross-correlation Operator (mathematics) Symmetry (physics) Modulform Genetic programming Nichtlineares Gleichungssystem Game theory Condition number Newton's law of universal gravitation Virasoro-Algebra Model theory Algebraic structure Correlation and dependence Vector potential Population density Voting Function (mathematics) Diagonal Musical ensemble Coefficient Matrix (mathematics) Spectrum (functional analysis) Maß <Mathematik>
Topology Observational study INTEGRAL Model theory Constraint (mathematics) Curve Power (physics) Number Goodness of fit Invariant (mathematics) Dedekind cut Term (mathematics) Different (Kate Ryan album) Symmetry (physics) Matrix (mathematics) Nichtlineares Gleichungssystem Summierbarkeit Polynomial Virasoro-Algebra Model theory Moment (mathematics) Multiple Regression Category of being Physicist Right angle Matrix (mathematics) Recursion
Geometry Group representation Boundary value problem Nichtlineares Gleichungssystem Line (geometry) Metric system Curve fitting Group representation
Dependent and independent variables Expression Optics
Geometry Numerical digit Group action Existence Forcing (mathematics) Weight Prisoner's dilemma Moment (mathematics) Line (geometry) Number Subset Prime ideal Boundary value problem Nichtlineares Gleichungssystem Recursion Position operator
Residual (numerical analysis) Explosion INTEGRAL Correspondence (mathematics) Weight Optics Maxima and minima Right angle Mortality rate Number Product (business) Directed graph
Metre Dot product Process (computing) Divisor Multiplication sign Prisoner's dilemma Graph (mathematics) Complete metric space Cartesian coordinate system Product (business) Residual (numerical analysis) Proof theory Finite element method Kritischer Punkt <Mathematik> Symmetry (physics) Natural number Summierbarkeit Nichtlineares Gleichungssystem Diagram Family Recursion Social class
Commutative property Hecke operator Cycle (graph theory) Prisoner's dilemma Archaeological field survey Invariantentheorie Modulform 1 (number) Exponential function Limit (category theory) Curve Parameter (computer programming) Residual (numerical analysis) Category of being Group representation Duality (mathematics) Function (mathematics) Symmetry (physics) Resultant Position operator
Metre Point (geometry) Commutative property Hecke operator Modulform Invariantentheorie Chaos (cosmogony) Barrelled space Curve Limit (category theory) Number Duality (mathematics) Symmetry (physics) Square number Modulform Rotation Area Cycle (graph theory) Exponential function Fisher information Category of being Proof theory Arithmetic mean Symmetry (physics) Uniformer Raum Function (mathematics) Object (grammar)
Commutative property Functional (mathematics) Hecke operator Invariantentheorie Modulform Curve Exponential function Limit (category theory) Fisher information Category of being Duality (mathematics) Many-sorted logic Function (mathematics) Symmetry (physics) Series (mathematics) Recursion Family Spectrum (functional analysis)
Commutative property Functional (mathematics) Presentation of a group Hecke operator Cycle (graph theory) Connectivity (graph theory) Multiplication sign Invariantentheorie Modulform Limit (category theory) Curve Exponential function Term (mathematics) Mereology Formal power series Proof theory Category of being Duality (mathematics) Function (mathematics) Symmetry (physics) Quantum Coefficient Physical system
Geometry Commutative property Spectrum (functional analysis) Functional (mathematics) Free group Primality test Hecke operator Scaling (geometry) Multiplication sign Invariantentheorie Modulform Sheaf (mathematics) Limit (category theory) Curve Food energy Rule of inference Power (physics) Duality (mathematics) Different (Kate Ryan album) Symmetry (physics) Physical law Nichtlineares Gleichungssystem Maß <Mathematik> Observational study Cycle (graph theory) Commutator Exponential function Price index Formal power series Category of being Function (mathematics) International Date Line Family Spectrum (functional analysis) Recursion Maß <Mathematik>
Covering space Polynomial Variety (linguistics) Variety (linguistics) Direction (geometry) Knot Volume (thermodynamics) Infinity Cartesian coordinate system Category of being Formal power series Estimator Goodness of fit Topologische Invariante Voting Logic Integer Game theory Resultant Directed graph Asymptotic analysis
Computer programming Slide rule Transportation theory (mathematics) State of matter Logarithm Direction (geometry) Invariantentheorie Tube (container) Knot Survival analysis Limit (category theory) Student's t-test Infinity Number Power (physics) Force Many-sorted logic Different (Kate Ryan album) Cuboid Circle Nichtlineares Gleichungssystem Series (mathematics) Asymptotic analysis Newton's law of universal gravitation Covering space Polynomial Matching (graph theory) Variety (linguistics) Prisoner's dilemma Forcing (mathematics) Weight Thermal expansion Division (mathematics) Statistical physics Degree (graph theory) Partition (number theory) Plane (geometry) Arithmetic mean Topologische Invariante Circle Order (biology) Configuration space Nichtlineares Gleichungssystem Fiber bundle Family Spectrum (functional analysis)
Computer programming Forcing (mathematics) Coordinate system Limit (category theory) Open set Plane (geometry) Partition (number theory) Morphismus Mathematics Invariant (mathematics) Liquid Circle Order (biology) Nichtlineares Gleichungssystem Nichtlineares Gleichungssystem
Spectrum (functional analysis) Commutative property Topology Functional (mathematics) Conformal field theory Curve Limit (category theory) Stress (mechanics) Open set Number Power (physics) Goodness of fit Tensor Cross-correlation Square number Series (mathematics) Asymptotic analysis Process (computing) Classical physics Point (geometry) Expression Food energy Function (mathematics) Perpetual motion Spectrum (functional analysis) Recursion Maß <Mathematik>
Covering space Spectrum (functional analysis) Commutative property Computer programming Topology 12 (number) Conformal field theory Classical physics Point (geometry) Stress (mechanics) Curve Stress (mechanics) Food energy Limit (category theory) Asymptote Tensor Many-sorted logic Function (mathematics) 5 (number) Recursion Directed graph Asymptotic analysis
Spectrum (functional analysis) Random number Topology Model theory Knot Thermal expansion Curve Rule of inference Order (biology) Mathematics Invariant (mathematics) Thermodynamisches System Matrix (mathematics) Polynomial Regulator gene Moment (mathematics) Expression Cartesian coordinate system Partition (number theory) Mathematics Category of being Voting Function (mathematics) Helmholtz decomposition Object (grammar) Matrix (mathematics) Recursion Maß <Mathematik>
Spectrum (functional analysis) Random number Geometry Polynomial Model theory Invariantentheorie Knot Quantum field theory Curve Perspective (visual) Mathematics Partition (number theory) Invariant (mathematics) Object (grammar) Helmholtz decomposition Modulform Matrix (mathematics)
Partition (number theory) Random number Invariant (mathematics) Polynomial Matching (graph theory) Model theory Multiplication sign Helmholtz decomposition Knot Quantum field theory Matrix (mathematics)
around we time and again right so 1st I would like to thank Geoghan missiles forehand it's this workshop on that I really like you really enjoyed them was lectures by prophesy designer the move from the new formulation of Folgers but so the I'm going to be to me to try to do just that nobody reads a it's not going to be very well in Westminster in the final of the time I just wanted to explain Morris what it is about the only thing that were obtained from because formed in support of this relation even from faces relations since the end of the Oculus football tree and I don't think there is really a good understanding all of white district and intermittent predicts that is but we don't really understand what so the prison so the proposal because him as return of an equation small this morning a rejection of the equation in thousands of and brother my way of writing the question looks very different but it's the same and it should be Crivello and it is equipped with an
unlawful and our way of reprimanded by the question which I like very and it's just a picture of the German always means the same thing what would be really interesting is to see whether over pictures of this that the only a way off on condition questions with U.S. ashamed to each this of the picture you associated dominantly Christian just across durable efficient integration what we would be really nice but which does not exist at the moment is is very legitimate treatment and converts teacher and is there really a way of surfaces into disuse which would be natural old was irrational and mother-in-law Raymonda at during or in any of our framework we for which we would have naturally such the composition which would imply that approach rejection this is unknown so I would I was told by the introducing introducing what is while took that temperature here rejection at the moment was state of the art so it's not serious not the and it's not finished and the of newborn series it's not it's not a completion of established but some the Burgett a rejection is mostly on agrees on a recalcitrant agrees on which associates at the most sequence of forms of informs on also number as a means to expect to recover what 1 and we we willing need to define all those things but what is the spectral care of infected danger is not really of the agreement on that that would strip steak for but the idea is that it's a kind of of Pankov bridge governs not necessary and bring but went on with some additional structure but Switzer incident of carved it said and to a carving associates forms on numbers by a rejection on this rejection is completely your city depends on the phone so the only thing you need to know Harvard in initial short-term so busy it would be the only does 0 1 as yours too and then everything is determined by Rick on the day she ends on messages which have good ended in violence of the correspond to what would this mean and variance once again deliveries from agreement that's why is this interesting what as the 1 . 2 ounces 1 is issued that these definitions so this very abstract aggression and you you may wonder where it came from but I should think this definition it although that does not mean dutiful mathematical properties on I think this is a sufficient reason to students to be interested in it suggests it does really magical properties foreign suggests that if you get the equation here the blundering number 1 seems to play a
role in very different from the Oval but in fact the only if the result of a competition is always symmetry on the it's it's not obvious from from the definition and in addition the once-feared observes only there on the trip as a special war but you can't prove their extortion that is always or on that is already magical another reason why the both things are interested is because they appear
at this reclusive structure appeals in many many other petitions on the it's not fully understood why it but it is so cancel orders conference is about half numbers in recent data on so in particulate applies to have its members there are so and among extensions of our harvest numbers high W. Williams ,comma Frieden involves over recent seemed is not putting them yourself or silver Jones putting them knows what you'll it upsetting that story it's not no 1 the on the it's not it's natural proved it's a new conjecture but uh but would like to understand on the also may never produce continent to the Austrian politicians which areas so let me introduce
you to read by the I'm a very simple example on amateur computation this definition may concocted tidbits of try to see how to compute in fact so his serve to attraction is no about taught in many places and must offices of in Indian China and Australia on their personal Boeing Melbourne when vintage suits it is says that there is a way it 1 way to really understand that equations try an example on computer computer culture this is the only way to really understand what through with this so that's what this is this but there is more to it as so the very Bodelson volumes but I am just don't let it introduces so they had the home which I should say that but just the idea is when you have but you want to compute the volume was what when you have a romance office of Jews of some with some number of
boundaries Avery is a unique and treats which consultant it'd go overture for which were among the result due this it's on it said that say that the demands of agendas boundaries of given and 1 and 2 and and rights under you want to computes how many of in some sense how many romance officers can have Of all the reasons given and formats it's useful to it so once you have your metrics with constant needed to go can cut your self-esteem into bands of bonds the way to get them into piles of sponsors not so the same surfaced indicating many ways because of false but once you have to get it into bills of policy does upon signings and whose boundaries are due this expenses so I but viewed the romance of the agendas boundaries there is almost most barrels of polls Obama tries some whole of a romance office you can obtain this way on that in fact when you want to obtain a romance office with given Mondrian's you York going to bluebells of post give on that should have been and the nestled on the boundaries have to match that you can rotate them by an own goal
this is the Nunavut to die at the end I used to guys are Code centuries and coordinate development of space undivided also former please just the sum of our Ohio from the and I do to die as they say that the the composition to bills of sponsors not to meet so a given romance office-bearers nothing unique definition of allies to eyes but this form is when right other but former can compute the volume of that space on so this is just a number on the principle Kennedy computed on it by using high-poverty geometry what I'm going to mention 2nd from enough of them but just so you just some number so some functions of the enterprise which
you want to computing Prince where you can't abide by just and I probably geometric In fact I prefer to consider Muller pressed false forms but I pressed also in tables Williams multiplied by the 2 minus the I and II II With Bayelsa by LI on integrates Over and I Switzerland has transformed so you get some functions of farmers the 1 his CNN on you would like to conclude those functions many examples have been computed by half what the 0 3 names Yevgeny 0 on 3 border is so it's the power prompted says there is no
structure and its silvery in volume is why not on few compressed Putin oppressed for so long it's just about to burst and Russian fought the use of the money is not for the moment following wants it's a functional later attributed the form fall and I will be right at all me down when I when I met buy beverages the II would could on media which would differ from what I wanted to be introductory on it's just functions Russian functions right the day before 1 1 1 1 it's this 1 honoree untarnished 1 1 0 you can cut
it into barrels of pop into 1 balls which the group back together so that there is not a unique way to do that sits a little bit more subdued how to compute the volumes but the results is that 1 it's opponent during any 1 which was not abuse from the
definition the opening of ritual you compute pressed for installments of various timber Russia a fraction of the 1 OK so true competition rights Over example of this 0 for it's it's a putting on your dignity and eyes so you have for the last 4
boundaries so forehands support in Europe for a response from is again the Russian for fraction version answer uh what's was formed
by 9 and it's in 2004 is is that said there is a very nice reduced Steve relationship among Williams on a very nice rejection region while news that only 1 3rd the Egyptian aeration direct in terms of volumes but at issue that best for on this is very rejection equation you get for overrode a full proposed transforms the so this is just the press false almost Nirvana's equation on the dangerous form so you see what I would show you reiterated corresponds to its specialization of taste but so it's just an involving so you can compute some WGN in terms of dubbed some WGA is solvable reviews which you have already computed for all right so this is so but I would show you explicitly how to use this equation in 1 minute so how to so and I idolize should not too complicated to people wrong that used to be but they want to show that it's in fact extremely simple With the
interpretation of this question
others In terms of this picture White is not known I analysts say it would be very dutifully someone would do between top and bottom of but it is not enough and nowhere as I was proved proved it is really she started to to study all the Geophysics going from 1
boundaries whenever understood was election relation and so it's very indirect proof that the press is very indirect and there is no picture corresponding to that it is not writes so 2 1
or write so let me show you how this of the sameness of the sameness of this equation
what does it do that way let's try to apply it when she calls 1 Antigua's yourself so that's complete W 1 1 so it's just an example of competition so I
just rewrote was immigration on in the right alongside you have only 1 term so busy doing this the composition of series
in the summer I always excluded the days where you were 1 of the sites can do this OK so it is the composition varies only about Tom but I was absent for Jeep was 1 and for the real 1 1 so you have a purposely default for the B 1 1 I'm just means this residue with a bad sign function but in tower on referred the reader 0 to unbridled side and remember about the your tweezers justified to be 1 of Maine is primed to square so it's so varies you have some very simple functional that's
doing it so this is just 1 of in my means ministers which would be 1 of forces square which 104 the square right now we shall expire on so since we need to call to computer is 0 shall expound every function at the end of the year but we should do with the noise function 0 so let's do they expansion of scientists around the noise function of very and compliments so 1 so this is just the same as before the sign starts with 2 -minus Tobias Christians squirrels the on we have lawful and enforce square so let's put so by as we hear River to form this is your square it's put into on faults right so Vorhies you have 2 computers just bats let's exponent 1 0 over while it's just this residue on so we have the Trebor reports so it just means we need to extract a question of squaring but some so you can read it off it's extremely it's 1 of ours is your pressed to buy squirrel 6 so what you find is just but on this matches with known but the this match in general and he doesn't prove too but it's always correct so why was it so I would like to respond so it worked so let me assure you however this is new values rejection so which I rewrites you here again is assigned to bases in Virginia tell how is it a special case of that 1 explained so us here the very we have a 0 0 Wednesday and almost all of our borrowing which integrate city are Vora boards in some new ones to form a former Mondale virus in the country explained or am romance share of a project to complex plane at once so they are complex numbers but in principle or Virginia station would correspond to date and number 2 rumors of right then another thing is I like this 1 since you need to integrate all the time it's better in fact to use of shorter reforms of and functions small natural down the WGA which were functions that me tell them in 2 of the shorter forms symmetry different reforms all right onto its Kameko vital Sheehan so it's a Sumatran different reform so it just means that I think the question I'm agnostic I hereby disease your wonders the an unbridled side also I'm experiment as he wonders if and when this is 0 is at present there because a 0 appeals so it could a rare 1 unbridled side to see that this isn't the only and only John this is not the only job but I understood this is undermined this is committee and the media while I haven't is here but they don't have the mind is this so net so it is safe so that people do this they undermined but since it was missing only to put 100 in the mind this is amended right but it's just because Nova snakes and all media I'm thinking for the evidence so no 1 can replace that by Olmedo so you but what you have is you have only just here only there but you have to do is inventing it may look strange but to remember that this is a point or different for you have to here and the numerical of 108 also offered the product is indeed 1 1 4 for the program right now and to explain why we take mine as why we take this many of the 1 where we think signed a law that so on what with code words would be between vintage illustrations sold the 1st not just let me explain this 1 of 0 Stromain is a square was just observe the foreign change 1 of ours is yours square 1 mitigated breakthroughs in the 0 the new observed that this is indeed for minors citizen off the workforce but you have a simple reported city was a 0 Anderson purported sequins miners 0 so it's Korean intercourse thoughts traditionally on this quantity is precisely what they called on the so replaces amended dossier but this interval so this is what they do they replace the unit of medicine to go on now I have a truce it isn't that the or this sign function invented new right so now let me explain why we take a residue of 0 we have both factors inventing new this introduced a function which would be signed to your fault by which would be obliterated to but another function which would be the square you see that tourism is is just for the 4th of X just because it is adjusted from 4 of the square right answers and never remark is that some additional troops this Tuesday dizzy it vanishes and the poor 0 while this is why I mean we need to take the reason you at 2 points were very supportive of course that's all we know very nervous or was going to be a part of zeros of the X so general situation would be to replace this by DX and this 0 byte with heroes of the 6 this is what expenditure so on a of remark is that the mind is here is such that minors he was quietly because to square so far the 1st not which is it that's a shipment is the soreness nickel but not would be such that X so I know we could either locally version so it's just somehow the over
solution at the of ideology which corresponds with an ax rights so let's move replaced so here I replacement was is invited
to next I replace the Our assigned to Beijing by twice lives Maine this way of mine is the 1 0 is an odd function so waves 1 of mine is is just twice while lives it but if we want to play to a more generous tradition this is what we really on now this in you also sought 1 just a member remarked a few wide yes you can click on media 0 1 on understood and this is why X of wide x of mine so you can read or write to dissuade cancel the last remark is so as I say this not as those equals money is so which was there uh well let's replace every were administered by this as slaves and now we get the general departure Egyptian so the barista rejection would mean chooses a function of X choose a function y the truce and we get to Olmedo 0 2 on the privacy and formed good s would be the solution of this equation but this is where I think where his use of the heroes of the so this is situation so we have done over the Vera retraction of the on his rejection into specialization of the general the Prijedor Egyptian soul what this is 1 that he has something of the sort cover letter so you had actually was square on why was signed 2 advisers over 4 by a it was so
it's not a Russian for and that the is on the complex plane was so and depends on an underlying character is a Russian occur but the function waves is not rational so depends what you mean by that so
don't general diversion Eureka from I'd like to defend this way so it's quite different from over the Phoenician which were prospective only gave orders this my way of doing it and it should be equivalent I prefer to do it that way because there is a symmetry between X and Y which you don't see noble fulminations right now that which is not abusive or from the tuition but said that I so this is approval to raise this way but there are many many other ways and I think it as everyone was was this a rejection has written a different definition even even myself and think in many different because they use different definitions so but what this 1 and so soon I will be sold this way through a spectrograph for me would be the date of the open romance office but it's not necessary compact or connected With the fundamentals of I was that somehow what which I reneges on Johns offline and Sakura functions at that point so when you don't really need a romance office like you just need some Jones often so so so that's why it's equivalent to prevent the violence of formulation which is that you have which is just locations in the north function look at so that's events all of which need and active shot in the yard so I would go to the branch .period sold number of arrangements will be the end which Professor Arkansas and was talking about at the this and somehow so I we assume branch .period would zeros of the but 1 impromptu soccer violence still give you extra was somehow you use a local coordinate s on the X was always X square 1 expert square Saudis coral to
affect his condition there but so it's it's equivalent as we just a choice of regard Golden and somehow as his growth of 2 weeks and in fact is you don't really need more X function underwriter
right the way incomes of a sort of escalates that's all I have sold fewer the inversion in In this case was just as to minus s but in general you need to compared version like that the answer the why is more as the equivalent of the Dow of prospective on villages 0 2 is more as the equivalent of the city of both the oral histories of conscience manifold and it's an good based whole or in the public to form 1 isn't as your president of brothers forms we have to make a choice once you have that you can computers richer Top take it as a definition finally did she and by this definition rights so ways it interesting to do that yes so up and often
is that it is definition what you define here used to various activist while this is harbor is always at dismissing the jurors always at East once this equation does not does never defined Allmendinger 0 this aggression is never a defining so to define its I
will use exactly what was written this morning the city I take fire such that the fight is a amended 0 1 and guy only did she 1 which was completed by very careful aim debated by 5 bakeries used debate 2 minutes to GE unsafe but it is the definition of Evgeni wheel which is also the same as that of their this is just around while in fact I also used to define some have 0 on F 1 but the definition would take me to take a real time to explain it's not so complicated that it was really take care of that expense I don't want to explain and so on this definition is only 14 out of too just just telecommunications building here this was gabled font ,comma combat as Canadian called the new players that this is the start of the new you only need to look at your vote neared the views of also looked Nova "quotation mark doesn't matter because this is ready to resume duress it's always risen units you can pretty bearish 1 of the conditions of the world just what what is this will hold may be revenge because that figure um so just a few rows of holes this initially was defined by its somewhat Matrix models was initially matrix models what's more people were interesting sort in enrollment interest is which to have it is you have a on this on the set of matrices To begin the Hermitian matrices undertaken nearest the beginning the end of the debate or components expenditure of trace of simple enough and so this is a typical example of how not to muster more but this is an example of 2 born in Runnemede tresses were interested in large Indiana Jones or about executives of onstage was saying what is in London New York and that when start to expand the log of a politician function you breeze fills an instructor right then should go to next older refunded the next older is a consultant then the next old origin and menace to on infected goes by you balls of an uh it's because it's more and because of the Comission symmetry which resulted from an ensemble of none hair mission which is like a glorious retreat it would also find all positive and but for her merchandise you and you have a and on what you don't have such an expansion for everybody but it's a form of very large just of of murals you have such an expulsion under question 0 which exists in chooses was how to computes to conclude negotiations so let's assume that you have such an expulsion Alcan computer questions on 0 or so you're not and interesting interest the partition function itself but you are also interested in expectation values of the spectrum of watches the video over Vinod individual perspective that's distributor and are 1 the spectrum is mostly and put into the reserve on some of our eye of 1 of our ex-minister and I were overrun dies of and in the news of and so what is the statistical distribution of this quantity you see that it would assume we are exactly on the spectrum so that's where user information with respect to a dysfunctional on denouement to compute it's not an expulsion went older refined but it did like and bans some functions white it is a sum of and so it's not very astonishing but they have like to 1 In 1 of and pens some would have I would have written the next correction is in and to remind us once again that goes by and it isn't because of and then the felt correction Over the corrections with and 3 and so on and 1 minutes to the bill would you want Alex on the day and the questions what the question was how can you compute was 1 of those functions on the sequence of functions once you can buy for high operation functions will brandishing function is to take a product of reserve also what they procured alters or to say whether it is but it's just a technical things which is the answer is defined as an endpoint correlation function on again still to observe how we did such large and 110 gives like and to a tremendous and this is not so obvious but but what you observed and then again goes by even balls of an but today on what you want to compute no obelisk or was good Haitians the bluejeans so people are interested in competing was coefficients the answer is in the literature of machismo and there has been many methods to stop trying to computers was coefficients the 1st show them what only did 0 1 a W 0 once so which just it's of the trace of a result it was already known by people like you know based on it .period the very beginning of Rhona Mitra said so it was known from the very beginning and it's very complete it's Windows examples then the two-point function 1st correction to a F 1 of the partition function on some days people were able to which we have to have 3 and 4 on the right was computed Morris case-by-case and some of them stays in 95 ontological focused unlike in co-produced and injuries and 2 in principle computer every patient but it wasn't good reason which was read Monday opponents and for instance they were able to run it only in the 2nd when it together with Britain and potential for a switch was not returning the union also away 1 of the general structure was notes from industry then the started to be able to find generates portions for a two-point function on Form if 1 but over works and that's that's so in 2004 the new the and was the paper by a symbol of the 104 miles off on some flaky and make our minds of what we uh 0 stories to different methods but so versions used Alonzo constraints and to find a relationship among but the region's I road through be questions which is overrated to the constraints but did but so we were able to find a ocean for prescriptions on the Swedish and looked very little sort it was noted that 1 the only difference is that the denominator was not this 1
it was only 1 term would spend they too because we were standing on the high-purity good days or visit was meant as an integrated a trial stones that good generalization was to do that and this is what we we did with checkoff on all but a more fundamental we understood that the former now which can generalize to Armatrix names is when you
replace the stunned by this difference rated twisted for the but it and then in 2007 with which all powers said OK let's study this equation this regression equation by itself in the party of any relationship between the matrix mingled on the most of these properties will for instance prevent it's always did some conciliatory but it does
some of the southern endurance properties our said questions related to which could disillusioned and so on so we knew we started to reduce from Massenet's coterie of integration and this is what worries altered to because the Petrobras Georgia right on a number of surprises but just when we finished that physicists manual when in fact we started to write a paper in 2006 and I was already talking about with my new so he rebuts wasted he promises something as before but money you will realize that this rejection was also confident ,comma within months but at that moment it was thought that ministers were and at least 2 million for my Egypt using this chance serial more about that uh but said but sending 1st assuming that the chance Sunseri was equivalent to a matrix model which was of reforms sought by the departure fishing was a very strong assumption which was blocked at the border for the but take what I will go the traveling that's
uh I just want to say something about the geometrical representation Virgin metrical representation as I said what this equations written a Vatican looks
complicated that it's very easy to remember if you think fit in the graphic away
so you Cheryl pelted each other with guns here by the
picture on thrust you only get Jeanne adjusted prices represented as a teacher
representative face of with an onerous for just a picture which represents that option sometimes I write as a repentant with boundaries on sometimes with punches it's a matter of what you prefer all this picture Olmedo 0 2 which would be something we've of June 0 with 2 Belarus it's or it's also assume with 2 punches but various Tonight Show represented just by a straight line between 2 between the X next to which appeals here let me repeat
what it ours what it's nearly
a pair of polls it's more the Belafonte would be that it would have 3 independent variables 1 2 this 3 cancel so somehow battalions of 2 independent variable and it corresponds to saying that it's a pair of policy but where you have we moved to the legs 1 sometimes we couldn't do short in handling the silver so it's like various 1 bothering but which is changed just a picture very often which would be repealed the desert liner with over 300 optics depending on the 1 of my old Desouza Okura and it would be very nice to find an extra interpretation of that but from what does not know the phones it's only in Munich to remember that expression just through the nite mnemonic forward to remember the question so that's show how it works in practice for competition but in practice for competition it's a all 1 1st there are different ways of writing the questions says so British immigration so ivory were presented with things with Bonner is on somehow
it stays but it says that you won't want to obtain his office of ginger with and onerous you wouldn't say you start from the 1st monitoring on go out somewhere but when you get out there short on much remains is either of us face so is either connected minutes of genes she minus 1 that's just 1 more moderate or give a possiblity when you remove short is that you get something disconnected then you have some Jews here was the tuition this year age from prime summons a of the number of words here and that the number of forces that picture number of boundaries Wisconsin and on our way to write it is when you were presented with punches 1 say that
carries only that straight line made by 300 not takes on new to to puncture all of this is the disconnected case all of this is very modest when it and so this is not just a ways of a plant in that equation that again it's only cruise Fresno geometry and but for moment I believe there should be a centimeter behind them but it is not not right so let me show you how to do this on computer every issue to after should the effigy remember to compute the FTC what you have to do is take it all on the once or something with synergy on 1 1 1 puncher but on the contrary we will dysfunction 5 which was know by derivative of as your 1 means you integrate if the jurors due on July vitamins to cheat on the defensive just 1 this is only a special case of all of the digital migration veteran questions in general issue of women did she and then into the quarter to did she had just 1 where have we would fight 1 of her legs integrate the debate tomatoes 2 achievements and decided she so this is true for every and positive and so in fighting this case is just a case and costs Europe 1 resulted in with boundaries it just means that it should data the function with and press 1 bomber is on disk on 1 of the boundaries divide between the ministers immense and when you get this suffers we've only and moderate so it is just a nomadic way it works right so let's do some examples so if you want to compute only does your only does your 3 so it's something with 3 3 patrols Charles often ponderous industry position at the authority this recursion acceptance of this some of two-term so effect is over to this brought to and that some prisoners dominant but cultural freedoms existence that so what do we do vote this would there's this
picture in this picture just means you might supply the weights associated to each leg so serious that day and is yours the Z Bar X 1 I'm sorry please only deserved to so just take the product the Wallstreet things on integrated optics integrate things I think the residue of branch at and you have a number of grass tensions with spread weights of the edge integrated at all of officers on the gives you exactly what is written so just limited quake but is very convenient is a nice this product is that in fact there is a signatory to zooms and if I lost 2 quantities so which reporters often residue when you reach reprieve for something to grow at the same venue in here so you could just say it is twice but 1 when you computer this new animal wonderful on the 1 1 so what's this rejection means is exactly that but is this picture so I'll use it they could take year throughout Windows Europe 2 so here again you think the product of whites associated to a Roland edge corresponds to a day this non-Arab edge corresponds to a when is your too I think this product integrates source at the branch of texts and that this media 1 1 this is just a rating of the integration rights let's do
not complicated example it's
contradictory as to which she want contradict to what you have to divide by 2 minutes to G which as many as 1 half example multiplied by 5 so this is this red dots means Friday at of this is only down to 1 soldiers to 1 with a 1 country but is still needed to 1 the plight Egyptian to 1 and was a sum of 2 downstairs here in what was obtained budget was a K some only down some the giant
that some on India 1 1 on the media when 1 another time which was only got 1 too and
this is just what you think about this but this was the only guy 1 1 we have already completed this year so let's replaces all media 1 1 but his brother you and for this 1 again which I computed by so this is what we don't want to which should be computed by the from so Soviet media 1 1 advocated by the grass on the want to wages British again so what this introduces a number of and then you have 0 0 meters 0 3 Ontario only Jesuit-run 1 1 why don't we deserve 3 I replaced it but his graft there will be factor too on only down 1 1 here prison but his brother and serve your media 0 through into big I just replacing to remember very distressed so we have to it is just some of those 3 graphs if class means you take for product of so to each edge so if it is our old recession today if there's no narrowed associated take just take a product or ban on Denver is used and the justices but remembering which although computers things so it means this residues was computed us then this 1 on hearing that brought this residues complete Mendes 1 than this 1 so you have to be careful about job over news that you to again control although residues and also the old all of our roles with great to remember this was written in the Russian which the interested persons and its importance in which older you take residues because because well boss Atkinson .period soldier although of competing residues doesn't it so 1 so it just went so this is just the way this is just the value of those brass become secretion so for me it's much simpler to remember Wallstreet graphs for F 2 were overrun or this and on the so you have arisen said the undefined but there is just this so this is just the most just enumerated in the way of writing but in fact it is very useful because most of the and you can prove a very easy to prove gratitude performances the symmetry all over this usually light equations ovaries prove gratitude on also developed an eye for men's and it comes out very easily congressional wins so so it's not only pneumatic tubes that actually really useful when you want to Incredibles is the same political appointments for trilogy work no U.S. itself or its president you have to take a sum of all critique of the good it is also wanted to know about the nature of issue over an efficient and the critical points you have this year an the reason is and raises various here it would never let and guns In this means and to a fault on this trip but you can also say that he and importance of the sum of on and it's only when the core of some resistance substantive the move to so OK so I what they notion that finish the so sorry for his watch so we know what it so I just
want to prisons vary for his some of the properties in a known very well defined way but just to give you an idea of what it was the 1st property which is not written here but which is extremely important as that is I already mentioned is that
although her rejection does not cubes a treaty in all its arguments the ones instigated the default for sold the 1st seems to play a different but in fact you prove attrition but that the result is always intrigued conferences the refuge's representation is very useful
to do things it's it's really it's very easy from 1 varies but promised to take steps but I think it is from graphical positions so on and so this is not to abuse at all or that it's going to produce some even under 0 3 only 0 3 remember officers ones too is a 3 it was a residue
of chaos C 1 0 the only day 0 2 officers in the 2 meters 0 2 of the 3 what we need to talk to what does look of symmetry but computed from the refinery this is always
right so this is not the reduced but this is happening this is an informal property over ways all of this would not be where the what a number of properties whose injected barrels wanted to be want to share this military proved this isn't proved false specialties but the
idea is issue what if you have this excellent was so it
means you have the care of somehow immersed into the square persistently 1 to square their current Austin too 1 of circuitous crosses square or whatever that's why there is a very nice properties that issue making it safe for us as a rotation in the square you the notion of branch points over the zeros objects on bonds Robert entrances that John McEnroe and so when you make auditions for instance to you challenge excellent y or and if you do a retention made payable to the areas will be totally different and so on the competition the looks totally different but when you contribute fetches the said so they are unchanged when you do that and it is but then 1 proof is that they charges this is the example of the lumber here if you forget that some of the excesses of the graduates now so and so what do I mean by that is what when both sides are always the center of his conviction then villagers greenside 1 of our social and so that's what they said it's proved on Monday when the car is really IG break Swiss Re proved larger break so you mean I mean involves Exxon widening on both excellent to have branched out and then it is Sonatrach it proved on inducted so for instance it is not proving the case correspond to cobalt 1 bit of song all true or even a uniform ,comma fruit and it does not apply Fogel freedom because does not apply at home but in the form of freedom we believe that it is reducing the treaty but it is not approved by this proof sold for in fact we are trying to draw is approved for 900 case but said its help and fulfillment we have not succeeded another property is well that's you produced this way always some modular forms but by uniforms and in went around with respect to so when discovered that some genes
it's 0 Maduro respect with the institution of the vengeance yeah yeah but so it's a soldier majorities executive Desue would you like to so I don't want to extend this is this is a property which you can Primerica from uninfected
for but 1 of the Russian for this gives you an extremely approved the is always it's so everything which subsidizes recursion we always had said the and this should is their I wonder if we don't really know what's majority meaningful fashion authorities but what it is to be explored wanted what integral beauties again his notes free proved displayed on a case-by-case for cellphones .period and families of spectral skills but it is
believed that the budget should hold in a very very generous said informal spectrograph for instance very softly and with illegal method proved that works for kitchen light spectra goes of what we formed to each insistence on we are working on the wrong end which consistent so we can prove it it's the provision that we suppressed its for long 2 and that's the issue taken all of a series of images make this series won over a sort which vaulted to humans to while this is about the function of something to Boston 1 of the the future
of the the what it I don't fall to me to be true to explain that to . several natural traces of coordinates the corresponding so if you have had you break function as neuromorphic functions here the coordinates with the Evo coefficient of energetic part flaunts function in each board
on that would correspond to the teachings of 2 component 2000 I mean basically gets out of its which it or for
systems it would be natural for me that is issuance of cauldrons of cheese which is it's me not to go foreign indigent but so the idea is that the WGA and while the W donated shares with satisfy some the timing of the phone on the Nunavut In the procuring editions mineral such relations should be satisfied and it is not approved in general on this is strongly related to the notion of quantum calves about but there is no general proof various formal apologies that is proved so on visited 1 ministers proved his Quentin orbited the present through cellphone again and it's really specialization of this was God's on of soap Castle killed a very important property which there I would like to say that to become or is about to take 2
imagine that you have a family of spectral Cos so you have on very which opponents sometime 1 thousand could be assumption of maturity on the major indexes 1 at a certain value of was to see a critical time the car becomes the generated desert coast instead of having to functional so that it doesn't ongoing function in surviving the
punishment for instance you can have a at this very special value when you are not that it
was just the world when you already did it away from the was deceived by the guest is resolved and you have something like that on what we saw Syria cannot free Computervision but here again competitive G as a function of his time on your today Georgia's has 2 main assisted to some power you can compute verdicts today villages as the administration to Sumbawa answers the question so edgy with lodged with 2 minutes to subdue some bore which is always proportionate administration on so To assistant about appropriate but were up and 50 hasn't it and what is that needs America's 2nd it is also the energy of some carved on that golf is just blowup of a security so so if you so am not very rule rigorous way to write it is that the units of energy of St is the energy of a unit of US what and but wasn't interested this is the idea the idea is that this the effigies under the omitted Shenzhen but also integrated but Orville mutagens new commute which taken in its section on this matter this is an extremely useful purpose .period there many many overall nice properties but I don't have time to explain to our relationships with send with and equations is 1 of the additional missions which I electrical from I duty which some people corner of special geometry properties of 1 it's a different mission questions I think you're going to talk about but were and that's all there is no real nice properties and why you know you're going also to talk about Vernon Chris Young the formalism ahead Georgia browse our sold there but they don't want to favor Nelson a treaty which you can also extend that to all noncompetitive spectral cuffs so you can where there is a notion of committed
girls were excellent don't commit any more but you can apply this formalism unwarranted works the same way and it's entrees Morrison property I don't want to to guru to
fall uh wanted to go 2 examples of reputations so now in its new mentioned some applications so I will be very sketchy we if precise definitions for instance about environmental was I wouldn't give precise definitions I just want to give you an idea on most of those things also conjectures and not everything is proved and many of those things on the prove that all but 1 example which she mounted a result is that the Pereda rejection something to do with not we not serious it's the completely totally mysterious almost nothing has proved yet but people raw Shawmut works the infects F in the 1st set and in the 1st place where it feels is by diving off from issued in 2000 I need a where he won some of the genes for say would transporting them you used to be raised no it's not say whether Jonesboro is too broad interest in obscurity have defined some variants associated to adults the past you over is 1 which could John Springer J and that it depends on lots case is not incidental Nevada virtue hundreds of pounds on an integer and is usually go Korea on Tuesday for more bombs so this is a pretty your it's been almost 2 thousand you of In a related citizens of Q of sorry necessary to cure instruments 1 sorry integers so it's a silken which is not on the boss altered to be interested in a holidays of the logic of this case when you sense a SoHo doesn't when you sent to a 2 to 1 on entry ingenuity in such a way that you will be a source and of Q so it's good new remains at 1 160 so how this behave what and it with 1 of at you find something press busy reduced some of note you to some 2 G minus 1 on that score aggression sorry sorry acute to attain this could be questioned SK of you so this agreement as minus 1 of you so there is a conjecture about estimates 1 of you which has been there for most of years which is not approved OK it's good volume conjecture so what's s minus 1 of you is it's voting conjecture it goes back catch it if you know I don't remember what that was but you know that's the recent thing is that they have some free claimed that's already estate's here they freedom and big game but all estate's are busy related to vote on the legends of at the butchery party British in reply to a certain carved on that here while serial presented few 8 militants were not 1 not so fishermen it's not good for the it's not on various notion of John variety associated to another don't want to govern these days but through direct authority of another is a cover once immigration Regent ex-wife wars like that like that Sweden question raised in excellent excellent white stated was just a
quirk of applied at a particular attraction computes was intervals of Medici she and up to you all of them to integrate all of our the
allies so orders the eyes integrate them up to you
divide by sectorial some young "quotation mark such fat 2 g prison ministry questions what good SKW the conjecture is that this computes listened takes of the transporting so as I say the and believed army is an approved this fall from improved soul and we are very far from proving rest of expulsion but uh bits you won't want to say we have aware for instance with my students frequented the data will we computed with for this not to be in the computed things up to sell power offer notes to on the matches Karen so yeah mean Dorit suffered an American agent in fact we can fruitful Torres nods and full toss notes redundant not only the transportation was the Bronx to bring but for but sold for even but not so far from improved format notes the just minus 1 has been proved by year's end here and there but not the rest of the expansion so on it's talk and mysterious why it works but it seems to us some people are already shortages works most of the recent because they go far but that so on so it's still it's legal to a of example which is you want to go through the politicians was through the politicians are or people courses ,comma but also playing politicians are you boxes in the gone off room 1 but sometimes you want the room which has called off the land 1 in that state as a chenille or did you what you want to rule which is not necessarily a square and undistorted puts boxes on the floor from from this direction before from the sky from this direction and they go to the butt of rumors I so it's a statistical physics program so it's coming problem and the awaits you put to a country aeration is just the number of boxes I also use the weight of acculturation is just that say you have a barometer of you so you have just tuned to look for a number of With Kuroda skewed more 1 so this is just how Yukon configurations so you wake every consideration by that of his young you're on In politicians on you want to compute their views the some which is what guarantee does number of freely politicians of different size but somehow this is equivalent to save you some overall or politicians with that for Due to another abduction on you won't find yes and that when the size of a room become large so which is Morris equivalent suggests takes when she was 2 1 in bowls of rescue so you want to go congestion tactics on again observe Jacinta to take schools by bottles of to work to ministry G. on won't contribute creations FG 1 Ave 1 for industry origin to bring the Internet is this is conjecture um so so we just sort when you put a lot of tubes so serious confusion with the issue tubes that when you put a very large number of then it starts to look like that and if you observe throw some regions were but there is no cure on the wars on awards here on sharing the country's you have reached the opposite wall of the room In the vocal yeah and so 1 series Shiva is a kind dissolved but she on most golf the varies annually since there is no Cuba or here and there is no and in Kingston New somehow here it's frozen it's got frozen and there is a chance which separated frozen region from where he quipped region and typical of the Arctic Circle the Arctic Circle of surprise the decreed from the frozen region viewership from home as well as on the shape of that started circle has been completed his forces versions the public unions could go from shifted the unfold but example Revay a would find it is diluted but you know what injured or what they find is discovered is the smallest of degree algebra coverages dumped onto a so on the decree derision said sizable boggles equation sold to renew old Alonso here we have a cover without a computer messages but in this make India's 1 infected slightly more subdued what's at St. I have a good news is that we did that get sentenced legislating most of them take any slice all I don't think the density of locate brought about whenever there is a Vatican edge and also was on the books since it was .period set up this happens the spectral of 1 which depends on which sliced through your true so witchdoctors Fletcher you truce which was slides bundles of survival but when you robots it's kick so you you have a family of bolstered his butt since they are related by goggles equations this means that they are order simply to get equivalent some Holly
correspond to themselves to from the old computers and efficiency thanks to boggles
equation so the conjecture which you can really the head the coordinator is that the evidence should be here to Compton made at the particular applied to those of adults cities and some hold what I would say there are some cases were misses proved it's because if I had listened to some of your politicians is of 3 politicians Morris Verwood comes as the some domestic invariants this opener come inferior for solvent until many forces which is equivalent to go Moffett and violence on somehow edges of offered and on in some way involved in the case which guru actually correspond to an enraged drainage program somehow is approved they're going to go from there when co-founders corroborate but engine order for an arbitrary trio the shape of the main this is not true proved this was the unfazed there is a branch .period here Bridgeman no more kids it's not protrude from the computer and convinced that happened at the would erode that would they just wanted to show his issue changes of that upstairs slice of the changes but to changes in the leg by simply morphism which consults lifted right for this is just an example of mediation that'll of what I want to say she worlds of at I
was just not just the number 1 recently see know is because the of you he come from merchants series becoming very very very important subject nowadays because of what's called educated conjecture which is I think military union more conjectural but I mean that's what you could buy digital but bids which say that's a good correlation functions some operate also in new vehicle from the history I don't want to explain on what this really means are very closely related to what was going on because of In the course of Bochum functions on that's on a bit so 1 thing is getting contraband imparting curator new computer than in this and needs of 1 billion reconfirmation surety bonds on such workshops which is usually written this way 1 persist through square on this say can compute was things in the units it's going to heaven and we like to computers to the enlarged Union aren't so if you want to expand but you compilation series function In 2 balls of Cuba at Celtic jobs the gains we go with powers of king would review involves of you and you would like to contribute kids on again there it is conjectural that the FDA's share Archer concluded by the 2 procedure rejection from the specter of golf on the spectral can also be 1 1 1 y square sought to use way square white square
given by US Open expression which is in fact the
euro was quoted don't surrender over dressing and don't sort in Munich for so distressing Najat also spectral carved on Winston stress ineligible and the disagreement of stress knowledge source get the speaker cover on the issue credit push for a person to meet you should get gestures correspond to the cultural expunge so this agenda conjecture that what we have good reasons to believe it's correct and
analysts way this year this program yeah yes
noncompetitive but didn't want to so the but it didn't want to look at and then there is the way to the final details in 1 committed to days such medicine when you
consider it is something which we have done before pot of Chekhov's on the region option which is here but what is
international assistance programs but we have to do this is 1 of the characters and you break but the humidity communities well but it wouldn't just
concluded that also initially the Depression-era Regulation was introduced in on indeed rose to find that out an expression of matrix lost by Jennifer expressed anger isn't so somehow it solves the problem of competing not an expression of interest on the issue of obstructed from explores the issue a played to an arbitrary carved then defines some interesting objects which could be environs of that covers on behalf of itself varied interests and fantastic mathematical properties on it to do and I think adjusts itself to be didn't just because of that but they have many applications and for instance was the fact that they conquered ,comma freedom violence for something passes of Canada also was looking story CAN END as improved but so did he was open but other foreign social rules of storage can look at it and knows it was proved that all but 2 of units it's notion is renewing a vote it's a fact that it applies to John Springer the total industry at the moment but instead and so on many of many of the interesting things to do with the or was controversial and indeed the having the general framework to do that would be very nice but also on the phone with his picture means urinals for French which
also 1 2 holds a clear here "quotation mark was going to be a long long way so as to 1 they
also called on OK and then it would come home from work I know it's full of strangers and this is really and the form of the question really is JIC which but it's and there isn't something else which is cool to perjury rejection worries you really have 1 perspective to also left but it's not the deal is due the 1st for the 1st quarter
of this year in fact this is this is for instance how we proved exploitation which we will do it twice on this also prevented birds from
already agreed time In stronghold of new regretted matches
2 the few around


  937 ms - page object


AV-Portal 3.21.3 (19e43a18c8aa08bcbdf3e35b975c18acb737c630)