Resurgence’s two Main Types and Their Signature Complications: Tessellation, Isography, Autarchy
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00:00
Derivation <Algebra>Divergenz <Vektoranalysis>Singularität <Mathematik>Stokes-IntegralsatzGeometrieFunktion <Mathematik>ModelltheorieDivergente ReiheFormale PotenzreiheAnalytische MengeBorel-MengeEbeneFaltungsoperatorStandardabweichungKategorie <Mathematik>Algebraisches ModellQuotientDifferentialoperatorNumerische MathematikPerspektiveRelativitätstheorieMengenlehreSingularität <Mathematik>ModelltheorieWellenpaketInvarianteKategorie <Mathematik>Ausdruck <Logik>Derivation <Algebra>UnendlichkeitPotenzreiheDeterminanteNebenbedingungAlgebraisches ModellAnalytische FortsetzungBeweistheorieDifferentialFunktionalGruppenoperationIdeal <Mathematik>IndexberechnungLeistung <Physik>MaßerweiterungOrientierung <Mathematik>Potenz <Mathematik>Superposition <Mathematik>TheoremVertauschungsrelationWinkelZusammengesetzte VerteilungFlächeninhaltTeilbarkeitFormale PotenzreiheExogene VariableStochastische AbhängigkeitNichtlinearer OperatorGewicht <Ausgleichsrechnung>Faktor <Algebra>Formation <Mathematik>VollständigkeitAdditionExistenzsatzQuotientOffene MengeSortierte LogikFaltungsoperatorBimodulDivergenz <Vektoranalysis>NichtunterscheidbarkeitAbgeschlossene MengeObjekt <Kategorie>DickeMultiplikationsoperatorStandardabweichungGeometrieÄquivalenzklasseKonstanteTurm <Mathematik>Computeranimation
07:32
BeweistheorieSummierbarkeitFaltungsoperatorNichtlinearer OperatorKoeffizientStokes-IntegralsatzNichtlineares GleichungssystemParametersystemAlgebraisches ModellÜbergangParkettierungNebenbedingungInhalt <Mathematik>E-FunktionAnalysisLokales MinimumFunktionalNegative Zahlp-BlockModulformSchlussregelDesintegration <Mathematik>KonstanteAuflösbare GruppeAnalysisDifferentialoperatorNatürliche ZahlNumerische MathematikModelltheorieParkettierungModulformInvarianteVariableKombinatorDerivation <Algebra>UnendlichkeitNebenbedingungAlgebraisches ModellArithmetisches MittelBeweistheorieDifferentialEbeneFunktionalGeradeGerichteter GraphLeistung <Physik>Lokales MinimumMereologiePhysikalisches SystemPotenz <Mathematik>ÜbergangswahrscheinlichkeitLinearisierungKonstanteNichtlineares GleichungssystemReelle ZahlÄhnlichkeitsgeometrieNichtlinearer OperatorParametersystemGewicht <Ausgleichsrechnung>ExistenzsatzPunktKoeffizientFaltungsoperatorDifferenteObjekt <Kategorie>MultiplikationsoperatorSchlussregelStandardabweichungZweiInverseDifferentialgleichungFunktionalgleichungRationale ZahlMengenlehreRenormierungGanze FunktionDelisches ProblemDifferenzenrechnungLineare RegressionPartieller DifferentialoperatorFlächeninhaltEreignishorizontDivergenz <Vektoranalysis>p-BlockEinsVorlesung/Konferenz
15:05
PolynomAnalogieschlussLineare DarstellungWurzel <Mathematik>Nichtlineares GleichungssystemSingularität <Mathematik>Einfach zusammenhängender RaumStokes-IntegralsatzKategorie <Mathematik>Inhalt <Mathematik>Stochastische AbhängigkeitRelation <Mathematik>ParametersystemUnendlichkeitAnalytische MengeKoeffizientModelltheoriePhysikalisches SystemBorel-MengeEbeneInnerer AutomorphismusVariableDualitätstheorieResiduenkalkülÄquivalenzklasseGrundlösungLineare AbbildungNormalvektorNichtlinearer OperatorAutomorphismusDifferentialKontraktion <Mathematik>Divergenz <Vektoranalysis>Derivation <Algebra>Formale PotenzreiheAnalysisDifferentialgleichungPolynomRelativitätstheorieSingularität <Mathematik>ModelltheorieFrequenzTourenplanungTopologischer VektorraumProdukt <Mathematik>StörungstheorieVariableKategorie <Mathematik>Derivation <Algebra>FaserbündelUnendlichkeitPotenzreiheÄquivalenzklasseAnalytische FortsetzungAnalytische MengeArithmetisches MittelDifferentialFunktionalGrenzwertberechnungIndexberechnungKohäsionKontraktion <Mathematik>Leistung <Physik>MereologieNormalformPhysikalisches SystemTaylor-ReiheKonstanteNichtlineares GleichungssystemSuperstringtheorieParametersystemGewicht <Ausgleichsrechnung>NormalvektorWurzel <Mathematik>PunktNegative ZahlIrreduzibles PolynomKoeffizientRiemannsche ZahlenkugelKomplexe EbeneKnotenmengeDifferenteObjekt <Kategorie>MultiplikationsoperatorDifferentialoperatorAnalogieschlussComputeranimation
21:36
Borel-MengeStatistische SchlussweiseFunktion <Mathematik>ModelltheorieFaltungsoperatorInverser LimesSinguläres IntegralNichtlineares GleichungssystemKonstanteStokes-IntegralsatzParkettierungKoeffizientProdukt <Mathematik>GewichtungSchlussregelTotal <Mathematik>EbeneAusdruck <Logik>Desintegration <Mathematik>VertikaleIntegralMultiplikationÜbergangswahrscheinlichkeitPrimzahlzwillingeSummierbarkeitGewicht <Ausgleichsrechnung>Algebraische StrukturNatürliche ZahlNumerische MathematikPerspektiveRelativitätstheorieSymmetrieTopologieFunktion <Mathematik>ParkettierungProdukt <Mathematik>ModulformKategorie <Mathematik>IntegralResonatorAusdruck <Logik>Derivation <Algebra>UnendlichkeitRekursive FunktionHyperbolischer DifferentialoperatorAnalytische FortsetzungArithmetisches MittelMorphismusEbeneFunktionalIndexberechnungInverser LimesKomplex <Algebra>Kontraktion <Mathematik>MereologieMultiplikationPhysikalische TheoriePhysikalisches SystemSinguläres IntegralÜbergangswahrscheinlichkeitKonstanteNichtlineares GleichungssystemÄhnlichkeitsgeometrieTranszendente ZahlNichtlinearer OperatorParametersystemGewicht <Ausgleichsrechnung>Coxeter-GruppeVektorpotenzialFeuchteleitungSymmetrische MatrixKreisflächeKartesische KoordinatenKoeffizientFaltungsoperatorRiemannsche ZahlenkugelNichtunterscheidbarkeitKonditionszahlRechenbuchObjekt <Kategorie>MultiplikationsoperatorStokes-IntegralsatzStandardabweichungCliquenweiteZweiRhombus <Mathematik>EigenwertproblemAlgebraisches ModellAutomorphismusPartielle DifferentiationComputeranimation
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Feasibility-StudieSummierbarkeitDesintegration <Mathematik>VertikaleGewicht <Ausgleichsrechnung>GewichtungVorzeichen <Mathematik>IntegralMultiplikationÜbergangswahrscheinlichkeitFaltungsoperatorÄhnlichkeitsgeometrieProdukt <Mathematik>GruppoidHill-DifferentialgleichungTermParkettierungKoeffizientCharakteristisches PolynomEbeneDerivation <Algebra>Singularität <Mathematik>Borel-MengeKombinatorikIdeal <Mathematik>Dichte <Physik>Stabilitätstheorie <Logik>Algebraische StrukturFunktion <Mathematik>Kategorie <Mathematik>Folge <Mathematik>Divergente ReiheKompakter RaumFlächentheorieMathematikNumerische MathematikSymmetrieTransformation <Mathematik>MengenlehreSingularität <Mathematik>IntegralDerivation <Algebra>FinitismusArithmetisches MittelBinärbaumEbeneGeradeIndexberechnungKette <Mathematik>MultiplikationSuperposition <Mathematik>Thermodynamisches SystemÜbergangswahrscheinlichkeitNichtlineares GleichungssystemNichtlinearer OperatorGewicht <Ausgleichsrechnung>Faktor <Algebra>Formation <Mathematik>DistributionenraumPunktSymmetrische MatrixResolventeFaltungsoperatorAbgeschlossene MengeKonditionszahlObjekt <Kategorie>SondierungMultiplikationsoperatorZweiIntegraltafelAuflösungsvermögenSekantenmethodeComputeranimation
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KraftSingularität <Mathematik>ZahlensystemSummierbarkeitAlgebraische StrukturDichte <Physik>Stabilitätstheorie <Logik>FaltungsoperatorFolge <Mathematik>Divergente ReiheKompakter RaumFlächentheorieFunktion <Mathematik>Kategorie <Mathematik>Borel-MengeEbeneProdukt <Mathematik>MultiplikationGewichtungKombinatorikArithmetischer AusdruckAnalytische MengeRelation <Mathematik>Markov-Ketten-Monte-Carlo-VerfahrenWechselsprungSchlussregelIndexberechnungAusdruck <Logik>BestimmtheitsmaßSpieltheoriePolareRekursive FunktionKomplex <Algebra>WärmeausdehnungHyperebeneLogarithmusBilineare AbbildungTurm <Mathematik>Algebraische StrukturGeschwindigkeitNumerische MathematikFunktion <Mathematik>MengenlehreSingularität <Mathematik>ZeitbereichAusdruck <Logik>Derivation <Algebra>Meromorphe FunktionAnalytische MengeDifferentialEndlichkeitFunktionalHyperflächeIndexberechnungLogarithmusMultiplikationPhysikalisches SystemProjektive EbeneStatistische SchlussweiseSuperposition <Mathematik>TabelleTermThermodynamisches SystemWechselsprungKonstanteGewicht <Ausgleichsrechnung>Faktor <Algebra>Unitäre GruppeKoeffizientFaltungsoperatorArithmetischer AusdruckPartielle DifferentiationPolstelleMultiplikationsoperatorSchlussregelStokes-IntegralsatzOrtsoperatorSymmetrieZahlensystemModulformArithmetisches MittelHyperebeneInverser LimesNichtlinearer OperatorBasis <Mathematik>PunktFlächentheorieStandardabweichungComputeranimation
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GewichtungFaltungsoperatorEbeneLineare AbbildungPhysikalisches SystemStokes-IntegralsatzHyperebeneLogarithmusKonstanteKoeffizientSchlussregelProdukt <Mathematik>KombinatorikSymmetrieSingularität <Mathematik>ParkettierungProdukt <Mathematik>Kategorie <Mathematik>IntegralAusdruck <Logik>Gerichteter GraphKomplex <Algebra>TermGewicht <Ausgleichsrechnung>SummierbarkeitKoeffizientFaltungsoperatorDickeNumerische MathematikHyperebeneWasserdampftafelObjekt <Kategorie>MultiplikationsoperatorComputeranimation
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MathematikStellenringWärmeausdehnungParkettierungKoeffizientPunktrechnungStatistische SchlussweiseTermSkalarfeldTeilbarkeitVorzeichen <Mathematik>Kanonische KorrelationEinflussgrößeSummierbarkeitInvarianteFolge <Mathematik>IndexberechnungGewichtungFaltungsoperatorDerivation <Algebra>Folge <Mathematik>MathematikNumerische MathematikZahlensystemProdukt <Mathematik>Kategorie <Mathematik>ZeitbereichAusdruck <Logik>Derivation <Algebra>Rekursive FunktionAuswahlaxiomFunktionalIndexberechnungSinusfunktionTermWechselsprungNichtlineares GleichungssystemEinflussgrößeGewicht <Ausgleichsrechnung>Unitäre GruppeTranslation <Mathematik>SummierbarkeitPunktElementare FunktionKoeffizientFaltungsoperatorBeobachtungsstudieArithmetischer AusdruckPartielle DifferentiationObjekt <Kategorie>DickeMultiplikationsoperatorSchlussregelOrtsoperatorSymmetrieParkettierungUnendlichkeitHyperebeneLogarithmusResultanteStandardabweichungComputeranimation
46:56
Theoretische PhysikÜbergangKlumpenstichprobeNichtlinearer OperatorParkettierungDerivation <Algebra>PunktrechnungKombinatorikNichtlineares GleichungssystemKoeffizientUniformer RaumRiemannsche ZahlenkugelIndexberechnungLagrange-MethodeFunktion <Mathematik>Stokes-IntegralsatzKonstantePunktIdempotentParametersystemAsymptoteUnendlichkeitWärmeausdehnungTranszendente ZahlEinfacher RingKreisflächeAlgebraische StrukturAlgebraisches ModellGruppenkeimSchlussregelPaarvergleichOperations ResearchAlgebraische StrukturGeschwindigkeitHolomorphe FunktionNatürliche ZahlRelativitätstheorieSymmetrieDeskriptive StatistikMengenlehreSingularität <Mathematik>ZahlensystemParkettierungAsymptotikModulformVariableKombinatorKategorie <Mathematik>Ganze FunktionDerivation <Algebra>UnendlichkeitMeromorphe FunktionUniformer RaumTotal <Mathematik>ÜbergangNebenbedingungAlgebraisches ModellAnalytische FortsetzungArithmetisches MittelDifferentialDivergente ReiheEbeneEinmaleinsFunktionalGerichteter GraphGruppenoperationHyperebeneIndexberechnungKlumpenstichprobeKomplex <Algebra>LogarithmusMultiplikationPaarvergleichPartieller DifferentialoperatorPhysikalische TheoriePhysikalisches SystemTabelleTermWärmeausdehnungWarteschlangeKonstanteNichtlineares GleichungssystemTeilbarkeitTranszendente ZahlNichtlinearer OperatorParametersystemGewicht <Ausgleichsrechnung>NormalvektorRandomisierungGammafunktionSummierbarkeitQuaternionengruppeKoeffizientFaltungsoperatorEinfügungsdämpfungArithmetischer AusdruckVerzweigungspunktKonditionszahlDifferenteRechenbuchObjekt <Kategorie>SondierungMultiplikationsoperatorStandardabweichungZweiInverseRechter WinkelDifferentialoperatorNumerische MathematikGebäude <Mathematik>ResonatorAusdruck <Logik>GeradeMomentenproblemRechenschieberZentralisatorFlächeninhaltÄhnlichkeitsgeometrieUnitäre GruppeDruckverlaufAdditionRiemannsche ZahlenkugelSpiegelung <Mathematik>Computeranimation
Transkript: English(automatisch erzeugt)
00:16
Well, clearly this is quite a pleasure, quite an honour actually, to be able to give a talk, an opening talk at this place.
00:27
To me it also feels like a throwback to my beginnings. The fact is that 40 years ago, almost to the day, René Tomme, the late René Tomme and Yvonne Kupka, who is very much alive, over
00:41
close to 90, invited me to this place to hold forth about resurgence and related topics. The two gentlemen were kind enough to submit it in my Lucubrations and even offered practical support. René Tomme offered to get my things typed by the secretary here at the Institute.
01:04
There was even Denis Sullivan on that day in the room. He showed his appreciation in his own way by reclining on the floor full length but with eyes wide open, mind you, so that he could follow my scribblings on the blackboard at an angle.
01:25
So nice to be here and nice to reconnect with these memories of long ago. Before starting, for good, let us get a few reminders quickly out of the way.
01:45
Research and functions, of course, live in three models. In the formal model, as formal power cities, or more general things, train cities, a mixture of power cities, of powers of z and exponentials, of towers of exponentials.
02:06
And then we have the convolution model, with the operation being path convolution. And then in the geometric model, as sectorial germs at the, excuse me, I've got a sore throat, as sectorial germs at the infinite.
02:32
So we start from the formal model, we want to get to the geometric model and put all the work and all the difficulty manifest in the intermediate model, in the convolution model.
02:45
So the singularities in the convolution model are responsible for the divergence. Moreover, they carry important information because they carry the Stokes constants.
03:04
So it is important to measure them precisely. And the tool for doing that is the so-called Allian derivations, which are the hallmark of research and functions, actually a research and functions or functions which can submit to Allian differentiation.
03:26
The Allian, they have indexes, indices, omega, which can, which belong to C or actually the C, the ramified version of C. And by definition, the Allian derivation of a delta sub omega applied to
03:46
phi of zeta is a weighted superposition of various determinations over zeta plus omega. This is defined with our difficulty close to the origin and then we move, it is extended, thank you, I'll use that.
04:06
This is extended in the large by analytic continuation. So they are defined like everything else in the convolution model, but by pulling back the extent to the multiplicative models.
04:21
And in those two models, we have the additional invariant Allian derivations denoted by this double struck delta. They are defined as the usual Allian derivations with, in front of them, an exponential factor.
04:43
And that makes then those invariant Allian derivations commute, which are with ordinary differentiation relative to z, which is very useful. And of course, they have no equivalent in the convolution model.
05:06
So the most two important notions, objects, that can be attached to a given resurgent functions are the active Allian algebra and the display. Take a reversion function or an algebra of such functions, then take the completion of that with respect to Allian and ordinary differentiation.
05:32
Then take the ideal of all Allian derivations that kill that completion, that completed algebra.
05:41
This is a bilateral ideal. And then take the quotient of all Allian derivations by this ideal. This by definition, this object might actually reflect the concrete mode of action of Allian derivations on that algebra. So to that extent, it is essential.
06:02
The thing is that, although by themselves, on their own, Allian derivations are totally free, totally unconstrained, in concrete situations, they are anything but free. Most of the time, they actually turn out to be equivalent, to be isomorphic to algebras of ordinary differential operators.
06:28
So we'll see many examples of that. And then there is a display, it is a sort, as you can see on the formula, it is a sort of Allian Taylor formula. But the main property and the main use is actually that of extending any relation between any number of
06:51
resurgent functions, take any relation r, involving the plus, minus, composition, ordinary derivation, etc.
07:02
And it automatically carries over to the displays, which is extremely useful because this second identity involved a huge number of constraints. So it is extremely useful to prove independent theorems and that sort of things.
07:22
So now let us start in earnest for orientation and for perspective. Let us comment on, let us examine on a double column the main similarities and differences between
07:42
Equationary resurgence and Co-equationary resurgence. This is going to be our main topic today, especially the second column. Equationary resurgence is relative to a critical variable, the variable of an equation. Co-equationary resurgence is relative to a critical parameter.
08:02
In both cases, the active Allian algebras are isomorphic, as I said, to algebras of ordinary differential operators. But in the second case, for Co-equationary resurgence, this algebra actually splits into two parts.
08:21
Correspondingly, for Equationary resurgence, there is one system of Allian difference equations that describe the whole situation, one bridge equation. For Co-equationary resurgence, there are two of them, two systems and very different ones.
08:44
Then, for Equationary resurgence, we have real Stokes constants that can assume any complex values and they are usually transcendental. With Equationary resurgence, this is perhaps the main surprise, the main difference.
09:03
Instead of that, we get discrete Stokes constants, which we call very specific objects. After renormalization, there will be integers. We call them the tessellation coefficients.
09:28
Then Equationary resurgence, as I said, all the work, all the proving has to be done in the convolutional model. But with Equationary resurgence, convolution is ordinary convolution.
09:40
For Equationary resurgence, it is not a binary operation, but a multiple convolution and a weighted one. There are weights attached to it, and it is a much more, quite different operation. And then, as I said, with Equationary resurgence, the differential operators,
10:09
which are equivalent, to which the active value in algebra is isomorphic, these can be anything, they are not constrained. With Equationary resurgence, they are heavily constrained.
10:25
They verify, they kill a certain two-form, which we call the isographic form. And then, with Equationary resurgence, the sums in the geometric models are ramified at infinity.
10:44
With Equationary resurgence, they are either not ramified at all, or finitely ramified. And so they exhibit this very unusual combination of divergence, of being a resurgence in various sectors, and being an entire function.
11:07
We call them, and there's more to that, we call them the autarch function. And then, the last event, of course, with Equationary resurgence, everything is ten times simpler and more straightforward than with Equationary resurgence.
11:22
OK, so let us go. So, very briefly, let us review Equationary resurgence, not for its own sake, but as a counterpoint to Equationary resurgence.
11:41
Take any equation, differential, non-linear, differential, difference, composition equation or general functional equation, and then form its full solution with the maximum number of parameters in it.
12:02
Then, the rule of thumb, we can say that the existence, side by side, in this equation of powers, inverse powers, z, and exponentials, either of z or powers of z, points to the existence of reversions.
12:21
Actually, there will be as many critical times as there exists blocks, let us say powers, inside the exponentials. Mercifully, in most situations, there is only one critical time, but if there are more than one, then we can live with that. There is an apparatus to deal with that. Simply, there are many, we have to consider to go through many moral planes,
12:46
and we first classify the critical times from slower to faster, and then we go through the moral planes by means of operators, which resemble Laplace operators, but with a different and more sophisticated kernel.
13:06
So, and then, once we've got hold of the formal solutions, there is a purely formal manipulations,
13:21
which tell you which are in the deviations are going to act effectively on that, and for forming the homogeneous, linear homogeneous differential equations, which the Euler derivatives are going to verify. And following this line of reasoning, we arrive very quickly at the bridge equation, which consists of three things.
13:47
You see, on both sides, the full solution with the maximum number of parameters, on the left-hand side, the invariant Euler derivations,
14:01
and on the right-hand side, the ordinary differential operators in the parameters precisely, in the parameters of the equation, and sometimes also in the variable itself. So, these operators carry the source constants that are as coefficients.
14:21
Otherwise, they are subject to no other constraints than making sense. That is to say, they must pair off a similar exponential on both sides. So, you see, this reduces the part of analysis to the minimum.
14:40
It makes everything more or less formal. So, in this case, the nature of the active Euler algebra is obvious. It is simply the native Euler algebra generated by these r sub omegas in these ordinary differential operators.
15:01
So, in this case, just two remarks. In this case, the display, it looks like a magnified, actually, full solution,
15:22
because the power series in it are going to be more or less the same, but they are two different. First, they carry in front of them Stokes constants, and then that indexation. See, I recall the definition. Instead of being by simple frequencies, by simple omegas,
15:42
the indexation is by strings of omegas. So, it is a much richer object. And then it has this property, which I mentioned a minute ago, of extending to all relations. So, this is an object which has no classical equivalent,
16:00
and which is extremely, I cannot over-emphasise this enough, and this is extremely useful. And then, just a remark, a funny remark. Resurgent functions, in all cases, in fact, they have a cohesion, which ordinary solutions of differential equations do not have, and cannot have,
16:26
in the sense that knowing one part of the full solution can constructively lead to the reconstruction of the whole solution, even of the original equation itself. You can think as an analogy of an irreducible polynomial of a cube.
16:46
If it is reducible, knowing one root will not tell you anything about the other roots. But if it is fully reducible, of course, you know the full picture, you can reconstitute the full polynomial from one root.
17:03
So, the cohere in this property. Now, let us, to better understand the interplay between equational and co-equational resurgence, we are going to focus on a problem,
17:21
a model problem, which manifests both types of resurgence. We start from the equation, the blue equation, with a small t, time variable, a system of new equations, non-linear, and in front of the system,
17:42
and possibly in the other coefficients, a perturbation parameter, epsilon. Now, it is convenient to switch to the critical variable, and to the critical parameter, and to work at infinity, instead of at the origin. So, we move from t and epsilon to z and x,
18:03
and the system becomes the red system down here. So, let us take a long, we are going to work on that for the rest of the talk, so let us take a long look at this system. I reproduced it here, in any case.
18:24
And just one remark, actually. We can assume, without actually losing much, that the perturbation parameter is absent from the coefficients b. Its presence there would not add anything of substance to the analysis,
18:44
it would simply complicate notations. And then a second remark, there is a special case, a very special case, the coefficients b, that reduce to a negative power, mainly to one of the z, to a constant of the z.
19:02
In that special case, the two, the variable and the parameter, coalesce into a product, z times x. So, in that case, obviously, the two resurgences coincide. So, something of this sameness is going to survive in the general situations.
19:23
But on the whole, it is fair to say that the differences are going to dominate. So, instead of working with a full solution, actually, we are going to work with an object which is in formation equivalent,
19:41
but more flexible. It is a formal automorphism, which the normaliser, which takes the normal form of the equation and its trivial normal solution to the system itself
20:00
and its non-trivial solution. So, this normaliser, direct and reverse, consists of two ingredients, ordinary differential operators d, that encode the Taylor coefficients of our system.
20:24
And then, the main object, from the point of view of resurgence of analysis, they are bi-resurgent monomials. So, you see down here in blue, they have a double indexation, a two-tier indexation.
20:41
The things upstairs, they are weights, complex numbers. And downstairs, these are simply analytic germs at infinity, on the Riemann sphere, with analytic continuation to the whole Riemann sphere, and possibly any pattern of singularities there.
21:00
So, again, they are going to be our main object. So, just a short aside, this is useful terminology. These normalisers, I just said that they are automorphism. And this reflects there being the contraction of ordinary derivations,
21:28
the ds and a mold, these bi-resurgent monomials, which are symmetrical with respect to the indices.
21:41
Symmetra simply means that they multiply according to the shuffled product. And the algebra made equivalent to that is Altenol. Instead of multiplication, you get zero here. And this is contracting symmetrical with derivations,
22:01
you get an automorphism. And contracting Altenol with ordinary derivations, you get, again, a formal derivation. So this is all our indexed objects are going to fall into one of these two categories, and we are going every time to contract them,
22:21
so as to get either automorphisms or derivations. So, very quickly, these, as you can see, above the bi-resurgent monomials,
22:43
which I should say, this has to be justified, but actually the fact is that they carry, they are simple objects, but they carry the whole diversions once, and the reversions, once they have been understood, in a sense, everything has been understood,
23:02
so we are going to focus on them. They are defined by this equation two, and in this case, in their case, in this instance, equation research into simplicity itself. Borel takes you from Z to Zeta, the conjugated variable,
23:23
and the recursion equation simply, actually simplifies. You can see equation two. So this equation three, may immediately give you all the information about the equation of reversions in this case.
23:40
It tells you exactly which eigen derivations are going to act on that and how they are going to act. They are going to produce elementary... Sorry. So equation of reversions is a trifle in this case. Now, co-equation of reversions is quite...
24:07
You see, in this case, the recursion equation, instead of simplifying, gets messier. We get equation eight transforms, and we perform Borel from the X to Xi,
24:23
the conjugated variable, it becomes a partial differential relation with suitable limit conditions. And in the case of depth one, the solution is obvious,
24:40
but for larger depths, there is no simple formula for solving this recursion solution. So we'll need a new operation which is precisely the weighted convolution. So, actually, again, for perspective,
25:02
let us say in advance what we are going to do. We need four things. We need a weighted convolution to express the bivariate resurgence modomials with respect to X. Then, to express their own eigen derivations, we are going to need a second type of weighted convolution.
25:23
This time, it will be a symmetral, an Alton with respect to the index. And then we'll need exact formulas for finding the eigen derivations of these two types of weighted convolutions.
25:41
And then, to express, actually, the eigen derivations, the eigen derivatives, we need a new object which is one of the stars of this theory, the tensile relation coefficients. So first, the symmetral convolution
26:04
is given by a very complicated multiple integral with a complicated multipath, and it is very nasty in a sense, very contorted,
26:21
but all the the fact is that all the calculations for for co-equation resurgence are going to be based on that. So we have to live with it. But I can, we should have, we should actually mark a pose
26:40
and describe this integral. Upstairs there are weights and downstairs there are germs in the Borel plane at the origin, which have infinite continuation, but usually highly ramified. These are the inputs. The output,
27:01
this is a statement, is exactly of the same type. The germs, the analytic germs at the origin with endless continuation and iron mitigation. So we cannot describe this integral in more detail, but we can give more telling actually characterisation,
27:21
just like ordinary convolution is the Borel image of ordinary multiplication. In the same way, weighted convolution is the Borel image of, let us say, a weighted multiplication, which is a simple integral kernel,
27:42
this one here. And this formula, although it is not, it doesn't, it is of no practical use actually. All the work has to be done in the convolution plane, the Borel plane. But from there it's a nicer picture and it has the advantage, the merit at least, of making a
28:00
symmetrality of use, because the kernel itself is quite obviously symmetrical. So, now, as I said, we need a secant to get a closed system that expresses everything. We need a secant type of convolution, this time with respect to the indices.
28:22
It can be defined in any of three ways, either as a superposition, as a finite superposition of symmetrical convolutions, or by a direct
28:42
integral formula, which is even more complex than the last one we've seen. Or again, by, in the multiplicative plane, by a simple kernel. And this kernel, as you can see, I mean, the first convolution is called, we call a secant willow. And I'm sorry, but we have to
29:00
introduce these objects. We cannot do without them. And going from one to the second, from the first to the second actually doesn't involve significant complications. again, I have to be content with the somewhat schematic
29:21
survey, but the aim is to actually show that there exists a machinery for dealing with the old situation, so that we're not condemned to deal with special case after special case after special case,
29:40
which is not a very inspiring way of doing mathematics, but I think something that deals with equation resurgence once and for all. So, what is the relevance of these two convolutions? Well, the first one, the symmetric one, it's quite simply that the by-resurgent
30:01
monomials with respect to x can be expressed in the Borel transform, in the xi plane, can be expressed as weighted convolutions of what? One of these germs. Now this is rather strange. You see
30:22
these germs, ci, xi is in the Borel plane. But it is built from data which originate from the multiplicative z plane. So we have this very unusual and rather unpleasant
30:42
jumble of two structures multiplicative and convolution. This is of the essence actually of equation resurgence. We cannot, this is how it is actually, we cannot help it. And the relevance of the second weighted convolution is simply
31:02
that we need that to express the alien derivative of the first type of weighted convolution and also the second type. This time we don't need a third one. The bug stops there. We get a closed system.
31:21
Now all this is fine as far as it goes, but it leaves the main difficulty unresolved. The thing is, how do we calculate the alien derivatives of these weighted convolutions? Now the first idea would be actually
31:42
well we might look at the integral formula and then take the endpoint close to a singularity and then see what happens. Actually it would be the height of madness to proceed in this way. Because just look, even in the case of ordinary convolution, as soon as we move
32:01
away from the origin, the integration path has a way of getting impossibly contorted and convoluted. And this is because we must not only dodge the singularities themselves, but also the mirror images of this singularity with respect
32:21
to the midpoint, to the point halfway between the origin and the endpoint. And this is for ordinary convolution, just imagine what it can be for a weighted convolution because it is a binary, the r-ity r, the number of the depth is
32:41
irreducible, meaning that a weighted convolution cannot be broken down into a chain of binary operations. It has to be taken directly in its full horror. So we must come up with something else.
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And the only salvation actually is to find a set of test functions which meet three conditions. They must be numerous in our factory to model the general ramified functions.
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They must be stable enough to self-reproduce under all the operations, convolution and weighted convolution. And then they must be, again, simple enough to lend themselves to alien differentiation in terms of
33:42
themselves, again, to get a closed system. Fortunately, there is such a system is at hand, they are the hyperlogarithm. And, but again, I mentioned this feature, this interference of the multiplicative
34:00
and the convoluted structure and the coefficients. In actual fact, the hyperlogarithm was specially suited for that purpose because they are stable with respect to convolution and with respect also to ordinary pointwise multiplication.
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But there is a hitch here. We need to differently, we will have to juggle two types of indexation because convolution adds singularities in projection. Whereas pointwise multiplication obviously keeps singularities
34:41
in place. So we need to use both the so-called positional indexations which reflect the position of the singularities themselves and incremental indexation which measures, indicates the gap between one singularity and the next
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when they are on the... Okay. Now, unfortunately, when everyone is very familiar with hyperlogarithm, but here it's a question of whether they are capable of many indexations and many forms. Here we need a very special two indexations and
35:21
we cannot spend too much time on that, but the formulas are there. There is a whole set of formulas which we will constantly use in the stakeholder. There are formulas for
35:40
finding the alien derivations when they are elementary. You see, when under alien differentiation, when the hyperlogarithm is taken in this basis, behaving exactly the way they produce elementary Stokes constant, the red objects, and they are, the logarithms
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themselves, the hyperlogarithm are symmetric with respect to their indexes and the associated monics, the Stokes constants are alternate with respect. So that we need all the time and we also need
36:20
to know exactly how they behave under ordinary partial differentiation with respect to the indices, there are formulas for doing that and then the JUMP rules, the way they have been defined these monics are piecewise analytic functions defined on a finite number
36:41
of c to the r. And we know the hypersurfaces which limit these various domains of analyticity and we need JUMP rules to describe the passage from one domain to the next. So
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these are the tools. Now, the first main set of formula. When we consider a weighted convolution with the simplest possible inputs, ordinary simple poles, what we get is far from simple. You see, depending
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how I wrote here the first for the depths 1, 2 and 3 we get a number of terms, we can express everything in terms of a superposition of hyperlogarithms. In total there are quite a lot of them. The so-called odd factorial 1 times 3 times 5
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etcetera, 2r minus 1 and you see that the u's and the v's the u's in blue, the v's in red, behave in a very specific way. There is of course an induction behind all these, there are actually two inductions, one which adds indices and one at the end and another
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forward going which I think is at the beginning etcetera. So, but this is not enough, I mean our inputs are not simple poles I mean of course this would be enough to deal with the case of meromorphic inputs, but we want to deal with all cases, so we want
38:22
to have a general hyperlogarithmic inputs. Now there is again a formula for that, I didn't write it here, but it's in the notes which I posted on my homepage in various papers. But there is an important remark here.
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The inputs themselves, the hyperlogarithm and the output is also a superposition of hyperlogarithms, but the inputs have to be inserted in positional notation, in positional indexation, whereas the outputs have to be noted in incremental
39:05
indexation. While there are tables in my paper, I mean the number of terms which you get is even is even larger. The formula in blue up there tells you how many if each of the inputs has depth dI,
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this tells you the number of terms. You see, for just to understand how complex the whole thing is, how devilishly complex it is, just take a weighted convolution of length four with inputs which are themselves
39:41
hyperlogarithm, each of them again of length four. What you gain is close to 10 billion terms. Of course there is a combinatorics behind it and the symmetry, symmetry makes everything rather manageable, but this is just to say that following the integration path,
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the multipath wouldn't have got us anywhere. We have to develop singularity combinatorics. So there is a formula for calculating the weighted convolution of hyperlogarithms
40:23
and since they can approximate anything we can take the weighted convolution of anything. now here comes the main surprise I want at least to be able to describe this very
40:42
strange object. The source what replaces the source coefficient, the tessellation coefficient and the essential discreteness. So you see in a sense we have everything, we have expressed the convolutional products in terms of
41:01
hyperlogarithm and we know how to differentiate them, but here comes a surprise when we take the sum, the huge sums and there is a gap in complexity and usually such gaps are attended by emergent properties and here the emergent properties are the
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discreteness. So let us take an example perhaps at length 3 we get exactly 15 terms and they are all hyperlogarithm and all the indices on top of them have the same length. So if we apply
41:42
an alien derivation of that u1 times v1 plus etc, u3 times v3, each of the terms is going to contribute something and we have to get the result, we have to replace each of these 15 hyperlogarithm by the corresponding
42:01
monic. So the x disappears and we have this sum. Now, and the surprise is we know how to partial differentiate these monics with respect to the indices. Now take any u or any v and apply the rules for partial differentiation and you find zero.
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This is not obvious on this formula but this is what you get. All the way to infinity. But they are not constant, they are not zero. If we apply the jump rules we find this recursion equation
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in red which I have to explain the notation but I can take my word for it. There are a fine number of domains in which the function is constant and there are jump rules which amount to explicit recursion
43:00
which can describe the whole thing. And this leads to, again I cannot explain in due detail the formula but this recursion rule leads to an explicit and simpler
43:20
expression for the desolation coefficients. You see, they have one effect actually. They are highly polarised. They are simple, they are elementary in the sense that they involve only sine functions of elementary functions which are homographic in each of the U's
43:40
and each of the B's. You see, we have the choice here between the locally constant and discrete functions of these desolation coefficients but we have the choice between expressing them as sums of hyperlogarithms which is slightly incongruous
44:00
obviously but intrinsically and an expression which is much simpler and more appealing and also from the practical point of view infinitely preferable but highly polarised. This is a situation which is not unusual in mathematics but it is
44:22
a rather extreme instance of such a situation and in fact the simplicity of these of these desolation coefficients is deceptive actually. They are rather mysterious and highly interesting objects. There are lots and lots of properties. I just mentioned three.
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They are invariant under they are symmetric with respect to their indices, their double indices. They are, excuse me, they are altenol and then they are invariant under a very important
45:01
involution of the so-called swap which exchanges upper and lower indices and that makes them be-altenol, mean twice-altenol. This is a very important property in arithmetic and the study of multisitters and all that. And then again they are discrete and for
45:22
depth one they are trivial they are always equal to one but at a higher depth surprisingly they are not constant but they are almost always zero. If you mean by any measure and what else?
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Many other properties. So they are quite highly interesting objects. Now we are in a position with the help of these two weighted convolutions and with the help of these desolation coefficients to express to do what we set out to do at the
46:01
outset to express the alien derivations of any convolution product. There are formulas for this involved three things the express I cannot describe them in detail but just I'll say it in words
46:22
we can express any alien derivation of a weighted product in terms of three things a new weighted product with new inputs which are either the translates of the old inputs
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or alien derivatives of the old inputs and new weights. this formula exists I'll have to skip them anyway if you care for that in the slides which I posted
47:01
on my homepage and now at last we can describe the we can give a complete description of the two types of research the first in terms of not of the complete solution
47:21
but as I said of the normalisers which are more flexible the first bridge equation is quite simple it is one of the standard forms up there and it involves as usual ordinary differential operators
47:42
which in this case depend on all parameters including this parameter x and it is a constant in z but it is an entire function of x
48:00
so it describes everything because it can be iterated in terms of itself so it gives all the information not so with co-equation or resurgence because the first bridge equation which you find the second one looks formally the same but actually the qs which
48:20
on the right hand side do depend on x but not as the entire function because here x is the resurgence variable the resurgence variable so to describe the resurgence in x of this new object it takes a third bridge equation which
48:41
thankfully is closed, it expresses the eigen derivations of q in terms of q itself now this is in the case of holomorphic for the first equation it is absolutely general no assumption at all for the second equation, for equation 2 and 3 we must assume that the data, the p's
49:03
are meromorphic if they are not then there is an added complication there is a new level of complexity instead of only qs you have two series of new objects new resurgence objects, qs and p's
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and they are related by the tessellation coefficients so again we would have to describe to look at this calendly we cannot do this here but the object again was to give a survey to show that there exists a machinery for
49:40
dealing with the general situation so you see I'll have to skip the end very quickly in words, but let us pause a little and take stock of what we've seen or achieved so far
50:01
you see, contrary to equation resurgence which is simple enough although it covers a huge ground but it is simple conceptually in terms of calculations equation resurgence there is a stratification of
50:21
four level stratification you see, the inputs themselves let us say they are hyper-logarithms weighted convolutions of these inputs they are huge combinations huge clusters of hyper-logarithms, again with emerging properties, quite unexpected
50:41
and then we have the qs which are again large sums of such clusters and then in the general case with ramified inputs b we have a fourth layer which again means the p's and the q's and the p's
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the last layer and the passage from the q's to the p's is again mediated by the tessellation coefficients so then again some
51:21
differences for equation we don't need to assume that the inputs b had to be germs at infinity no in the case of equation resurgence we had to assume that they were in the multiplicative plane they admitted endless continuation
51:42
with again uniformity conditions and then for the first bridge equation the index reservoir is simple enough it is generated by the multipliers by the lambdas here in this equation not so with equation resurgence
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it is much more complex it is linear expressions of the weights and the singularities of the inputs so you see but to make up for that the source constants simplify they become
52:21
the tessellation coefficients which are there is a trade-off between the complexity of the set of omegas of active alien derivation and the complexity of these source constants and then many other properties
52:41
but I want just to say in a few words there are only five minutes left the two what is the matter actually with autarchy and with isography and autarchy you see in the case of equation resurgence
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we have the q's are the q's and the p's are resurgence functions but you can imagine that they stand in close relations with the a sub omegas of the first bridge equation
53:21
because remember there is a case when both coincide and so in the first case they are entire functions of x in the second case they are a resurgence function and they are not exactly the same but there is a close relation we go from one system to the other so
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they have this double property of being entire and a resurgence in sectors so obviously this imposes on the active alien algebra certain constraints because otherwise there would be an effective ramification at infinity if there is to be no ramification
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or only a finite ramification there has to be some constraint but the interesting thing is that this constraint assume the form of this form and each operator in the active alien algebra each ordinary differential operator
54:22
kills a certain differential two form which I called the isographic form why, because it had to be given a name and so I gave example, I gave three examples and I'll produce a paper hopefully with many more but of so called
54:41
such isographic forms and then autarchy, well autarchy is precisely this property of well I'll finish on that autarchy autarchy is the opposite
55:00
of anarchy autarchy is usually spelled with a key but I wanted it to rhyme with anarchy you see it is you see roughly the autarch functions or entire functions whose asymptotic behaviour in the various sectors is fully described
55:21
by resurgence asymptotic expansions which in turn generate on the alien differentiation a closed finite system their autarchy relations and so you see despite being transcendental the autarch function have a strong algebraic, I mean finite
55:41
and algebraic flavour about them and more of they are not freaks of nature they are quite common actually if you take something at random of course it will be anarchic but in practice
56:01
it will be if it is something of any use it will turn out to be autarch and the showcase example of autarchy is of course the inverse gamma function you see on its own it is an entire function but on the right side is asymptotic expansion which is well known
56:21
and which combines with the first ramified factor actually to destroy the ramification and to produce something which despite the resurgence turns out to be entire and then the showcase is the instance of anarchy is of course the xi function which
56:42
is associated with the zeta function, the Riemann zeta function because there is a strip which defies accurate description and the two things
57:00
isography and autarchy are closely connected and I gave three examples but the first one in keeping with our philosophy we deal with the bioresurgent monomials, we want to systematically as a matter of principle we want to study and to solve
57:22
the difficulties at the most basic level and also the first example is devoted precisely to the bioresurgent monomials and to this double genes faced nature being entire and being resurgent so one moment if I may
57:42
so I wanted to say something about other types of resurgence but we have no time for that I just want to mention in 30 seconds a spin-off a fallout of this theory when we describe the weighted
58:02
convolutions of hyperlogarithm or the tessellation coefficients you observed certainly that the u's, the weights and the v's, the singularities in positional notation interacted in a very specific way. The u's
58:22
got added cluster-wise the v's got subtracted pair-wise with additional constraints and this gives rise actually to an interesting
58:41
theory. This gives rise to four to the so-called flexion structure which is actually made very informally. This is the sum total of all interesting operations which you can obtain by using four basic flexions operations which are four flexions which combine
59:01
the u's and the v's the w's are two indices mean v's and u's and the flexions combine them transform them in precisely this way adding the u's and this flexion structure contains
59:20
the dual inside the flexion structure is a set of a certain algebra and a certain group which Adi and Gari which are the properties of preserving double symmetries which is extremely useful for in arithmetic for studying what I call dimorphi. Dimorphi you can think of it as the property for a
59:41
set of transcendental numbers of a q to be stable under two different multiplication tables which are difficult to combine. So the multisitters and the hyperlogarithm form precisely, they are highly interested highly interested
01:00:00
and highly useful fundamental transcendental numbers, and they are dimorphic, and this structure, the reflection structure, is extremely useful actually to study that, and mostly over the last 15 years or so I've been basically into this into this dimorphic business, and so
01:00:21
I wanted to mention this spin-off, but actually these transcendental numbers are precisely the typical ingredients, the building bricks of these Stokes constants for equational research. So in a sense we have come full circle, and we have here a nexus of objects which is very
01:00:45
closely knit, and in my view rather harmonious. So time to stop, thank you. Any questions?
01:01:02
A couple of questions here. This is a graphic form which you mentioned. Is it bilin, it's a non-degenerate bilin form? Yeah, it is actually, it is even in the case when it is not even, when the number of variables is odd. Even in that case it turns out to be a symplectic form, but it is not given naturally in that way.
01:01:29
It is not, this is interesting because there is a quite different, there is also resurgence when you consider symplectic systems.
01:01:44
In certain cases where on top of the resonances, the standard resonances implied by symplecticity, you have other resonances. Then resurgence gets grafted into this, but the underlying symplectic structures get grafted
01:02:02
onto that, and so that the A's you see in the bridge equation, they derive from a potential. But they are real, they are continuous. So in that case also you have, the similarity is superficial, because in that case also
01:02:23
you have resurgence constants which derive from a potential, but they are continuous, they are indexed, there is no discreteness about them. So, but there is an interplay, but yes, there is a symplectic structure behind it,
01:02:42
but it presents itself naturally in an unusual form. I gave three examples in the notes. There is also a short question, there is also deceleration coefficients which are zero, plus minus one, plus minus two, yeah?
01:03:01
Yeah, after not, yeah. Yeah, after normalization, you said that very often zero can be back measured as a function of some complex parameters and... Let us say, well, in every sense you can think of. Actually, if you take the coefficients at random, it would be zero most of the time. Or if you consider them to be functioned on the Riemann sphere two time,
01:03:24
and you take the Borel measure, it will be... Although I did not actually compute the exact Borel, but this is absolutely spectacular actually. I mentioned the values for depth four actually, they are almost always zero.
01:03:43
Again, the simplicity of these creatures is highly deceptive. I have to say, well, at least I do not understand them fully. Although they are there, they are described, but we would like actually to get other formulas for them
01:04:02
and to know more about them. So there is a lot of work to do in this line. More questions? Thank you.