The Euler Characteristic of Out(Fn) and the Hopf Algebra of Graphs
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Number of Parts | 28 | |
| Author | ||
| Contributors | ||
| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/51276 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
9
10
13
22
00:00
Graph (mathematics)AlgebraReceiver operating characteristicTheoryQuantum field theoryField (agriculture)Algebraic structureAlgebraic numberGraph (mathematics)Musical ensembleProjective planeGroup actionMultiplication signReceiver operating characteristicMereologyConnected spaceSinc functionProof theoryPoint (geometry)Graph (mathematics)Right angleMoment (mathematics)Hopf algebraAlgebraComputer animation
02:31
Group actionLocal GroupBijectionAutomorphismSubgroupNormal (geometry)PermutationInterior (topology)HomöomorphismusSurfaceMassTorusObservational studyGeometrySpacetimeIsomorphismObject (grammar)Closed setHomomorphismusSet theoryMany-sorted logicLoop (music)Symmetry (physics)Connected spaceCategory of beingPoint (geometry)Pole (complex analysis)Group actionSpacetimeInterior (topology)QuotientProduct (business)String theoryPresentation of a groupGenerating set of a groupInversion (music)MultiplicationInverse elementMultiplication signMusical ensembleSurfaceMetric systemGraph (mathematics)MathematicianAsymmetryPhysicistDegree (graph theory)Line (geometry)CircleMereologyRight angleAutomorphismKlassengruppeAutomorphismRiemann surfaceElement (mathematics)Conjugacy classIdentical particlesDifferential equationOrientation (vector space)Free groupQuotient groupNormal subgroupFundamentalgruppeResultantObservational studyTorusSocial classAnalogyModule (mathematics)Computer animation
11:03
Graph (mathematics)HomotopieLocal GroupSurfaceCurveManifoldInvariant (mathematics)ModulformAlgebraic numberObservational studyReceiver operating characteristicHomomorphismusSequenceHomologieGraph (mathematics)ManifoldGroup actionPoint (geometry)LengthInvariant (mathematics)Complex (psychology)CurveReceiver operating characteristicElement (mathematics)HomologieHomotopieModule (mathematics)Cartesian coordinate systemProduct (business)AutomorphismQuotientObject (grammar)Different (Kate Ryan album)Right angleMany-sorted logicAutomorphismHomomorphismusBoundary value problemSet theoryStatistical hypothesis testingComputabilityStochastic kernel estimationNichtlineares GleichungssystemInfinityOrder of magnitudeMathematical physicsGraph (mathematics)Interior (topology)SpacetimeExact sequenceMereologyCategory of beingVibrationFundamentalgruppeElementary arithmeticFree groupMathematicsAbelsche GruppeSymplectic manifoldGauge theoryCohomologyGenerating set of a groupKlassengruppeLecture/Conference
19:36
HomologieTerm (mathematics)Dimensional analysisTheoremThermal expansionAsymptoteCoefficientParameter (computer programming)Proof theoryFocus (optics)Local GroupAnalogySimilarity (geometry)SpacetimeWell-formed formulaTheoryGraph (mathematics)Mathematical modelTheoryNetwork topologyFraction (mathematics)Receiver operating characteristicModule (mathematics)HomologieSummierbarkeitChi-squared distributionMany-sorted logicSpacetimeCombinatory logicGraph (mathematics)Parameter (computer programming)Category of beingPower (physics)Gamma functionPerspective (visual)Nichtlineares GleichungssystemHomologiegruppeMortality rateResultantConnectivity (graph theory)Hopf algebraObject (grammar)Bernoulli numberDimensional analysisPoint (geometry)AutomorphismTerm (mathematics)Group actionTheoremExpressionKlassengruppeCurveCalculationAnalogyNumerical analysisProof theorySurfaceGraph (mathematics)Thermal expansionMathematical modelTorusQuantum field theoryCoefficientAsymptotic analysisVertex (graph theory)Square numberFunctional (mathematics)RankingSocial classMultiplication signMereologyRootBoundary value problemCombinatoricsParity (mathematics)Stochastic kernel estimation2 (number)InfinityLinearizationDifferent (Kate Ryan album)Latent heatIdentical particlesPhysicistDivisorTopologischer KörperAlgebraAnalytic setMatrix (mathematics)Oscillation
28:09
Mathematical modelGraph (mathematics)Parameter (computer programming)TheoryWell-formed formulaSequenceSpacetimeVertex (graph theory)ForestElement (mathematics)Graph (mathematics)AlgebraQuantum field theoryRenormalizationLinear mapProduct (business)ConvolutionLocal GroupInverse elementTheoremLogarithmCycle (graph theory)ForestSummierbarkeitGraph (mathematics)Numerical analysisAlgebraGraph (mathematics)Functional (mathematics)AnalogySign (mathematics)Associative propertyMany-sorted logicGroup actionCombinatory logicModule (mathematics)Algebraic structureTerm (mathematics)SpacetimeInverse elementReceiver operating characteristicLinear mapTopologischer KörperTensorproduktQuantum gravityVector spacePower seriesSeries (mathematics)Proof theoryParameter (computer programming)ResultantWell-formed formulaTheoryPoint (geometry)Element (mathematics)Mathematical modelQuantum field theoryExact sequenceKlassengruppeCalculationLinearizationObject (grammar)CurveRing (mathematics)Formal power seriesSubgraphSet theoryMultiplication signSubsetMathematicianVertex (graph theory)RenormalizationNegative numberNetwork topologyIdentical particlesRight angleMultiplicationDifferent (Kate Ryan album)Arithmetic progressionNormal (geometry)Social classDifferenz <Mathematik>Slide rulePhysicistMereologyRandomizationPerspective (visual)Graph coloringVolume (thermodynamics)TowerSequenceSurfaceComputer animation
36:42
LogarithmDuality (mathematics)RenormalizationLie groupTheoremNumber theoryReceiver operating characteristicRational numberElement (mathematics)Conjugacy classInfinityAssociative propertyCalculationINTEGRALCommutative propertyComplex (psychology)Graph (mathematics)Well-formed formulaOpen setHomologieMathematical analysisInvariant (mathematics)Group actionFinitismusWell-formed formulaReceiver operating characteristicRational numberGroup actionElement (mathematics)ExistenceGraph (mathematics)Price indexQuotientCentralizer and normalizerIsomorphismSummierbarkeitExtension (kinesiology)Power (physics)HomomorphismusDuality (mathematics)Thermal expansionTheoryRenormalizationNumerical analysisInvariant (mathematics)CurveAssociative propertyCommutatorRight angleTotal S.A.INTEGRALDimensional analysisHomologieTopologischer KörperConnectivity (graph theory)Arithmetic progressionComputer programmingMany-sorted logicCalculationComplex (psychology)SpacetimeLie groupOpen setDifferent (Kate Ryan album)Cycle (graph theory)Table (information)MereologyModule (mathematics)CoefficientIntegerExpressionFeldtheorieProjective planeTerm (mathematics)Musical ensembleProof theoryNetwork topologyCondition numberRandomizationObject (grammar)InfinityComputer animation
45:14
Mathematical modelGraph (mathematics)Parameter (computer programming)Well-formed formulaReceiver operating characteristicHomologieDimensional analysisSummierbarkeitMany-sorted logicFree groupRight angleFunctional (mathematics)Algebraic structureCombinatory logicThermal expansionMathematical modelAssociative propertyCalculationGraph (mathematics)CommutatorLogarithmInverse elementResultantWell-formed formulaExpressionComplex (psychology)ComputabilityParameter (computer programming)Sign (mathematics)Power (physics)Potenz <Mathematik>TheoryInstantonMultiplication signMereologyInversion (music)1 (number)CoefficientSpacetimeMathematical singularityMusical ensembleConnectivity (graph theory)Numerical analysisSquare numberDerivation (linguistics)Receiver operating characteristicFilm editingGroup actionIdentical particlesRenormalizationGraph (mathematics)DivisorConservation lawDifferent (Kate Ryan album)Asymptotic analysisBeta functionRepresentation theoryINTEGRALConvolutionProduct (business)Rational numberSequenceVertex (graph theory)IntegerAbsolute valueTheoremLie groupProper mapNegative numberDirection (geometry)Position operatorSeries (mathematics)Similarity (geometry)Computer animation
Transcript: English(auto-generated)
00:16
First of all, yeah, thanks for for the great organization of this conference. So I'm going
00:21
to talk about a project that, of course, wouldn't be possible without Dirk. So, probably, I hope you can see it. But this is a picture of the group a couple of years ago of Dirk's group. So I was in the middle of my PhD back then. And I mean, I cannot really
00:43
point at them at the moment, but you probably can recognize Dirk there on the right, and a couple of other faces. And, yeah, it was a great time. And Dirk was a great supervisor. And he, I mean, already, like Bob DeBorro wrote in his letter, he really implemented these ideas
01:07
of letting everybody develop their own scientific style and develop themselves individually in the scope of the school in Berlin. And so it made a very colorful picture, just like this picture
01:22
there in the summer of 1915, I think, of 2015. And a couple of months afterwards, Dirk, he introduced me to Karen Fuchman. And there, she first introduced me to this problem
01:41
that I'm going to talk about today. But it turned out to take quite a while. Until I realized then that the tools that I'm going to, or that I was going to develop in my PhD, were actually capable of solving this problem. And, yes, so this is the story
02:05
that I'm going to talk about, connected to this Euler characteristic of out of n. And naturally, this wouldn't be possible without Dirk because he made this connection, but also because it's connected to the work of Dirk via this Hopf algebra of graphs
02:25
that plays an important part in this path to this proof of this problem. All right, so I'm going to introduce this problem or try to introduce this in a very like pedagogical way. And, yeah, so if you don't know anything about out of n or outer space,
02:47
you should also sort of at least understand something about it. So it would be very slow. And if you don't understand something, please just unmute yourself and ask a question or put it in the chat. I don't know if I can see it here with the chat,
03:02
but probably someone else can just ask the question then instead. So this is about groups. And especially, it's about automorphisms of groups. So there's a
03:23
an automorphism of a group is just a map from the group to itself, an isomorphism, which respects the group property. So it's pretty basic. And for every group, this group of automorphisms has a normal subgroup, namely the inner automorphisms, which are sort of the
03:41
boring automorphisms of the group. And these automorphisms are just given by conjugation of elements in the original group. And the group of outer automorphism is now the quotient of the group of inner automorphisms, or this is now the quotient group where these inner automorphisms
04:06
are quotient out. And so, yeah, and sorry, I forgot to tell you that if this is too basic for you, please wait until the end of the talk. There's going to be some new results that I
04:20
didn't present before. So also, if you have heard this talk by me before, which could be at some occasion, then bear with me in the end of the talk, I will give some some new results that are very fresh. Alright, so we have this outer automorphism group of a group, and we want to apply it to a specific group, namely the free group. And the free group
04:44
is something you also probably very well know. So it's just the group of n generators that can be thought of some lectures that you can just multiply by writing them in a string. And there's one identity that just says that the inverse of some letter times the letter
05:04
itself is nothing. And therefore, you can just form a group by multiplication of two elements, and you can also form inverses. So it's sort of the simplest group you can imagine in a sense. And this automorphism group of this free group is our main object of interest in this talk.
05:27
This object is also something very concrete, so you can write down some generators. I think Karen even actually wrote them down on Wednesday in her talk. So for instance, you can take as generators the isomorphism of the group where you just replace the first letter by a
05:49
product of two letters, and another generator of the outer automorphism group is replacing a letter by its inverse, and you may permute the letters non-trivially. So this is sort of
06:01
a generating step for this outer automorphism group. Another example of an outer automorphism group is the mapping class group, which is the group of homomorphisms of a close connected orientable surface of genus G. And you can think of this as the outer automorphism group of the
06:21
fundamental group of the respective surface. I mean, you also probably know these kind of objects where you think of the symmetry of some surface, but you can think of this also as automorphisms of the fundamental group of the surface. At least for surfaces, this works. An example is this group of homomorphisms of the torus, which is of course very important.
06:45
It's a modular group, essentially. And it has two generators, so you can cut the torus in two kinds of ways. They are indicated here as red and blue circles that you can hopefully see. And you can cut the torus along these lines and then turn it by 360 degrees,
07:03
then glue it back together, and you get a generating set for your mapping class group of the torus. And yeah, you can identify this with SL2Z. Right. So part two, so now I've spoke on groups. So now why spaces? And the reason to talk about spaces is, of course, a very
07:26
basic philosophy that you probably also all know, that this question, how do you study such objects, such groups as the mapping class groups or out of n, can be answered. And for physicists,
07:44
this is probably more intuitive even than for mathematicians, that to study sort of a group, it's always best to realize the group as some set of symmetries of some concrete object. And this is the philosophy. So we want to study these groups by inventing some object which is symmetric
08:04
under an action of this group. So the idea is to realize the group as symmetries of some geometric object. And in this case, so the group would be either out of n or the mapping class group. And of course, this is an idea that has been developed by many people.
08:24
And for the mapping class group, this construction works by the Teichmuller space. So for a Teichmuller space, what you need for as a data is some close connected and orientable surface S. And you make a space from this data by taking the space of where each point
08:44
is a Riemann surface, x. And at each point, you also have a marking. This marking is just a homomorphism from the surface, which is given as data to the Riemann surface. So you can see in this picture that so you have to imagine like this big space, each point
09:04
is a marking and a pair of Riemann surface and this marking, which identifies the surface with the Riemann surface. So an alternative way to think about this is just to think about each point to give some kind of metric on the surface,
09:21
but you can also think about it as the marking, just a homomorphism. Okay, now the mapping class group is a symmetry of the space just by composing with this marking. So you can just modify this marking by an homomorphism of the initial
09:47
surface, which you started with. And so you permute the points in the space in some way. And this is this action of the mapping class group. And of course, this action would be interesting if it wouldn't have a bunch of very nice properties. But this is just the basic idea
10:04
behind this construction. For RFN, this approach is sort of mimicked. And this was invented by Kaller and Karen in 1806. They sort of invented an analogous construction for RFN.
10:26
And this construction replaces the surface, which is data, or which is given as input data, with a very simple graph with the rows with n panels. So it's just a self-loop graph,
10:40
which has n self-loops or n tadpoles. It's a basic tadpole graph. And so in this case, we have here R3, which is the rows with three patterns. And then you construct a space from this by taking each point in the space to be a graph
11:06
with length assigned to each edge, just like in the talk by Francis on Wednesday. And at each point, you also have an additional data, which is a marking, which is in this case a homotopy from the rows to the respective graph.
11:25
So the marking now gives sort of a deformation, or it gives sort of the information how you have to deform the rows to get the graph. And just as in the mapping class group picture,
11:45
RFN now acts on the space, which is therefore called outer space, by composition with a marking. So there's a tricky point here that I'm sort of skimming a bit over that you have to identify your element of RFN with an element of the outer
12:08
automorphism group of the fundamental group of the rows. But the fundamental group of the rows is just the free group, which is quite easy to see. So this works this way. All right.
12:24
Okay, so this is a picture of outer space, just as an illustration by Karen Tufel-Nate. And you can see that it's made up of these graphs. And so there are different cells. And
12:44
in different tests, they can be the same graph. And in the different tests, these graphs, they only differ by the marking. So you can imagine now RFN permuting these cells, but always mapping like the graph on the same graph, but changes this marking.
13:04
And in the boundaries, so in the smaller dimensional faces of this thing, there are then the graphs of one of the edges is contracted. All right. So you already heard about a couple of applications of outer space.
13:25
So of course, you can apply this to study the group RFN. There's also applications to the modular space of punctured curves or tropical curves. You can use it to study some invariants of symplectic manifolds. And in Marco's talk, you heard about applications in mathematical physics.
13:47
And Francis talked about applications to graph complexes. All right. So now we want to apply all this stuff that I have been discussing now.
14:04
And the application is, of course, we want to study some invariants of those groups. And what we want to do, we want to study the cohomology, of course, of RFN, for instance. So that would be one invariant that we want to study, a topological invariant.
14:23
And the trick is that we can identify the cohomology of this group with the respective object of the quotient of outer space with the group. And this quotient is this modular
14:42
space of graphs. And this works because outer space is contractible, as Karen and Marco showed. So one of the simplest invariants that one can imagine that one would like to study, there's, of course, the Euler characteristic. And so we can study this Euler characteristic
15:05
using this quotient. All right. I briefly want to give a further motivation to look into this Euler characteristic. So why is this Euler characteristic especially
15:23
interesting for the case of RFN? The reason for this is that there's a very natural map from the free group to the free abelian group. So FN is the free non-abelian group, of course, and ZN is the free abelian group. And this map, of course, just works by forgetting
15:45
that the generators of the free group, they are non-commutative. So it just slashes them all together, and what you get is effectively an element of the abelian group. And you can derive from this homomorphism, you can just get a homomorphism of groups,
16:04
also from RFN to the outer automorphism group of ZN. And the outer automorphism group of ZN is a very well-known group. It's just GLNZ. And now one is interested in the kernel of this
16:23
map. So the outer automorphisms of the free group, which go into GLNZ, are the non-interesting or the commutative automorphisms of the free group. And one wants to study these
16:42
inherently non-commutative automorphisms. And these are in this mysterious Torelli kernel here, which I call calligraphical TN on the left-hand side. And one can just make this into a little short exact sequence. And yeah, so like I said, this non-abelian part of RFN is
17:04
now interesting. This is this Torelli kernel. And by this short exact sequence, one can relate the Euler characteristic of RFN with the product of GLNZ and this Torelli kernel, just by the vibration property of the Euler characteristic. And so this Euler characteristic
17:27
of RFN here on the left-hand side of this equation down there, it gives sort of a leverage to look into or to peek into this interesting kernel. But there's something really interesting
17:40
happening there, namely that the Euler characteristic of GLNZ is zero for all N larger than three. So with some very basic arithmetic, you can see that there's something funny going on, except for the case where the Euler characteristic of RFN vanishes. So
18:02
if the left-hand side also vanishes, this is all fine. And then it just means that the Euler characteristic can be anything. But computations, they actually show that the Euler characteristic is not zero for most cases. And one can actually prove, or I mean,
18:21
we actually proved eventually that it's not zero for any N larger than two. But from this, one can actually deduce information still on this kernel, namely that it has one finitely generated homology for N larger than three if this Euler characteristic of RFN is not zero.
18:42
So this is interesting that one can leverage information on this Euler characteristic into this kernel and deduce that it has not finally generated homology. So the conjecture that John Smiley and Karen had in actually seven is that the Euler characteristic does not vanish. And they also conjectured that the Euler characteristic grows exponentially or the magnitude
19:05
grows exponentially for N to infinity. And they performed some initial computations of this Euler characteristic up to N to 11. And these computations, they were later strengthened by
19:20
Sagie up to 100. But yeah, so the question was, of course, if one can prove this in general. There's a related conjecture, a stronger conjecture due to Magnus, which states that this Torelli kernel is not finitely presentable. And this says in topological
19:44
terms that the second homology group of the Torelli kernel is infinitely generated. Which, of course, also implies that the Torelli kernel does not have finitely generated homology.
20:01
And also, Bess Wiener-Waks and Margalit in 2007 proved that this Torelli kernel indeed does have a non-finitely generated homology. But they didn't touch any of the other two conjectures up here by doing this. They used a different argument to actually show that it has
20:21
non-finitely generated homology. So the upper two conjectures remained open. And yeah, so the result of Karen's and my work was now to make the upper conjecture, the initial conjecture on the Euler characteristic into a theorem.
20:42
So the result is that the Euler characteristic of autofin is not zero. And yeah, that could be the end of the talk. But I will explain some details in the rest of the talk on this result and give some further aspect and newer results too.
21:02
So the theorem explicitly is that the Euler characteristic is always smaller than zero for n larger than two. And that it has a specific growth rate, which is given by the gamma function for n to infinity. And there's this weird log squared term here in the denominator, which I don't really know how to interpret. But this is how the
21:25
oscillator matrix actually looks like. So this settles the initial conjecture. But there are a lot of immediate questions. For instance, the large growth rate of the Euler characteristic, it indicates that there's huge amounts of homology in odd dimensions of these groups. And nobody knows where this homology is coming from, because there's
21:45
only one odd dimensional class known in rank seven. And this is due to a large computer calculation by Bertoldi in 2016. And the question is, of course, what generates all this homology. So I will talk a bit more about this later. But first of all,
22:07
I will talk about how we obtained this theorem. So we proved this based on an implicit expression for these numbers chi out of n. And this expression looks as follows. It's a bit complicated,
22:20
but bear with me. So it's an asymptotic expansion of a function that you see there on the left-hand side, this 2 pi i square root times e to the power of minus n times n to the power of n can be expanded asymptotically for very large n in terms of the gamma function.
22:42
And if you do this, you find that the coefficients, they encode these numbers, this Euler characteristic of n. And this is what we proved. And one can deduce from this that one can deduce properties of these numbers from this theorem. So there's
23:04
analytic arguments, which is quite complicated, that goes from this asymptotic expansion expression or the implicit expression of these numbers to the actual result that the Euler characteristic is always negative and the growth rate. But in the rest of the talk,
23:25
I will mainly talk about how to prove this implicit equation here, because it's connected to theory. And one can also use this Hopf algebra of Dirk to actually prove this, or it's used to prove this. So just to put this a bit into perspective, an analogy to the mapping class group,
23:48
or the analogous calculation of the mapping for the mapping class group was performed by Hara and Zagier. And in this case, things are much more beautiful in a sense,
24:03
because there you get an explicit result for the Euler characteristic. And they just found that the Euler characteristic of the moduli space of curves of GNSG or mapping class group of a surface of GNSG is given by the Bernoulli numbers divided by some fractions,
24:25
divided by some stuff. And yeah, the original proof of this fact was due to Hara and Zagier, also in 86. But there quickly came an alternative proof based on quantum field theory or topological field theory methods later by Penna. And this proof was then eventually
24:47
simplified by Koncevich in 92, who also used topological field theories. And this simplified proof was sort of the blueprint for our proof of the Euler characteristic for chi out of n.
25:03
And the only thing that we sort of added was this inverse philosophy that comes from the Hopf algebra approach of Dirk to quantum field theory. So how does this prove by Koncevich of this Euler characteristic for the mapping class group
25:24
work? So the first thing that Koncevich did was prove this identity, which relates on the left-hand side, the mapping class group or the Euler characteristic of the mapping class group with a sum of graphs. So this is mgn. So you've also seen this in this talk
25:45
in this conference already. So it's not the modular space of surfaces of GNSG, but there's a generalization with n punctures, and you probably all know that. But the ingenious thing to do is to form sort of this weird linear combination
26:04
of different numbers and package them together in this generating function with the strange 2 minus 2G minus n factor popping up there. And if you package it up this way, you can calculate this linear combination of numbers in terms of a generating function that
26:25
or in terms of an expression, which every physicist immediately knows how to evaluate. So it's just a sum of our connected graphs. And I have to add that. So you have to take all graphs which are connected and which have neither zero, one, and two valid vertices,
26:48
but three or higher valid vertices are allowed. And for each graph, you take the parity of the number of vertices, and you divide by the automorphism group of the graph. And you take
27:01
into account so that you mark the Euler characteristic of the graph, which is just the number of vertices minus the number of edges of the graph. And so the difficult part, of course, is to prove this identity and to consider this proof using a combinatorial model of mgn
27:20
and based on this model by Penobet, which is a model from ribbon graphs. And you probably also know the story how this works. So it's a little picture of this. So I unfortunately don't have a pointer. But let me see if we can do this. So on the left hand side, you see this
27:40
ribbon graph gamma. And you can convince yourself that it has one boundary component. And it has Euler characteristic minus one. So from this data alone, it follows that you can embed this ribbon graph on the torus and stretch out the ribbon graph, such that there's
28:01
only one point left on the torus. And so therefore, you get sort of a model of a surface of genus one with one mark point. And this gives you sort of the model for m11 for the mapping class group, genus one with one mark point of the modular space.
28:22
All right, so consider this model to prove this identity. And then the easy part for physicists, maybe for mathematicians is a bit more complicated, is to actually evaluate them sum on the right hand side, which you do via topological theory, argument. And if you do this,
28:42
you see immediately that you get these negative zeta values that result in each order and that. And then sort of to recover the Hara-Zaghi formula for the mapping class group Euler characteristic, you just use a short exact sequence argument, which relates modular spaces
29:01
with different numbers of punctures. So you can just forget one puncture on your surface. And this is a nice map, so you can use it to relate the Euler characteristic of these spaces. So this is the proof of the Hara-Zaghi formula in sort of one slide, due to
29:23
which is pretty nice that it is so short in a way. And the proof of our results sort of works analogously. The only thing that we added is this sort of this, this randomization argument.
29:41
So to actually make this argument work, it's useful to take sort of an algebraic perspective, just in the philosophy of Dirk. So we take H to be a vector space, which is spanned by all kinds of graphs. And so all the graphs, they are connected and
30:02
three-valent or higher. And you can just now formally write these kind of objects as this linear combination here of the Euler characteristic of the modular space. As sort of an evaluation on an infinite sum of graphs,
30:24
which is sort of an element in the formal ring of power series on this vector space. And this map phi, it maps from this vector space to Q. So it associates to every graph just this alternating sign on the number of vertices, which you can see below.
30:46
So this phi is very easy to handle via topological field theory. But if you look into the analogous picture for Artifan, you have sort of a similar, you can write it down
31:00
in a similar way. But this map that you need to evaluate on all graphs is much more complicated. It's not just the sign of the number of vertices of the graph, but a non-trivial sum. So here,
31:21
in the top, I wrote down this generating function of the Euler characteristic. And you can also express this as sort of an evaluation of such a character on this formal sum. I call this character tau. And this x is again the formal sum over all graphs. And this tau
31:41
maps again from the vector space of graphs to Q. And to each graph, we associate a sum over all forests of the graph. So a forest is just a subgraph with no cycles. And for each forest, we take now the alternating sum over. So we take this alternating sum over the forest,
32:03
where we take into account the number of edges in this forest. So this character or this function on this vector space is harder to evaluate. But the Hopf algebra comes to the rescue. So this is not directly approachable via TFT.
32:23
And also, I forget to mention how to actually prove this. So this was already proven by Smiley and Karen, by John Smiley and Karen. And you need this forest collapse construction, which is due to Mark Haller and Folkman, to prove this kind of identity there.
32:45
So to make progress there, one can employ this Hopf algebra of graphs. So we can promote this vector space to an algebra by just allowing for multiplication and just freely multiplying
33:03
graphs. So we also have to take disconnected graphs into our vector space now. And we can define a coproduct. And probably you all know this coproduct that takes a graph and maps it onto the formal sum over tensor products of the graph times the respective
33:23
tensor products of subgraphs. So there's a subgraph on the left-hand side. And you contract the subgraph on the right-hand side. And you specify which subset of subgraphs you allow. And in this case, it's the set of all bridgeless subgraphs that you want to sum over.
33:43
And if you follow this construction through, you obtain the core Hopf algebra of graphs. And Dirk invented this or generalized this renormalization of algebra in this sense in 2009. And of course, this core Hopf algebra is very related to the normal renormalization
34:05
of algebra. And it's even the renormalization of algebra of quantum gravity. So this is an example of a coproduct calculation of this algebra. So you can see that if you hit the wheel with three spokes with this data, then you get a formal sum over
34:24
graphs. And in each tensor products summoned, you end up with respective prefectures giving you the number of these kind of graphs that appear there. All right, so the important thing
34:45
now with this Hopf algebra is that using this construction, you now have a group structure on these kind of linear maps from the graph to something else. So linear maps as this very simple phi map for the moduli space, of course, or this more complicated tau map
35:03
that I had before for out of n. And so under this map or under this group of multipliers, so you can multiply them, but they actually form a group under this multiplication.
35:21
And the interesting thing now is that this map phi, which is associated to the modular space of curves and the map tau associated to out of n are mutually inverse elements under this group. So this core Hopf algebra or this Hopf algebra of bridges graphs, it sort of
35:45
dually relates these two characters. And this is sort of key to this implicit formula for our Euler characteristic. So in a way, this means that tau is the renormalized version of
36:02
phi or the respective counter term to phi. So we can write this down in a more physical way. So remember that we can evaluate the strange linear combination of Euler characteristics for the moduli space case via this topological field theory here in the top on the right-hand
36:25
side. So you can see that you integrate over an action or a functional sort of thing. It's just a one-dimensional integral in essence, but you can interpret it as a quantum field theory.
36:43
And the action is this 1 plus x minus e to the power of x up here. And you take the log of this whole thing to get the connected graphs. This gives you an explicit formula for the Euler characteristic of the moduli space.
37:02
But this duality between this phi and tau map implies that these chi-autofan terms, they are encoded by the renormalization of the same topological field theory. So by renormalization, I mean that you can set up an implicit
37:22
renormalization condition, so to say, where you say that this topological field theory is supposed to vanish if we add certain counterterms. And these counterterms, they are just encoded by our numbers chi-autofan. And this is the trick.
37:41
And this way, you get this implicit asymptotic expansion for our numbers chi-autofan. All right. So this is it. So if there are any questions to this initial calculation,
38:01
please ask them. I will still briefly talk about some extensions of this work. So as an outlook, what I talked about was just the rational Euler characteristic of these groups. But you can, of course, also just go into the naive Euler characteristic,
38:22
which is just the normal alternating sum of the batch numbers of the group. But this thing is much harder to analyze than the rational Euler characteristic, because it does not behave nicely under isomorphisms or homomorphisms.
38:43
But there's an explicit formula for this naive Euler characteristic, which is due to Kenneth Brown. You can express it as a sum on finite order elements of the initial group
39:00
chi-autofan in this case. And the normal, the rational Euler characteristics of the centralizers that correspond to these finite order elements. And so our investigation so far, they indicate very strongly that the quotient of this naive Euler characteristic
39:25
with the rational Euler characteristic, it approaches a finite constant for n to infinity. So this will then automatically also prove that this indication of all this existence of
39:43
homology in odd dimensions is actually there. So that would prove this existence of this homology. And yeah, we didn't quite prove this yet, but we are almost there. It's a bit technical to do this kind of stuff. What this also leads to and how it connects
40:03
also to Francis' talk is that this is sort of all part of a bigger picture of these graph complexes, of these conceived graph complexes. So there's a trinity of graph complexes. There's the associative graph complex, the commutative graph complex, and the
40:23
Lie graph complex. And I already indicated here in this table that the associative graph complex, it's a lot of belongs to the world of modular spaces of curves. And so it's these very coarse invariants as Euler characteristics. They have been studied by Hara Zagir already,
40:44
or one can already associate them to it. There's this commutative graph complex, which is already a bit more mysterious. So the rational Euler characteristic of this is really simple. It was
41:02
written down by Koncevich in 93. And Wilwecher and Zivkovich, they computed the integral Euler characteristic up to some order. And this commutative graph complex is also the one that Francis talked about on Wednesday. And, yeah, as you learn from that, it has also
41:24
very complicated homological structures. And it's kind of mysterious where the cycles come from in this object. But there's another graph complex, the Lie graph complex, which is sort of related to autofan, in a sense. And you can calculate the rational Euler
41:47
characteristic of this graph complex, also with Koncevich methods. That's a bit of a different expression for it. But the integral Euler characteristic for this is unknown, or there's no formula for it. There's only some calculations by Murita, who used a supercomputer to calculate
42:05
the integer Euler characteristic up to order 11. But recently, I figured out a way, with Carol, we figured out a way how to actually also write down a closed formula for this
42:27
Euler characteristic, for this naive Euler integer Euler characteristic. So this is the formula. And so it's sort of explicit. It's an infinite dimensional integral over an infinite dimensional
42:41
space. And so you can expand this and actually calculate these numbers. And when I implemented this formula, I was able to get 14 coefficients out. And I was really happy because I got more coefficients than the initial Murita calculation up to order 11. But then I showed this program
43:07
to Jos Vermaseren. And he sent it back to me, and it was much faster. And then it was 40 coefficients. And yeah, just before the talk, Jos emailed me and told me that he found
43:22
another trick to make the calculation faster. And now there are 70 coefficients of this integral Euler characteristics known. So this is quite some progress there as well. And this is in preparation, so to say. Yeah, so this brings me to my summary. So the short summary is that
43:44
the Euler characteristic or the rational Euler characteristic of autofence non-zero. And there are lots of open questions exactly on all this homology and where it comes from in this group. So this rational Euler characteristic itself, it indicates that there's
44:01
much homology in odd dimensions, but it's not known where this comes from. And our investigations into the naive Euler characteristic, they support this totally. But the question, of course, remains what generates this? Yeah, and so there's also
44:23
other questions. So can one sort of explain this kind of weird duality between these two things? So there's already in the original paper there by Consevel, he indicated that this is like a casual duality. But it would be interesting, I think, to try to understand this on this
44:45
very graphical level, on this level of the software. And of course, yeah, one can also think about generalizing this to things like right-angled team groups that Karen was talking
45:00
about in her talk. All right, so thank you. I also would like to take the opportunity to also still thank those two people here in this picture for organizing this great conference. So this is a rare sight of them from the front. So usually if you go hiking with them,
45:24
you only see them from the back because they are so fast usually. So yeah, thanks, you two, you are great for organizing this great conference. All right, let's thank Mishi for his great talk. Are there questions?
45:47
Yes, I have a question if I'm allowed to ask. Absolutely. You have explained that Koncevich formulated his theorem by this formula where he takes the sum over graphs, and these graphs are
46:03
ribbon graphs, right? Yeah. And later you formulate the renormalization, the Hopf renormalization, also with ribbon graphs, or is it with other graphs? I know the trick there is that you can forget about the ribbon graphs.
46:22
So this is sort of this trick, how this identity works. So you take this non-trivial linear combination to be able to forget about the ribbon graph structure. But you're right, so you go from ribbon graphs to connected graphs.
46:43
Okay, because ribbon graphs and the other graphs have different automorphisms, and the groups are different, right? Or is it? Yeah, but you can project down to graphs if you forget about, so there's an easy map from ribbon graphs to normal graphs by just forgetting about the orientation of the vertices.
47:06
Okay. So this is a nice function, and this expression up here on this left-hand side, this just encodes sort of this forgetful map, which forgets the individual
47:22
orientation of the vertices. So this is part of the trick. And yeah, you're right, I skipped over this because of time. Okay, so there's some subtlety there, that's true.
47:44
All right, while other people are thinking of a question, I just can't resist to ask, what does the beginning of the sequence for the integer Euler characteristic look like? Okay, yeah, it looks, I mean, I can, I don't know if I,
48:07
it's a couple of numbers. I mean, it's not really, there's positive and negative ones, so it's not exclusively negative or positive. And it also vanishes,
48:20
I think, for a couple of initial numbers. I don't know, I think I have some coefficients here. Yeah, I can send you the paper about the, with the initial 11 numbers by Mobita. And I can also send you a list, but yeah, sorry, I didn't include them in the presentation.
48:45
And like I said, was recently calculated 70 of them, I think yesterday. Francis has a question. All right, thank you. Forgive me if I've asked you this question before,
49:00
but you had tau and phi that were inverses under the convolution, convolution product. So I wonder, so you've got, say tau to the power of one and tau to the power of minus one, but you can also look at tau to the power n under the convolution. Have you, do you know, have an idea of what that might mean?
49:25
Um, no, sorry. No idea. Yeah, I think we, yeah. Yeah, I have to think about this. No, I don't. I, from, from the back of my head, I don't know anything about this thing.
49:49
Yeah. So maybe there's something better directly. I don't have any idea. Sorry. I was wondering since, since you mentioned that you have a, you have this picture with the phi
50:05
and tau for the out of n and, and the MDNs, the moduli spaces. So do you also have a way to, to the commutative graph complex and it's order characteristic? I mean, you mentioned the papers where people have computed this with other methods, but is there a way to formulate this in a similar way in your approach?
50:23
Um, yes, yes. Character in that case? The character would be, um, well, yeah, so yeah, actually, I mean, it will be the same one for the rational order characteristic.
50:43
It's the same one because, uh, it's surprisingly the associative and the rational associative and the commutative, uh, graph complex, they have the same order character, the same rational order characteristic, but this is kind of an accident. Um,
51:00
so for the integral order characteristic there, you have to work more and yeah, I, uh, also because Francis was talking about this, I already started to look into this and try to actually extend these methods, these new methods for the, for the integral order characteristic,
51:21
for the Lie graph complex to the computative graph complex, to sort of get in similar expression that I showed in the back, um, in the, in the end of the talk for, um, for the commutative graph complex. But, um, yes, I, I, uh, wasn't able to do this now, um, on the side of the conference, but yeah, I think it's totally possible. It should
51:44
be actually not so hard. And it's just a question to get the science right eventually. So it's very just, yeah. So it's, uh, so this is sort of the challenge with this things to get, get all the factors of two and science properly. So this is the, the challenge and, um, yeah.
52:06
And also, I mean, this works by, you can interpret this as, as, um, some doing some representation theory on graph and, um, yeah. So one can totally also do this for the commutative graph complex there, but they have already been different methods by, um, by Komsiewicz
52:25
and Zivkovic, who, who did this calculation, but they got sort of a different formula and they don't have any asymptotic results as far as I know. Thank you. Are there any final questions?
52:40
Can I ask a question, Karen? Yes. So Mishi, do you have any sort of easy way to explain the appearance of this strange asymptotics inverse square of log n? Yeah. I mean, I, I know where it comes from because, um, this is how I calculated it. Um,
53:02
it comes from, uh, the, so the Lambert W function, it has a singularity, um, on the minus one branch. So, um, the Lambert W function that has a branch cut, and if you go around the branch cut once and you go to the minus one branch, then you see
53:24
a logarithmic singularity. And, um, if you take them the first derivative of this singularity, you get a one over log singularity. So it's sort of a log to the power of minus one. And, um, this is the type of singularity that pops up there.
53:42
It's a pretty funny one. And, um, it has this kind of. Okay. Or is that Lambert W sneaking up, isn't it? Yeah. Okay. Yes, it's guilt by association because I remind people that Lambert W occurred in a paper by
54:03
Deacon and Karen, when they made a toy model of beta functions. And it, uh, it's the absolute of the essence of the, uh, of the all of the summation that, uh, Reimar talked about. And now you see it here in this context. It's, uh, it's also in the stuff Gerald and Mishi did together, uh, that I'd love to understand better. It's related for an idea of a video
54:24
costume called trans asymptotics, where you, where you re-sum these trans series in the other order. So you re-sum all orders of the instanton expansion for each order of the perturbative expansion. And that naturally leads to this, um, Lambert W because you're changing from exponentials to powers of the, um, small parameter. So it's, it's deeply buried in this
54:48
story as we all know. All right. Well, let's thank Mishi again.