Transcript

On the Topological Invariance of Helicity - Oliver Edtmair

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· 1 h 9 min · 9,531 words · 72 timestamped sections

This is the full, timestamped transcript of "On the Topological Invariance of Helicity - Oliver Edtmair", an episode of Institute for Advanced Study published January 16, 2026 (1 h 9 min, about 9,531 words). It mentions 14 people, works and ideas, including Ed Oliver, General Hap Arnold, Calabi, C corp, C0.

Who speaks in this episode?

What is discussed, and when

  1. – 10:00: General Hap Arnold, Ed Oliver, C corp, Euler
  2. – 20:00: Gamb, David J. Novak, General Hap Arnold, Giriwan
  3. – 30:00: Calabi, C0
  4. – 40:00: Calabi
  5. – 50:00: Calabi
  6. – 1:09:23: Ed Oliver, General Hap Arnold, Puren

Notable quotes from "On the Topological Invariance of Helicity - Oliver Edtmair"

  • Oliver today who tell us about the topological environment of Felicity.
    about Ed Oliver · at
  • That's a fluid that's incompressible and moves without friction that's described by the oiler equations. So in such a fluid one has a velocity vector field that's a time…
    about Euler · at
  • The origin yeah. So now Arnold gave an alternative description of helicity that illuminates a little bit better what this quantity is supposed to measure. and this works as follows.
    about General Hap Arnold · at
  • Does this extend and just for homorphisms with no extra regularity and then the second question is an extension question so one wants to know whether one can define a helicity…
    about C0 · at
  • I should also mention some previous works. it's a particular nice one is this work of Gamb and there they answered the invariance question for disk map suspensions. So one…
    about Gamb · at
  • This work is connected to these kind of ideas and then also there's a very recent work that appeared in the archive actually this week. so need a bit more time to digest it. It's by Giriwan and Novak. So these are people working in fluid.
    about David J. Novak · at
  • Very nice relationship between helicity and the calabi homorphism. So in some sense one can think of elicity as generalizing the calabic homorphism.
    about Calabi · at
  • And this in some sense is very it's like very explicit. really it ties in very well with the works of Puren on polations. this old work from the 70s and well our kind of arguments it kind of continue this.
    about Puren · at

Who and what is mentioned in this episode?

People

  • Ed Oliver — Person · Main speaker (guest presenter) · 4 mentions · first at
    Delivers the talk on helicity, its invariance/extension to C0 flows, and outlines the paper’s main results and proof strategy.
  • General Hap Arnold — Person · Referenced expert (originator of key viewpoint/questions) · 5 mentions · first at
    Credited with an alternative (linking-based) description of helicity and with posing the invariance/extension questions under volume-preserving homeomorphisms/C0 flows.
  • Calabi — Person · Referenced expert (namesake of Calabi homomorphism/invariant) · 9 mentions · first at
    The Calabi homomorphism/invariant is central to the talk’s plug-insertion formula and the universal C0 extension of helicity.
  • Gamb — Person · Referenced researcher (prior related result) · 2 mentions · first at
    Cited for prior work proving helicity invariance for disk-map suspensions under conjugation by homeomorphisms (likely refers to K. Gambaudo & É. Ghys).
  • Euler — Person · Referenced expert (namesake of Euler equations) · 1 mention · first at
    Referenced via the Euler equations governing ideal incompressible fluid motion, the physical origin of helicity as a conserved quantity.
  • Helmholz — Person · Referenced expert (classical theorem in fluid dynamics) · 1 mention · first at
    Cited (with Kelvin) for the classical theorem describing vorticity transport, motivating helicity as a conserved quantity in ideal fluids.
    Right there's a very old theorem that goes back to Helmholz and Kelvin and that says that this vorticity vector field evolves in a very special way.
  • Kelvin — Person · Referenced expert (classical theorem in fluid dynamics) · 1 mention · first at
    Cited (with Helmholtz) for Kelvin’s circulation/vorticity transport ideas underpinning helicity conservation in incompressible ideal fluids.
    Field. So this fi dels one how the fluid particles move around in space and the consequence of this is that the helicity is a conserved quantity of the fluid and this follows…
  • David J. Novak — Person · Referenced researcher (recent related preprint) · 2 mentions · first at
    Appears with Giriwan as coauthor of a recent preprint on helicity for low-regularity vector fields and non-conservation in weak Euler solutions.
  • Perig Müller — Person · Referenced researcher (prior related result) · 1 mention · first at
    Cited (with Spath) for work connecting helicity-type questions to C0 symplectic/contact geometry via extensions to continuous strictly contact isotopies.
    Then Kamadoo she showed that one has the invariance. So the two vector fields have the same valicity. there also is work of Müller and Spath. and this already connects the…
  • Spath — Person · Referenced researcher (prior related result) · 1 mention · first at
    Appears alongside Müller in cited work on continuous strictly contact isotopies and C0 symplectic/contact geometry related to helicity extensions.
    Then Kamadoo she showed that one has the invariance. So the two vector fields have the same valicity. there also is work of Müller and Spath. and this already connects the…
  • Giriwan — Person · Referenced researcher (recent related preprint) · 1 mention · first at
    Cited (with Novak) for a very recent arXiv work extending helicity to low-regularity vector fields and producing weak Euler solutions where that helicity is not conserved.
    This work is connected to these kind of ideas and then also there's a very recent work that appeared in the archive actually this week. so need a bit more time to digest it. It's by Giriwan and Novak. So these are people working in fluid.
  • Puren — Person · Referenced expert (foliations background) · 1 mention · first at
    Cited for foundational 1970s work on foliations that the speaker says their constructive foliation arguments build on; likely refers to William Thurston (mis-transcribed) or another foliation theorist, but transcript says 'Puren'.

Concepts & topics

  • C corp — Concept · Core topic · 1 mention · first at
    Central to the discussion on helicity and its properties in the context of volume-preserving vector fields.
    Yeah well thanks a lot for giving me opportunity to speak here. I will talk about elicity and some C properties of it and this is all going to be joint work with soini and yeah…
  • C0 — Concept · Core topic · 1 mention · first at
    Discussed in the context of extending the concept of helicity to continuous volume-preserving flows.

Full transcript with timestamps

Oliver today who tell us about the topological environment of Felicity. Uh yeah well uh thanks a lot uh for giving me opportunity to speak here. Um I will talk about elicity and some uh C properties of it and this is all going to be uh joint work with soini and yeah just let me know if you have any questions or uh can't hear me well or anything like that. Uh right so I should start by uh explaining uh what helicity is. Uh so our setting is going to be a closed three manifold y that's equipped with a volume for mu uh and just for simplicity of the talk I want uh y to be the three sphere but this is not a necessary assumption one can drop this quite easily uh on this free manifold I want to look at a volume preserving vector field x and then helicity is a real valued invariant of such volume preserving vector fields and at the Most economic definition goes as follows. So first one forms the inner the product of or the

contraction of the vector field in the volume form. This gives a do form omega and because x is volume preserving this is a closed do form. Uh now because we under three sphere so that's where theification comes in. This closed do form actually is exact. So one can choose a primitive one form alpha that's this alpha here. And then helicity is defined uh by this integral here where one takes the integral of the free form alpha d alpha over the three manifold y and then there's a short argument that shows that this actually is independent of the choice of this one form alpha.

Now so this expression here this differential form here is familiar from contact geometry. So that would be the contact volume of a contact form. Uh but here we are more general than that. We really take any volume. So vector field and uh this might not be a contact type situation. Uh right. So the fundamental property of elicity that's very important for this talk is it's invariance under volume reserving the fumorphisms. uh if f is a volume preserving the fummorphism for three manifold uh then the helicity of the bullback of the vector field under thisorphism is the same as the original helicity so it's quite straightforward to prove this with our definition of elicity here one simply has to notice that if one replaces x by the bullback one can also just replace uh the one from alpha by the bullback because the ifent preserve the volume form and then it's a relatively uh short computation to check

that the list is invariant. Uh right so now uh what where does this construction come from? The origin is in hydrodnamics and in uh oh no this should not be in brackets just in hydrodnamics. Um and it was introduced in this context as a conserved quantity. An example is this is the motion of an ideal fluid. So that's a fluid uh that's incompressible and moves without friction that's described by the oiler equations. So in such a fluid one has a velocity vector field that's a time dependent vector field and then each time it tells tells you how the the fluid particles move at the different locations in space. H but it's a second vector field. That's called the vorticity vector field.

That's also important in a fluid. It's the curl of the velocity field. And that one tells you how the fluid element turns at a prescribed location and uh time. uh right there's a very old theorem that goes back to Helmholz and Kelvin and that says that this vorticity vector field uh evolves in a very special way. So what happens is that the vorticity vector field at time t is simply the push forward uh of the vector field at time zero under this map fi here. And this map fi is the the flow that's induced by the time dependent velocity field. So this fi dels one how the the fluid particles move move around in space and the consequence of this is that the helicity is a conserved quantity of the fluid and this follows from um the invariance right so because it's an incompressible fluid or is the volume preserving the fumorphism and then the helicity was invariant in the pull back or push forward and such

things so it's constant in time that's the origin Uh yeah. So now uh Arnold gave an alternative description of helicity that illuminates a little bit better what this quantity is supposed to measure. Uh and this works as follows. So this description is based on the flow generated by the vector field and in particular on the linking of flow lines. Now so what one does is that one picks an initial point P and an initial point Q in the three manifold and then we pick some uh so our goal is to measure the the linking of the two flow lines starting at these two points and one does as follows. One picks two large but finite times S and D and then one looks at the corresponding finite flow line segments. So the P flows for some some time and ends up at fi S of P and the Q flows for some time and then ends up at the five T of Q. Now so when one gets these two actually very entangled flow line segments to get a linking number we

want a link. So do loops. Uh and to get loops what one does is one simply closes up uh the flow line segments by some reasonable choice of short path. So for example one pick an auxiliary metric and make a choice of shortest segment. Then one gets a linking number for every point P and Q and for times S and T. And the next step is to let uh the time go to infinity and take uh an average. So there's an aotic theorem that tells us that if one does this, one gets uh a well- definfined linking number as linking number for almost every pair of points P and Q and moreover the result is an integraable function. So one can integrate it over the product of the manifold with itself. uh and then Arnold proved that this exactly recovers uh the helicity as defined previously in terms of differential forms and also should mention the work of Fogle here.

So Arnold was not 100% precise about everything and then Fogle uh later made this proof fully regorous. Uh okay. So now uh Arnold asked the following question. Uh the first question he asked is uh whether is also invariant under conjugation by volume preserving homeomorphisms. Uh so here's what this means precisely. If we have two smooth volumes of vector fields and x1 and then x2 they both generate flows and if we assume that there exists uh some volume serving homeomorphism also orientation preserving that's this plus here uh such that the two flows are conjugated in this sense for all times t. The question is whether these two vector fields then share the same helicity. uh so if f is a diffmorphism or any sort of regularity uh then this is this previous invariance that I explained and I wanted to know does this extend and just for homorphisms with no extra regularity and then the second question is an extension

question so one wants to know whether one can define a helicity for c volume precelling flows and what I mean by this uh here is a continuous groupomorphism from R to the volume preserving omorphism group given such a fi is there any sense in which any way in which we can define uh an extended helicity uh yeah the second question is supposed to be the harder and more general one uh the reason this one wants this extension to be reasonable in some form so in particular it should be invariant on the homeomorphisms if it's defined for zero flows it should be invariant in the dose so then it imply ideally the the invariance part.

Uh there are some uh motivations uh for this question. The first one is just from the point of view of dynamics. Um so we have felicity it's up smooth invariant of smooth flows. Uh so link can ask uh is it actually a C or topological invariant? That's a natural question. uh there also is motivation coming from fluid dynamics. So recall that in this context solicity was this conserved quantity on the excelicity of this vorticity vector field that's conserved. Uh and uh yeah this is a bit of a vague motivation but you could imagine a smooth fluid that evolves and maybe at some point runs into singularities.

uh and then you could wonder uh does solicity uh conservation maybe make uh sense beyond this point. So in this context it could be interesting to uh understand homorphisms. Another motivation uh just comes from zero simple geometry. So this is uh uh an interesting field geometry. So well one likes to find uh applications of that. Okay. Now uh why so what what is uh interesting or hard about the question? Uh right so we can first examine uh the original definition in terms of differential forms. Uh right. So here one one one forms this two from omega pick the primitive one from alpha and and takes this integral and uh yeah wants to to prove the or answer the question based on on this definition one runs into some immediate problems. So for the invariance question in the smooth setting there was this bull bag of alpha under diffmorphism f involved but if f is just a homeomorphism one doesn't know how to define this and

then for the extension question if if our c flow just was this continuous group homomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomorphism into the volume serving commorphism group then there might not even be a vector field. Uh so then yeah one doesn't quite uh know what to write here. Uh now the second description that Arnold gave uh here one can be maybe more hopeful at first and this maybe also was the motivation of Arnold for asking the question in the first place. So if one has some link with a linking number and the second link that's homeomorphic to it that of course has uh the same linking number. So the linking number of links is preserved in the homeomorphisms.

Uh but in our setting here of this asmtoic linking number there's an important subtlety that lies in these short geodessic arcs that one used to close up the finite flow line segments. So here I zoomed in into the earlier picture uh at this judacic arc. uh and the problem is that uh some conjugating homeomorphism it will conjugate flowline segments. So that's that's fine. So this green flow line segment goes to a nice green flow line segment and the the blue one to a blue one. That's all good. But this uh gestic arc here it's not in general going to the gestic arc. It's just going to some potentially very rough and windy uh C arc. Uh so I tried to sketch one here and uh yeah so our conjugating homeomorphism just a homeomorphism there could be some sort of infinite twisting going on. Uh so in principle this this JIC arc on on the left side could be mapped to something that that twists uh

arbitrarily much somewhere. Uh and and then there's just no good way anymore to uh control how the the linking number changes if we go from the left side before the conjugation to the right side after conjugation. So that's one subtlety here. And then there's a related subtlety for the extension question. So one one can you know find the C zero flow one can define one where there's a lot of of twisting near some flow line and in particular one can arrange it such that the asmtoic linking number is no longer an integraable function as then one doesn't know how to make sense of this formula uh right so it's uh yeah not not immediately clear how to approach this question right so Now here here is the results I want to talk about today. So I repeat that here our nodes question and uh then uh what we're able to show is that the answer to the invariance question is yes if we restrict to volume serving smooth

vector fields that have no zeros. That's a nice uh big class of vector fields and and for those uh one has 10 invariance and then we deduce this from an extension and our extension is to see zero flows that have first of all no rest points. So this no rest point assumption uh that corresponds to the no zeros in the smooth vector field. Uh that's the the C version of that. And then we need a second assumption and this one demands that uh there's no individual periodic orbit in the flow that has positive measure. Uh so this uh trying to rule out some rather pathological flows. uh it can can happen for such zero flows that one has one single periodic orbit and uh it's like it's like three-dimensional measure is 99% of the total measure of the free manifold uh and and this kind of thing you want to rule out. It's not quite clear whether there's uh much reasonable things to be said about that. Uh right.

And then yeah so the the relevance of this no zero so no rest point assumption is that we connect the whole story to foliations and if one has a vector field with no zero so has a zero flow with no rest points uh then the flow lines form a one-dimensional foliation uh and and that's useful for us right so this is the main result uh now I should also mention some previous works. Uh it's a particular nice one is this work of Gamb and there they answered the invariance question for disk map suspensions. So one would take a two-dimensional disc and a compactly supported characterism.

Then one can form a mapping Taurus and then one can completely convectify that uh to a voling flow on a closed manifold. And then if one has two smooth vector fields that come from this construction that are also also conjugated by a homeomorphism then Kamadoo she showed that one has uh the invariance. So the two vector fields have the same valicity. Uh there also is uh work of Müller and Spath. Uh and this already connects the question a little bit to C simplectic geometry. Uh the precise result is a little bit technical to state. So they have a extension to what they call continuous strictly contact isotopies.

Uh so for the expert uh this would uh it's very much related to uh the fact that uh heomorphism isotope has a unique Cz Hamiltonian generating it. Uh so this this work is uh connected to these kind of ideas and then also there's a very recent work that appeared in the archive actually this week. Uh so need a bit more time to digest it. It's by Giriwan and Novak. So these are people working in fluid dynamics. Uh and what they did is they uh they find an extension for helicity but not for Czero flows but for some uh very low regularity class of vector fields. And then uh what their work also does is they produce uh a weak weak solutions to the oil equations uh where then their extension of elicity actually is not preserved under the evolution. Uh now this whole work somehow goes a little bit in a different direction from ours. Right? In our work, well, we have C zero flow, but there might not be any vector fields in the

background and what they have these low regularity vector fields, but they might not have any good sense of flow they generate. Uh but but yeah, it would be interesting to compare these two works uh more in the future. Okay. Now uh yeah so now I want to say go a little bit into uh how how we prove this theorem on uh invariance and extension. Yeah. So the first the first step and I already uh briefly mentioned that is that we want to connect uh the theory of foliations. Uh and so here's what we do in the smooth setting. So in a smooth setting we have our three manifold the volume form and a smooth vector field volume preserving with no zeros. Uh and this uh structure induces first of all a one-dimensional smooth foliation. Uh so the leaves of that are just flow lines of the flow generated by the vector field x. Uh there also is a transverse measure present. Uh so there's this uh do form omega that's obtained by contracting the

vector field and the volume form. Uh now this do form omega its kernel exactly is is tangent to the the foliation and one has this uh smooth area form in the transverse direction and one can interpret this as a transverse measure. So the transis measure is a structure that induces uh a measure on every transversal surface to the foliation. Uh and there's a a version of tapus theorem here. So our structure is like is locally difforphic to uh a standard chart. So the standard charge is interval across the disk. So here the foliation would be tangent to this interval vector and the the transverse measure comes from the standard area form on the disk. Uh okay and so right so the idea is uh we want to uh so if we have such a flow it induces this structure and uh we will then mostly be interested in this smooth me measured foliation. uh my notation for these structures will be little omega the same as this do form. So this means

uh the whole thing this one dimensional fiation plus transverse measure uh subject to the property of locally being diffmorphic to the standard piece. Uh and this sometimes is also called a Hamiltonian structure in literature. It's just a a closed smooth maximally non- degenerate do form on a three manifold. Uh okay. Now in the C0 setting, one starts with the three manifold Y a volume form and the rest point three volume preserving flow five. Uh and this also gives a one-dimensional foliation by the flow lines of fi but now it's just the C zero foliation.

Uh there still is a transverse measure. One no can no longer can make sense of this do form but one can get a transverse measure. um defined as follows. So one just picks a transverse piece of surface to the flow uh and then one can uh flow for some time epsilon and measure the three-dimensional volume that's swept out by that and if one divides that by epsilon this induces a two-dimensional measure on this transverse surface piece. So one still has this in the CC world. Uh and uh yeah so this whole structure uh still will be locally homeomorphic to a standard chart. Uh and this I didn't forgot to write it here but here is where the assumption on no periodic orbits have positive measure comes in. Uh so if there was a period go of positive measure then this transverse measure would have an atom. So there would be a transverse surface and then the intersection point with the periodic orbit of positive

measure would be a point of positive measure on the transverse surface. But by assumption there's no atoms in this transverse measure. uh and then there's a theorem of oxtopian ulm that tells one that uh one silver one is locally homorphic to the uh the standard chart. Uh right and my notation for these uh cz measured foliations will be a capital omega as opposed to a lowerase omega. uh and uh in our paper we called these things also Cz Hamiltonian structures u in analogy to the Hamiltonian structure and the smooth case uh right and yeah so it is possible now to view halicity as an invariant of uh just Hamiltonian structure omega instead of this whole pupil of volume form and volumes in vector field. Right?

So if you recall the very first definition of elicity, one simply had to pick a primitive one from alpha of this omega and and the elicity was def defined in terms of alpha. So elicity only depends actually on on this structure. Uh so one can just remember this holian structure and forget this whole dupal of volume form and vector field and hicity will be an inant of that. And then if one does that one can reformulate the questions the invariance question becomes is helicity of the Hamiltonian structure invariant under uh now just orientation preserving homorphisms. So here one actually threw out the volume form. Now it's just any homophosis and it's just orientation preserving uh that conjugates these two Hamiltonian structures where this is understood as as measured foliations. Right? So one one can't pull back the do form but as measured fiations this whole thing makes sense and then yeah we one wants to know

whether these two Hamiltonian structures have the same elicity and extension well we want to define elicity for a C Hamiltonian structure. Uh okay. And so here is uh an important uh operation in our whole story uh that we call plug insertion. Uh so there's uh an operation that modifies locally our uh measured foliations. Uh so it works both in the smooth and in the CC0 setting. Uh so one starts with a trivial foliation chart. This would be uh just a cylinder of the form interval cross surface uh with uh some area form on the surface and then some embedding of the structure into into our Hamiltonian structure. Our surface will always be open connected. And in addition we take an Hamiltonian isotopy compactly supported on our surface sigma.

Uh and uh giving this information one can now modify the measured foliation by inserting this Hamiltonian isotope uh into the trivial foliation chart. Uh so one wants the new leaves that at first were just horizontal in the trivial chart one wants them to trace out this Hamiltonian isotopy. So an individual leaf in here would be of of this form a set of all points t comma 5 t of p and t ranges over the interval. Uh yeah so one can carry this out in the smooth settings. one takes a a smooth Hamiltonian structure, a smooth chart and a smooth isotopy and one gets back uh a smooth Hamiltonian structure and uh I will use this kind of notation here.

So that's the original Hamiltonian structure and then P stands for this whole data here of the trivial chart and the isotope and then one can also carry it out in the C0 world as a a C0 Hamiltonian structure C0 chart and then an isotopy in the Hamiltonian homomorphism group the surface where this group is the CZ closure of the diffmorphism group inside the homorphism group and then the new Hamilton CC Hamiltonian structure would be denoted like this. Okay. And now here comes uh a crucial observation that already is present in the work of Gambu and cheese that I mentioned earlier. Uh so there's a very simple formula in the smooth setting that tells one how uh helicity changes if one performs this plug insertion operation. So if we start with a smooth Hamiltonian structure and we put in a smooth Hamiltonian ispy locally in some chart it turns out that the helicity exactly changes by the calabian variant

of the time one map of the Hamiltonian is so calabi is this group homomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomorphism from the Hamiltonian complex supported Hamiltonian diffmorphism group of our open surface sigma to the real numbers. uh it's one way of defining it is in terms of this integral here over one choice h of uh complexly supported Hamiltonian generating our uh the femopism five. Uh so one has this homorphism and uh yeah one one gets this very nice relationship between helicity and the calabi homorphism. So in some sense one can think of elicity as uh generalizing the calabic homorphism.

Uh yeah there's this important result of banyaga that essentially saying that there's not that many more interesting group homorphisms out of this Hamiltonianism group. uh panyaga proved that the kernel of the calabium is equal to the combinator group of the Hamilton and the femorphism group and it also is a simple and perfect group. So there's yeah like any homorphism out of this group has a kernel which is a normal subgroup but then there is not that many normal subgroups basically only this commodator subgroup and this gives rise to this calabi homorphism.

Uh yeah, one can now actually re reformulate the observation up here on uh the helicity change on the plug insertion as follows. uh one can observe that if uh we put some black P into a smooth Hamiltonian structure then the amount by which elicity changes it uh did not depend on which Hamiltonian structure one puts the plug into it also doesn't depend on the embedding of the trivial chart where one puts the plug into uh it only depends on the residue class of the time one map of the isotope that is part of the pluck data inside the epinization of the Hamilton and hemorphism group. So this is the quotient of the group by the coma subgroup and if I put some plug into some smooth Hamiltonian structure the multicity changes it only needs to know this recipe class and nothing else.

Uh yeah and so in some in some sense our so I mean this this whole property as it's formulated uh well there's a a clear guess as to how one should generalize this to the C world uh one would want to say that if one puts in some C plug then theity change only depends on the residue class now andization of the Hamilton and homie move group. uh yeah so some sense our our work in the CZ setting is saying that uh if one uh demands such a property to hold in the CO world uh there's a unique way of extending helicity this will be the spirit of uh the next result I'm leading up to now before I can formulate it we need a little bit uh more preparation uh so yeah so Now let's look at now two open connected surfaces. There's sigma prime and the sigma and then some error preserving embedding of sigma prime into sigma. Now if one has such an embedding one can look at this diagram here. uh so we have on the top the homorphism from

the abilization of the Hamiltonian diffmorphism group to the real numbers induced by the calabi homomomomomomomomomomomomomomomomomomomomomorphism and this is a group isomorphism exactly by Banyaga's theorem that tells one that the kernel of calabi exactly was the commodator subgroup which is the thing that we quotiented out here one has the same kind of isomorphism for the other surface sigma prime and Then there's a map introduced by push forward of compactly supported defmorphisms under this embedding Yoda. Uh and this whole diagram is commutative. Um one can see this just from the definition of calabi.

One can take Hamiltonian here and then push it forward and then the integral of the push forward is the integral of the original Hamiltonian. So one gets this homorphism here and well because the di commutes and because both homorphisms induced by calabia isomorphisms this push forward by Yoda actually is a group isomorphism and also it is independent of the choice of embedding Yoda. So it says for any for any two open connected surfaces this uh abilinizations are isomorphic. So it doesn't matter whether it surface is a disk or has some genus they're all isomorphic and moreover they are naturally isomorphic in a way compatible with maps induced by inclusions.

So that's all in the in the smooth uh setting and uh it really it follows from Banyaga's result. So the proof of this is uh somewhat involved. Subanyaga's theorem is is hard to prove. Uh but yeah, so in in our work it was useful to have a C0 version of this. Uh so what we proved is that if one looks at the groupomorphisms induced by C0 preserving embeddings. Uh they also introduce an isomorphism of aizations of Hamiltonian homorphism groups. And again one has independence of uh of Yoda. So this means that if one has any surface open and connected it takes the Hamilton homorphism group and then they cinization the result is naturally isomorphic to the the group associated to any surface.

So one can for example pick the standard dis the standard area form like all of these groups are naturally isomorphic to this one here. So one can give it a name. We called it curly R in our in our work. And now there already is some interesting consequence here. uh so what this is micular saying is that and I will come back to this later but for those for the experts it it means that the one once one knows the simplicity conjecture for the disk one can using this proposition get it for any other open connected surface this already was known so there's uh yeah I will come I will I will see uh more this in a bit so this This is known but uh yeah one can uh get this results alternatively through this proposition once one has the case of the disk.

Uh okay now here is uh our theorem um introduce some notation. So in this whole diagram in the bottom row we simply have uh the smooth calabi homorphism from the Hamilton and the femorphism group and and now we can think of it as uh well first projecting onto theization uh and and then we had this natural isomorphisms between theization and the real numbers. So we can identify those two and we can think of this smooth calabiomorphism in this way. And then in the top row, I mean, we still have a natural projection from the Hamilton and homeomorphism group onto the abolonization of the Hamilton and homorphism group homeorphism group. Uh, and well then because all of these were natural isomorphic, we called them curly R and then this whole logical projection I want to give it the name Calbar. So in some sense we think of it as an extension although to theological of the smooth calavicomorphism.

So medicical has this natural homorphism from the real numbers to the curl r that's induced uh uh by by this homorphism here from the aonization of the smooth diffmorphism group to the abanization of the uh Hamilton and homorphism group. Uh okay. And now uh here's here's the theorem that we proved that really underlies this earlier uh main result I stated. Uh so we proved that there's a unique way of assigning uh an extended helicity taking values in this curly R in theization of the Hamilton and homorphism group.

assign such a thing to every C 0 Hamiltonian structure capital omega subject to three extremely natural properties. Uh the first property is uh saying that this uh extend helicity should or extend the smooth helicity. Right? So if one has a smooth Hamiltonian structure, one can take it smoothity which is a real number. uh and well one can also think of the smooth Hamiltonian structure as uh C0 Hamiltonian structure and then one would get this extended elicity that the theorem claims exists that takes value in the curly R and we demand that if we map the smooth elicity uh from the real numbers over to the curly R we exactly should land at the extended elicity. So this is natural to demand. Uh well any extension really should have the property of extending.

Uh now the second property is equally natural. Uh here we demand that our extenticity extension is C zero Hamiltonian structures uh should be invariant under orientation preserving omnioromorphisms. Well of course we want have this property. If it's defined for C0 structures, it should be in the homorphisms. So there's no negotiation about that. And then what we proved is that if one just adds one additional uh property uh then this uniquely specifies what the extension has to be. uh and this property is uh this counts the zero version of the uh observation of gambadu and cheese on the relationship between helicity and calabi. Uh so we want to demand that if um when it starts with any C Hamiltonian structure and puts in any CC black pluck so we insert some isotopy of Hamiltonian homeomorphisms.

uh then the extenticity should exactly change uh by this extended uh calabium invariant of the time one map of uh the the isotope and recall this was simply the the rescue class of the time one map in this abilization uh right and so we say that the existent extension with these properties and exactly one such extension. Uh there's uh more. So the theorem uh continues. So one could complain maybe that well this extension here it takes values in this strange group curly R theization of the Hamilton and homorphism group and not in the real numbers. So how about real valid extensions? Uh and the suggestion here is the following. Uh so if we assume that we have some real valued extension of elicity uh to C Hamiltonian structures and assume uh that this extension satisfies uh of the analogs of properties one and two above.

Uh so property one if it's a real valid extension of elicity well yeah it's simply just saying that the the hellbar of a smooth mult structure is exactly the same as the extended one and then the the version of two yeah this extension should be invar than the homeomorphisms. Uh and okay let's assume that our extension also has the property three prime. Uh it's a bit different from this one here. uh we assume that uh the helicity change under plug insertion uh does not depend on where one puts the plug. It only depends on the time one map of the isotope one inserts. Uh so this was this property that uh I uh observed above that holds in the smooth case and uh if one assumes one has an extension that has the zero analog of this property uh then this extension actually is of the form uh lb bar. So this is uh this um extension up here taking values in the curly R composed with uh a homorphism P from the curly R

to the R. So this is some sort of projection. U so we saying that if one has a real world extension uh that has these very natural properties then it must be of this form. Uh and actually more rule. If one defines the extension this way for some choice of homorphism from the curly R to the R uh then it also will have these three natural uh properties. Uh right so in some sense if one insists on a real value hicity extension to be well behaved with respect to this insertion of plugs. uh then real valid essential simplicity are in in one to one correspondence with such homorphisms P from the curly R to the R.

Uh yeah so in in this sense our uh list the first part of the theorem is universal like any any other one uh will just uh derive from that. Okay. So now uh right so we have this theorem uh but for this theorem to really be useful uh we actually would want to know a little bit more about this homorphism from the real numbers to the curly R might recall this was this homorphism that's induced by that is the map that's induced by the inclusion of the Hamiltonorphism group into the Hamilton and homorphism So one wants to know more about this map uh in particular our priori and this actually was a long-standing open question of the simplicity conjecture. It's not even clear whether this early R is a trivial group or not. Uh so if the R for example is a trivial group then uh well our extension was a bit boring then more or less we would have proved that uh it is not possible to extend elicity in a

reasonable way subject to good behavior with respect to plugs. Uh so we want to know that the r is non-trivial and also we would maybe want to know that this homorphism is injective for example. Uh right. So now what what yeah what is known about this homorphism? Uh there have been uh a lot of very interesting works in this in the series of black geometry in the recent years. Uh so what do we know? uh first of all there's the work of Christopher Gardner and Sephardini uh that resolved the simplicity conjecture and uh this is equivalent to what they proved is that that this group curly R is not the trivial group so that's that's good so our extension is not trivial and then uh later on they uh joined forces with Men Smith uh and uh in in this later work um there's this set of flur theoretic spectral invariants called link spectral invariants that plays an important role uh and what one gets out of the link

spectral invariance is a sequence of quasimorphisms uh from uh the Hamiltonian homorphism group to the real numbers so there's one such link spectral invariant for every positive integer d uh these are not homorphisms but they are basorphisms. So they're homorphisms up to some bounded error called the defect and uh what is also true for distin is that the defect actually converges to zero as the t goes to infinity. Uh so the the error in the CT not being homorphism uh gets smaller and smaller increases. And now one can look at uh this diagram and this diagram tells us pretty much everything that's known about uh this group curly R. Uh right on the bottom row we again have the smooth calabicomorphism.

Uh so that's this canonical projection to the abunonization and then the natural isomorphism to R. Uh on the top row we have this projection from the Hamilton homorphism group uh do the curly R. So this is what I earlier called overline cal. This was this extended calabi homorphism and using the link spectral invariance one can also define the group homorphism from the Hamilton and homorphism group to this uh vector space here. So one first looks at the vector space of all of all sequences of real numbers uh and then of course one can input aom homeomorphism and then output the sequence of link spectral varants that lands in this uh space of sequences. Now because the CDs only were quasiomorphisms, this is not a groupomorphism. But because the defects go to zero if one uh quotients out by zero sequences, so sequences of real numbers converging to zero, then one turns them the resulting induced map

into a grouporphism. So one gets this map up here. uh and then there's a diagonal map that inputs a real number and outputs a constant sequence. Uh and the remarkable fact is that this whole diagram is a commutative diagram. Now this means if we start with a Hamiltonian diffmorphism we take its calabian variant and then we look at the constant sequence where each entry is the calabian variant of the diffmorphism in this vector space that's the same thing as first thinking of the diffmorphism as a homeomorphism and then looking at the sequence of link spectral variants.

So this is the same thing and uh the reason for this being the same thing this is what is called the v law. So it turns out that if one starts with a diffmorphism then the sequence of link spectral invariants it converges to the calabi invariant of the diffmorphism and this exactly is a reformulation of the statement that this whole diagram here is a commutative diagram. uh okay so I claim that uh everything we know about the curly R uh essentially follows from this commutative diagram. Uh now the first uh important observation is that uh the map from the real numbers to the curl that's induced by including diffmorphisms into homemoromorphisms.

This is an injective map and one can see this from this diagram here because the diagonal homorphism that takes a real number and goes with a constant sequence. This is an injective groupomorphism and the diagram commutes. So if you go diagonally it's the same thing as going up and then going right. But if this composition up and then right is injective then uh the map that goes up must have been an injective one. Uh we can also see that uh the map from R to curly R is not a sjection uh because it is known that the curly R does sect onto this very big vector space of all sequences mod zero sequences. So in some sense one should think of the curly Rs being quite a bit bigger than the real numbers.

Uh yeah and so this uh this whole fact really is uh it is a very deep fact. So the only proof we have of this really follows from this whole story with the link spectral invariance. uh and uh it also really is what what makes this uh theorem above on the curly r valued universal helicity extension uh useful. So if one wants to for example deduce the invariance of elicity just for smooth Hamiltonian structures it really uses this injectivity. So if we so what we want is we take two smooth Hamiltonian structures omega 0 and omega 1 and if we assume that they're related by a orientation preserving comorphism f right we want to conclude that the the ordinary smooth utilicities are the same and now we can prove this quite easily right so we can well take the smooth elicity of omega 0 take the smooth elicity of omega 1 that gives us two real numbers. Uh and we can also think of omega 0 and omega 1. We can view it as a special

case of a cz hamilton structure and we get two extended solicities uh uh overline hell that both take values in the curly r. Uh now it was one of the properties in the theorem first one the extension part uh that uh if one starts with a smooth munic structure thinks of it as a C01 and uh um then moves forward the the smooth elicity into the curly R by the natural map. Uh then exactly one gets the extended elicity. Uh so uh the ordinary elicity maps to the extent elicity by the first part in the theorem and then by the second property in the theorem because the extentity is invariant under the homeomorphisms uh the two extentities on the right hand side here they must be the same right because our two smooth hon structures were conjugated by a homeomorphism and then because the natural map from R to the curly R was injective from and deduce from the extended helyses on the right being the same that the initial

helyses on the left uh were the same. Uh okay. So this uh this is how uh invariance follows. One also can deduce from this diagram that there exist projection maps from the curly R back uh to the the real numbers right then we also explained that uh such projection maps from the curly they are they are in one to one correspondence with real value deals in the extensions So if we want to know whether real validity extension exists then we have to know does such a projection exist. And from this diagram up here one can deduce that a projection does indeed exist. uh because one can choose a projection of this uh vector space of uh sequences of real numbers modulo zero sequences uh to the subspace that's a image of the inclusion of of ru can such a projection that gives a projection from curly yard to r and this gives a realisticity extension now it's important to point out that there isn't really a natural choice of

projection from this space of sequences to the real numbers. Really any such choice really uses the axum the axum of choice. So it's not going to be extremely natural object and moreover there's many possible extensions if one accepts the aim of choice. Uh right so in some sense it it really is the most natural thing to to stop at the curly r value universality extension. Okay, now this uh is the refined version of our main result. And then maybe in the last couple of minutes I have I want to say a little bit about uh how one starts proving this. And the the easier part of the proof is uh well so the so so here here is how I want to construct our extended elicity that inputs a zero Hamiltonian structure capital omega and outputs elicity and it's based on this claim here. Uh so I claim that if we start with any CC Hamiltonian structure uh then well this might not be homeomorphic to any smooth Hamiltonian

structure and this might just not happen. The examples of such cult structures that are not smoothable but I claim that one can always localize the obstruction to smoothing inside some CC block. Uh so I claim one can find a smooth Hamiltonian structure little omega and some C blocks such that uh this structure here obtained by putting the C block plug into the smooth structure is homorphic to the original structure capital omega and once one has this claim one is essentially forced to define in this way simply by the properties listed in the theorem. Uh so if our capital omega was homorphic to this guy here then we have to define its elicity to be the sum of the uh smooth elicity and it's a real number and we take the image in the curly r plus uh this extended uh uh calabian variant of the time bond map of the isotope in the plug. Right? So there's there's no choice. Now of course then the really

subtle part of of the whole story is to uh show that this that would be independent of choices. Um right but uh just to for the definition we don't know that we can find such a little omega by plug and then we have to divide it in this way. Now why uh why should one believe the claim? uh here is the idea. uh so the important observation is that if one has some C foliation and I sketched some picture of a steolation here then locally well locally it's homorphic to the standard so locally it's always smoothable and the obstruction uh to smoothing structure foliation globally really lies in the presence of recurrent leaps right so you can smooth locally and then smooth on a bigger and bigger set and they really only run into troubles if there's a leaf that is recurrent. Uh but yeah, so now our trick is to first of all we can uh just cut some holes into our manifold. So we throw away a bunch

of charts and we so that's just chooses a bunch of transverse little discs and one chooses sufficiently many of those such that there is no more recurrence in the compliment. So one really cuts every single leaf into small compact pieces. That's the first step. Uh then the second step uh would be to show that one can smooth the structure in the complement of uh uh of these blue flow boxes because there was no more recurrence in the complement. uh one can show that uh the restriction of a structure to the complement is going to be homeomorphic to some uh smooth Hamiltonian structure in the complement. That's step two. And and then so the the trouble now of course is that if one has this homorphism defined in the complement of the flow boxes, it might not extend to a global homeomorphism. So there's some illustration of that, right? So the there's some homorphism defined on the compliment of flow boxes.

Um so in both the left and the right it's a local picture where the foliation just consists of vertical lines. Uh and then so that the color is supposed to indicate where which leaf goes. So this orange leaf here would go to the orange leaf here. The red leaf up here go to the red leaf up here. And then yeah in this picture it's now very clear that one just cannot extend the homorphism that's defined in the complement of the flow boxes over the flow box simply because uh well leaf has got to leave and on the left side the red and the orange are the same leaf right there straight leaf going through here and on the on the right side red and orange don't line up.

So this cheat and structure extending edition uh but the point now is that uh right we didn't want to smooth globally our structure we wanted to smooth it up to a plug insertion. So if in our flow box now one puts in an appropriate C plug that reroutes the leaves then after this operation one can extend this homeomorphism. Uh and this is what uh what proves the the claim. Uh right so this was this claim here that any CC haldon structure is homop one with a C plug inserted. uh and this then forces what the holisticity extension has to be. uh right now as I mentioned uh then the real difficulty is to show that there was an independence of choices right I can I can write the same system structure maybe in many different ways as a smooth one with a plug inserted and have to figure out why it was that the exentalicities should agree and this actually this just the invariance is a special case of this

so if we just look at smooth structures with no plugs. Uh right, this invariant statement would be a a special case of this key difficulty here. Uh right, but yeah, so I I think I'm uh reaching the end of my hour. Uh so maybe I would uh just stop uh my talk here. Okay,

thank you very much. Thanks a lot, Oliver. That's very good. Um, any questions?

Oliver, every everything you have not explained is constructive by hands. no invariance and what what do you use more hammers or is everything by hand or what? Ah so well so there's um there's some two two parts to the story. Uh there's the the part on the foliations. Uh and this in some sense is uh very it's like very explicit. Uh really it uh it ties in very well with uh the works of Puren on on polations. uh this old work from the 70s and well our kind of arguments it kind of continue this uh well this this kind of line of argument uh so yeah this is very constructive

but then uh this alone is is not really so this is then not not really enough to get anything out of it. one really has to combine it with um these results from on error preserving homorphism groups that uh that do use uh blur theory.

So somehow yeah this really this the key fact that that shows up repeatedly here is uh this whole statement of the injectivity of this map. So what you're saying is if you take a diffmorphism Hamilton diffmorphism and you assume you can write it as a product of commodators of homeomorphisms then it already was possible to write the diffmorphism as a product of commodator of diffmorphisms. uh this statement uh I I'm not aware of any there's one proof I'm aware of and it really passes through uh these spectral invariance and flur theory. Uh yeah so that other part on folations is is more uh constructive by hand but then really one has to combine these two pieces uh for our story

but you just algebraically use what you have from the the spectral invariance you don't have to develop further there

uh yes one does not so yeah there we don't know of a way of defining spectral variance for the volume flow because yeah it could just not be of contact type. So it's super clear what to do there. Uh yeah so the the fur theory enters algebraically uh through there's connections between foliations and the algebraic structure of transformation groups. That's right okay since I'm already on assume you have finitely many zeros of the flow. Can you kind of open them to and approximate or is it hopeless?

So now it's not hopeless. So we were at some point working out the case of uh generic zeros. So as a smooth vector field with a a non- degenerate isolated zero. uh then I think one can take care of these uh yeah there there would be some

approximation procedure uh if one goes to full generality and as some some arbitrary vial set of a vector field we don't quite know yet and uh maybe it's not even quite clear which they do conjecture so we talked to some people for example it's up very conceivable things differently. Uh it's uh not clear but yeah so some uh certain class of singularities one can do.

Thank you. May I also ask a question? Yes. So um when you at the very beginning of the of the of the presentation you were uh giving Arnold's definition of the linking number

of two flow lines. Uh yes I just wanted to um to get back to the way you were explaining that you connect the end points of each um of the flow lines right um then you said that we use geodessics. Can you um can you clarify this a bit to me like geodessics with with respect to which metric and how do you define them actually? Right. So I mean there there are there are some subtleties hidden in there actually and and this is uh part of what later clarified more rigorously. Uh yeah, I mean so the well the overall idea is that hopefully uh these short segments uh in the fully in the smooth world hopefully should not matter too much but that we hope within has the flow lines flowing for very very long times then maybe this little segments here maybe don't affect the linking number too much. Uh yeah. So I think one uh way of doing it and maybe you have to look it up in in focal work is yeah we just pick some auxiliary

metric now smooth metric and then okay then if you have two points uh there exists well it could be of course in general it could be more than one choices of show that's connecting arcs. If there's more than one has to just make a choice. Mhm.

But then the idea is that maybe there's like not there's not. So I mean this whole thing only has to be defined for almost every pair of points. So somehow the idea is that uh the set of pairs of points where there was an ambiguity actually was a zero set in the end. Uh right. Um but yeah so there's there's some some uh that the challenge is to argue that in the end it did not depend on the choice of a reasonable short path.

Okay. Okay. I see. Thank you very much. Um okay so if there is no more question thank you Oliver Yeah.

Questions about this episode

Who speaks in "On the Topological Invariance of Helicity - Oliver Edtmair"?

Ed Oliver (Guest).

What is discussed in "On the Topological Invariance of Helicity - Oliver Edtmair"?

In order: 0:00 General Hap Arnold, Ed Oliver, C corp, Euler; 10:00 Gamb, David J. Novak, General Hap Arnold, Giriwan; 20:00 Calabi, C0; 30:00 Calabi; 40:00 Calabi; 1:00:00 Ed Oliver, General Hap Arnold, Puren.

What is mentioned in this Institute for Advanced Study episode?

14 people, works and ideas are mentioned, including Ed Oliver, General Hap Arnold, Calabi, C corp, C0, Gamb. Each one on this page links to its own page with every episode that mentions it.

How long is "On the Topological Invariance of Helicity - Oliver Edtmair"?

The episode runs 1 h 9 min; the transcript is about 9,531 words in 72 timestamped sections.

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About this transcript: the episode's spoken words as captioned on YouTube, split into timestamped sections. Reclip.Me locates every person, work and idea in the text, keeps the time it is first named, and quotes the words around it verbatim. Last updated January 24, 2026.