A Monge-Ampère Operator in Symplectic Geometry
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 | 13 | |
| 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 | 10.5446/48148 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
8
00:00
Vector potentialTheoryPlane (geometry)Harmonic analysisDerivation (linguistics)SpacetimeStatisticsQuadratic equationFunction (mathematics)ModulformPhysical systemExistenceMereologyLine (geometry)Negative numberGlatte FunktionPoint (geometry)Student's t-testPlane (geometry)Musical ensembleHydraulic jumpWater vaporNetwork topologySlide ruleConvex functionNichtlineares GleichungssystemHarmonic analysisComplex (psychology)Set theoryMatrix (mathematics)Quadratic formSubsetSign (mathematics)Vector potentialOpen setAnalogyRange (statistics)TheoryEquation2 (number)Vector spaceSubharmonische FunktionCompact spaceReflexive spaceLecture/Conference
05:10
Affine spacePlane (geometry)ManifoldHarmonic analysisFunction (mathematics)DreizehnTheoryConvex functionReal numberEquationHelmholtz decompositionManifoldFunction (mathematics)Linear subspaceSubharmonische FunktionTangent spaceMaxima and minimaHarmonic functionAffine spaceReal number2 (number)Convex functionSet theoryCondition numberDerivation (linguistics)Line (geometry)Harmonic analysisPlane (geometry)Boundary value problemVector spaceTerm (mathematics)Standard errorDirection (geometry)Right angleAutocovarianceDeterminantStatistical hypothesis testingComplex (psychology)MereologyQuadratic formNichtlineares GleichungssystemMatrix (mathematics)Sign (mathematics)Matching (graph theory)1 (number)SpacetimeFiber bundleMetric systemArithmetic meanGroup actionQuadratic equationAlgebraic structureOrder (biology)IRIS-TSpecial unitary groupCausalityStress (mechanics)Equaliser (mathematics)Computer animation
10:21
Helmholtz decompositionTheoryFunction (mathematics)EquationPlane (geometry)CoefficientLogical constantDirected setSymplectic manifoldAffine spaceManifoldContinuous functionTime domainModulformSymplectic manifoldSymplectic vector spaceFunction (mathematics)PotenzialtheorieRight angleSet theoryCategory of beingComplex (psychology)UntermannigfaltigkeitQuadratic formAlgebraic structureConnectivity (graph theory)SpacetimeMereologyVolume (thermodynamics)Nichtlineares GleichungssystemLine (geometry)Equaliser (mathematics)Geometry2 (number)Arithmetic meanLinear subspaceVector spacePlane (geometry)Element (mathematics)Social classGroup actionKettenkomplexArc (geometry)Matching (graph theory)Sign (mathematics)Sinc functionSpecial unitary groupVector potentialGenetic programmingOrthogonalityPrice indexWell-formed formulaGrothendieck topologyComputer animation
15:38
Limit (category theory)Maxima and minimaSmith chartInequality (mathematics)Function (mathematics)Computer programmingSocial classNegative numberKettenkomplexExistenceRegular graphStatisticsLine (geometry)Limit (category theory)Category of beingFamilyStatistical hypothesis testingMaxima and minimaUniformer RaumPlane (geometry)ForestPosition operatorDecision theoryContinuous functionHarmonic analysisDerivation (linguistics)2 (number)Point (geometry)Multiplication signMechanism designDifferent (Kate Ryan album)Quadratic formGlatte FunktionSlide ruleAbsolute valueConvex functionArithmetic meanProof theorySupremumLecture/Conference
20:54
Function (mathematics)Harmonic analysisEquationDampingPoint reflectionRegular graphVector spaceFunction (mathematics)Set theorySubsetSymplectic manifoldNegative numberQuadratic equationSummierbarkeitDerivation (linguistics)2 (number)Arithmetic meanPositional notationMultiplication signGrothendieck topologyMany-sorted logicMultiplicationClassical physicsLatent heatKettenkomplexStandard errorSpacetimeCartesian coordinate systemMortality rateComputer animation
23:41
Boundary value problemGlatte FunktionTrigonometric functionsFunction (mathematics)Time domainTheoremHarmonic analysisDifferential calculusOperator (mathematics)PolynomialIndependence (probability theory)Mathematical analysisOrthonormal basisProof theoryFunction (mathematics)StatisticsDrop (liquid)Set theoryPlane (geometry)SummierbarkeitExistenceHarmonic functionBoundary value problemEndomorphismenmonoidHelmholtz decompositionParameter (computer programming)CommutatorArchaeological field surveyNichtlineares GleichungssystemComplex (psychology)TheoremArithmetic meanTheoryExtension (kinesiology)Term (mathematics)Orthonormal basisPoint (geometry)MereologyNumerical analysisVector spaceUniqueness quantificationTime domain1 (number)Connectivity (graph theory)ModulformQuadratic formSecond fundamental formCondition numberDerivation (linguistics)Sesquilinearform2 (number)Translation (relic)SpacetimeDeterminantGeometryDifferential equationGlatte FunktionContent (media)Continuous functionSeries (mathematics)Different (Kate Ryan album)Special unitary groupLocal ringPrice indexStress (mechanics)Quadratic equationState of matterSimilarity (geometry)Radical (chemistry)Bounded variationTable (information)ConsistencyLine (geometry)Multiplication signMetric systemMusical ensembleOcean currentMatching (graph theory)Maß <Mathematik>Moving averageHydraulic jumpFundamental theorem of algebraLecture/Conference
31:42
Function (mathematics)Symplectic manifoldComplex (psychology)Line (geometry)Nichtlineares GleichungssystemOperator (mathematics)Negative numberSymmetric matrixCoefficientPolynomialHarmonic analysisViskositätslösungEigenvalues and eigenvectorsGroup actionExpressionMathematicsQuadratic formSign (mathematics)Order (biology)Symmetric matrixEndomorphismenmonoid1 (number)SummierbarkeitNumerical analysisComplex (psychology)PolynomialAlgebraic structureEqualiser (mathematics)Self-adjoint operatorMatrix (mathematics)Direction (geometry)Combinatory logicTerm (mathematics)Derivation (linguistics)Connected spaceHarmonic functionSet theoryMetric systemEquationLine (geometry)Plane (geometry)Identical particlesFocus (optics)Matrix (mathematics)Linear algebraFunction (mathematics)Symplectic vector spaceProduct (business)Mereology2 (number)Arithmetic meanOrthogonalityDimensional analysisMultiplication signCondition numberSpacetimeMaß <Mathematik>Proper mapTwin primeModulformWave functionChaos (cosmogony)Quadratic equationConnectivity (graph theory)Monster groupLink (knot theory)Different (Kate Ryan album)Nichtlineares GleichungssystemNegative numberTorusMatching (graph theory)Game theoryGreen's functionAsymmetryComputer animationLecture/Conference
39:17
ViskositätslösungInvariant (mathematics)Element (mathematics)SpinorGroup representationRepresentation theoryComplex (psychology)MultiplicationClifford algebraAlgebraic structureManifoldCompact spaceRiemannian geometryHessian matrixFiber bundleHeegaard splittingCanonical ensembleSheaf (mathematics)Operator (mathematics)Körper <Algebra>Vector spaceMetric systemConvex functionTime domainTheoremRootEigenvalues and eigenvectorsStochastic kernel estimationNichtlineares GleichungssystemCharacteristic polynomialHarmonic analysisGroup actionStress (mechanics)Regular graphMoment (mathematics)Metric systemModulformShooting methodDifferent (Kate Ryan album)Arrow of timeComplete metric spaceComplex (psychology)TheoryComputabilityAbsolute valueReliefState of matterGeometryMusical ensembleFunction (mathematics)Element (mathematics)Descriptive statisticsDeterminantEigenvalues and eigenvectorsClassical physicsGroup representationAlgebraic structureCharacteristic polynomialMultiplication signGame theoryNegative numberPrice indexEndomorphismenmonoidSlide ruleClifford algebraProduct (business)Quadratic formNumerical analysis12 (number)Algebraic functionOrder (biology)Time domainManifoldAxiom of choiceDimensional analysisDerivation (linguistics)Object (grammar)Parameter (computer programming)MultilaterationBoundary value problemIdentical particlesSymplectic manifoldConvex setDifferential operatorComplex numberArithmetic meanRootSquare numberDifferential calculusSign (mathematics)Schiefsymmetrische MatrixSymplectic vector spacePolynomialConnected spaceSkewnessQuadratic equationLecture/Conference
46:52
Regular graphTrigonometric functionsTheoryLimit (category theory)Characteristic polynomialFunction (mathematics)MultiplicationResultantMultiplication signSeries (mathematics)Exponential functionExpressionFundamental solutionCone penetration testTranslation (relic)Uniqueness quantificationSlide ruleLaplace-OperatorManifold10 (number)Figurate numberNatural numberMortality ratePhysical systemIRIS-TCollins, JohnMoving averageSign (mathematics)Lecture/Conference
48:53
Regular graphEllipsePoint (geometry)Hydraulic jumpMusical ensemble2 (number)Condition numberRegular graphNumerical analysisGroup actionModulformInvariant (mathematics)ResultantSpezielle orthogonale GruppeDerivation (linguistics)TheorySet theoryReliefOpen setMaxima and minimaSubgroupSymmetry (physics)Lecture/ConferenceMeeting/Interview
51:44
Boundary value problemDynamical systemAnalogyComplex manifoldTime domainDifferent (Kate Ryan album)Compact spaceManifoldFunction (mathematics)Beer steinComplex (psychology)Musical ensembleCategory of beingLecture/Conference
53:39
Harmonic analysisModulformMusical ensembleFlagLecture/Conference
Transcript: English(auto-generated)
00:15
So this is joint work of Blaine Lawson with Rhys Harvey, and
00:21
I'd like to, so let's make sense of what Blaine had to tell us all together, right? So, so here is the plan of this talk. You'll see that Blaine and Rhys, they discovered already maybe 15 years ago that
00:46
traditional potential and pluripotential theory had a wide range of generalizations. And this talk will mention them and then immediately go to a special case, which is the Lagrangian story.
01:03
And the new, the recent output is the existence of a Lagrangian analog of the Monjean-Empère equations. That's the point. So let's start with this rather general setting.
01:23
So you start with a compact set G of p planes in Rn, right? And then you consider quadratic forms A, whose restrictions on all these planes have a non-negative trace.
01:44
So that's a subset in the set of quadratic forms on Rn, denoted by p of G. And then imagine you have a C2 smooth functions on some open subset of Rn.
02:00
You want to say that it is plurisubharmonic with respect to that subset G. If its second derivatives belong to this set p of G at every point, in other words the trace of the restrictions of second derivatives to every, each of these
02:22
p planes W in G give a non-negative trace. So you probably, and the word for this is G plurisubharmonic. Yes, so the example will come on the next slide.
02:47
You're just, you're too fast, right? Too fast! So let me go first to the example, and then I'll get back to the general definitions.
03:05
Remind me that I've skipped several pages. Simplest example, this G set consists on the whole space, just one subspace, the whole space itself. So what are we dealing with really?
03:23
So these matrices or quadratic forms are merely assumed to have a non-negative trace, and now the trace of second derivatives, this is the Euclidean Laplacian, and the request of being PSH is Laplacian is non-negative, these are subharmonic functions.
03:44
And I'll get back later to G harmonic functions, right? So then the second example is when the set G consists of all lines, all one-dimensional subspaces, right?
04:03
So then what is required from quadratic forms is that their restriction to lines be non-negative. So, and from these reflex convexity functions, when you apply this to second derivative of a smooth function,
04:21
right? And the next example is now our Rn is in fact R2n and viewed as complex vector space Cn. And now the set G is the set of all complex lines.
04:42
And now what you require is that second derivatives restricted to complex lines give a non-negative trace. In other words, these functions, their restrictions to complex lines are subharmonic, and these are called Plurie subharmonic functions.
05:10
Are you happy, Jerry? Okay, so let's get back... Okay, so far. You will be my test student, right?
05:27
Right. Right, so here I didn't miss anything. So now again, these general G
05:41
Plurie subharmonic functions again mean that restriction of these functions to affine planes, G planes meaning planes whose direction belongs to that set G are subharmonic. And now you can generalize from affine G planes
06:03
to minimal G manifolds. A G manifold, it's a submanifold, all of whose tangent planes belong to this set G. Why minimal? Because when you
06:22
restrict a function to submanifold, then its second derivatives, covariant derivatives, according to the submanifolds, are not exactly the Euclidean second derivatives. There is an error term involving the second fundamental form, and
06:42
when you take a trace, what you recover is the mean curvature. So for a minimal manifold, hm equals zero, and so the condition of subharmonicity on G submanifold becomes equivalent to our previous definition.
07:05
Right, and of course affine G planes are special cases of minimal G manifolds. Well, there is a trace missing on the slide. Thank you, that's right. There should be a
07:24
trace here. Thank you. Now, what about G harmonic functions? Well, so the request here is that
07:43
the second derivatives, instead of belonging to this big set P of G, belong to its boundary. So it merely means that for every W in G, the trace of second derivatives on W is non-negative,
08:03
but for at least one of these W's, it has to be zero. So this is the notion of G harmonic function. So let's go through our examples again. So for the apparently trivial example,
08:24
with only one vector space, the whole space, then we got convex functions. Then we see that G harmonicity means that vanishing of trace of second derivatives for the only available
08:41
subspace. So this is vanishing of the whole Laplacian, just harmonic functions. A slightly less easy example, lines in Rn then the request for being G harmonic is first to be G
09:04
plurisubharmonic. So second derivatives on all lines are non-negative, but that there is at least one line on which it is zero. And so this non-negative matrix of second derivatives, its determinant will vanish because it's not invertible.
09:24
So this is known as the real Mont-Jean-Père equation. And now the inspiring example where G consists of complex lines in Cn,
09:42
then for G harmonic, we required G plurisubharmonic. So now I will go a bit more into it in detail. So what does this mean? So let's start again from this space, vector space of quadratic forms on R2n.
10:05
In presence of the complex structure, this splits into Hermitian and skew-Hermitian quadratic forms. It turns out that the skew-Hermitian part plays no role, it disappears, and
10:23
that this PG, this non-negative trace on complex lines precisely amounts to assuming that the Hermitian part is a non-negative quadratic form, Hermitian form, right?
10:46
So plurisubharmonicity, let's formulate this as follows. Take the Hermitian component of second derivatives, and this is a non-negative Hermitian form. And now harmonic means that this form vanishes along at least one complex line,
11:05
it's non-invertible, so its complex Hessian vanishes. And this leads us to the complex Mont-Jean-Père equation. So something that Harvey and Lawson observed very long ago is that there are natural
11:31
circumstances where you're given a handful of special planes, it's a calibration setting.
11:41
A calibration, this is a differential form, which has the property that on every, so it's a p-form, and you assume that on all p-planes, its restriction to the p-plane is less than the volume element on the p-plane. And you say a plane is calibrated by that form
12:06
if equality holds. And this is the case for complex lines, they are exactly calibrated by the Kelleform on Cn. So this gives rise to a notion of plurisubharmonic, pluriharmonic, plurisub potential
12:30
theory attached to each calibrated geometry. And as we heard, there are quite a lot of them.
12:40
But today's lecture does not follow this line of thought, but a slightly different one, which we reach now. So here, we are on R2n, which we view as a symplectic vector space,
13:03
except that in his notation, a plane will view it as a complex vector space Cn. A Lagrangian subspace, it's a middle dimensional subspace, real subspace, on which the restriction of the Kelleform vanishes identically.
13:28
Equivalently, it's a subspace W, which is exactly orthogonal to its image under the complex structure. So this is the space of Lagrangian subspaces.
13:44
And now, when we can define this P of lag, this set of quadratic forms whose restriction to all Lagrangian subspaces have a non-negative trace. And this gives rise to this notion of
14:02
Lagrangian plurisubharmonic function, which is kind of new. Well, it was new to me until this week. I don't know about you. How many of you were aware of this theory? Raise your hand.
14:25
Raise your hand high enough, please. Thank you. Right. So this is merely a recollection of the general definition in this special case. So the functions we are interested in
14:45
are C2 functions, whose restrictions to all minimal Lagrangian submanifolds are subharmonic.
15:02
So now, C2, not a good idea. You know that an important step in potential theory is to extend to a more wider classes of functions, which are not a priori smooth.
15:25
So how come this is possible? Here is the reason. So what is this P of G? So this P of G, again, these are quadratic forms
15:42
whose restrictions to a certain class category of planes have non-negative trace. So observe that if A' as a quadratic form is bigger than A, meaning that the difference is positive
16:02
semi-definite. And if A is in P of G, then A' is in P of G. These P of Gs are convex cones, which are kind of monotone in this sense. And the second mechanism is that if two functions u
16:27
and v satisfy u less than v, and u at x equals v at x, then this implies that the second
16:43
derivatives of v are larger than the second derivatives of u in this sense of quadratic forms.
17:06
So given a function u which is not smooth enough, well, you can at least consider all smooth functions which are larger and that are equal to u at a point x, and then formulate the assumption that for all these functions,
17:28
this convexity, this positivity property holds. And this provides us with a notion of pure sub-harmonicity for a function which I priorly did be merely continuous or even
17:45
upper continuous. So this is what is written on this slide. So this idea to generalize an inequality or sub-equation to a wider class of function is now classical. This
18:05
is called viscosity, being non-negative in viscosity sense. So is this enough for this slide? Let you time to read it precisely. OK. Maybe you wonder what happens when the
18:33
function u has no smooth function that gets a bit above, like the absolute value function.
18:42
Well, we all know that this is a convex function, right? So no problem, right? At this point, there is no test, available test function, but this does not matter. So this does not prevent the definition from making sense. So now, what is the profit we gain by enlarging this class
19:12
of pure sub-harmonic functions? So by its very definition, the notion is stable under taking the maximum of two functions. Furthermore, it's closed under decreasing limits
19:29
and closed under uniform limits as well. So as a consequence, given a family of g
19:40
pure sub-harmonic functions, provided that they are bounded uniformly locally, I can take the supremum of these, and this is again the g pure sub-harmonic function. So you see now that the Perron method for producing harmonic functions,
20:02
maximal harmonic functions, will generalize to these larger classes, these other classes. So it was really worth allowing a wider class of functions. Of course, we will then
20:25
have a prove existence of sub-harmonic functions and harmonic functions, but with a priori, very little regularity. So a regularity problem will show up, and that's a hard problem that will
20:43
be addressed at the end of the slides. So these are observations. What do we know as far as regularity, just from scratch from the definition? We observe that these
21:04
Lagrangian sub-harmonic functions, they are at least sub-harmonic in the classical sense. Indeed, if since any Lagrangian is orthogonal to its image by j, which is again a Lagrangian,
21:24
then a Lagrangian sub-harmonic function will, if second derivatives on w have no negative trace, jw have no negative trace, but since they are orthogonal, the global trace is the sum of these two traces, so it's non-negative again.
21:46
So that is the starting regularity we have, but not more without more effort. So now, what about harmonic functions? So here is the definition.
22:06
Which is non-trivial, and so take time to understand the notations that I will comment now. Maybe I keep this. So here, so int means interior. So this tilde here means complement.
22:40
So imagine this p of lag is a Euclidean, so it's subset in some vector space,
22:50
that looks like this. So interior, this is here inside, p of g, more generally.
23:01
Then you consider its complement, and then minus the opposite. So this will look like this.
23:22
I perform a central symmetry. So this is what is denoted by p of g tilde. And now by definition,
23:40
so this p of g tilde, it's exactly the set of quadratic forms, a, such that there exists one plane from the set g on which the trace is non-negative, going to complement as replaced for all w by there exists w. And that's exactly how
24:08
p harmonic functions were defined in the smooth case. So let's take this as the definition of lag harmonic functions for a general continuous or
24:23
upper semi-continuous function. You want that it satisfies this p lag condition, and that its opposite satisfies the p lag tilde condition in viscosity sense.
24:43
So this is a coherent definition. Here we are now. We are equipped, I think, with all notions. No, no, no, there are plenty more. So convexity, right? Maybe you've heard the word pseudo-convexity in complex geometry.
25:07
Well, we need a similar concept. So a pseudo-convex means for, let's say, smooth domain in Cn, means that locally, so near each point of the boundary,
25:24
there is a defining equation, a function whose zero set is the boundary, where the set is locally where u is non-negative, and which is purely subharmonic.
25:43
So let's take this as a definition in the general case. So again, this has a translation in terms of the second fundamental form of the boundary. So it exactly means
26:02
that the restriction of the second fundamental form to Lagrangian planes which are tangent to the boundary has a non-negative trace, but the trace now is not a number but a vector, and you want that this vector points inward. So that's the notion of Lag-convex domains.
26:27
And now I think we have all definitions to state the first theorem in that theory. A Lag-convex domain, then the homogeneous Dirichlet problem has a solution, meaning
26:45
you're given a continuous function on the boundary. It has a unique Lag-harmonic extension inside. And I think you can guess what the proof is. It will be a Perron-type argument. No.
27:10
Well, a Perron will give existence. There must be an argument for uniqueness, which I don't quite understand. So why? Because,
27:23
well, Blaine sent to me his paper concerning this concept this week, and I realized that this was a survey of one, two, three, four, five, fifteen, a series of fifteen papers by Harvey and Lawson in the last fifteen years. So I gave up. I apologize for this.
27:55
So now the new, so this solution of the Dirichlet problem perhaps is in paper number
28:08
two or three in the series. But now let's go to the contents of paper number fifteen, which is the Monjom-Perel operator. So we'd like to write a differential equation
28:25
that, at least in a smooth case, characterizes these Lagrangian harmonic functions. Up to now, we have nearly, we have given a geometric definition. And it would be so nice to
28:42
have an equation like determinant of Hessian, determinant of second derivative. And this is exactly what we are aiming at. So let's get back to the space of quadratic forms
29:01
on R to N. And again, it splits, thanks to the complex structure, into Hermitian and skew-Hermitian quadratic forms. And the Hermitian ones, you have the ones which are proportional to the metric and the others, which have trace zero. So this is this Herm zero.
29:28
So, Hermitian forms on Cn with trace zero. And the first observation is that only
29:42
two components in this decomposition in three terms will play a role. Indeed, the trace zero Hermitian part will disappear instantaneously. So let's see this.
30:04
Imagine E is a Hermitian form of trace zero. And let's show that the trace of its restrictions to all Lagrangian planes vanishes. So let's W be such a plane, ek an orthonormal basis of W. Trace of restriction of e to W, that's the traditional
30:29
sum e ek ek. And now, let's use the fact that J is an isometry. So the trick is,
30:42
on the top line, is to introduce the J. Since it's an isometric, it changes nothing. And write this sum as a half of the sum of two terms which are equal. So on the right, I've simply plugged in this J. And now, Hermitian means that e commutes with
31:07
J as an endomorphism. So let's do the commutation. And what I see now in the brackets, it's the sum of e ek ek over an orthonormal basis of the whole space now, W plus JW. So that's the trace
31:28
of e, which vanishes by assumption. So Blaine wants to stress the parallel and the difference
31:44
from the traditional holomorphic case. In the holomorphic story, the condition for plurisubarbonicity depends only on the symmetric part of second derivatives,
32:03
whereas in our Lagrangian story, the trace-free Hermitian part disappears, and it's the other part, the skew-Hermitian part, which plays a role. Well, that's not such a surprise. We know that in dimension four, for instance, complex lines and Lagrangian
32:28
subspaces are really kind of orthogonal. So this was the first step.
32:41
Now, let's go a bit deeper in this linear algebra. So now, we will for a while ignore the trace, so the quadratic forms, which are proportional to the metric. And again,
33:01
under the unitary group, we have these three summands. Let's focus on the third one, skew-Hermitian matrices and quadratic forms. So in terms of the symmetric operator, it's a symmetric operator that anti-commutes with the complex structure.
33:20
So it follows that this can be diagonalized, and that the eigenvectors have a J action. If E is an eigenvector for eigenvalue lambda, then its image by the J, the complex structure, is an eigenvector again, with eigenvalue minus lambda. So eigenvalues come in pairs of
33:46
opposite numbers. So now, let's denote by lambda one, lambda n, the non-negative ones in this order. And so, we see that the lowest possible sum of n of these eigenvalues one can
34:05
make is precisely minus the sum of the lambda i's. As a consequence, when you restrict b to some n-dimensional subspace, the trace will always be at least that number.
34:25
So now, in general, we start with a quadratic form. This will have a trace component, a skew-Hermitian component, and we ignore the Hermitian component, which we know will play
34:44
a role. So finally, we were, like in the previous slide, simply added to the matrix B lambda i times the identity matrix, or excuse me, trace of a divided by 2n times the identity matrix.
35:04
And so, the trace of the restriction to a Lagrangian plane will be at least mu minus sum of lambda i's, where mu is the trace divided by 2. So now, we see
35:21
that these expressions mu minus sum of lambda i's are the ones which will relate to our
35:43
restrictions to Lagrangian subspaces are non-negative. It's exactly assuming that this number, where mu is the trace divided by 2, and lambda 1, lambda n, the non-negative eigenvalues of the skew-Hermitian part, are non-negative.
36:04
The converse direction arises from the fact that the equality is achieved by an Lagrangian subspace. And so, this gives the idea of, this is our differential equation, in equation,
36:23
except that it doesn't look like a differential equation yet. So now, let us form all these combinations of mu and lambda i's. And now, you introduce all possible sign changes and then multiply all these numbers. Why?
36:47
Because when doing this, we get an expression which is polynomial in the eigenvalues and which is fully symmetric. Lambda 1 plus lambda n itself was symmetric only in the
37:04
non-negative eigenvalues, not all of them. Whereas, once you have introduced all the sign changes and multiplied, what you obtain now is fully symmetric in all eigenvalues of B. So this becomes polynomial in B, as we know. A polynomial expression which is symmetric
37:27
in the eigenvalues of an endomorphism is a polynomial on the space of endomorphisms. So here we are, and we define this as the Lagrangian-Maugin-Pere operator.
37:42
It's this polynomial. Well, it's not yet an operator, but it's a polynomial on the space of quadratic forms. So that's it. And now, by getting back to our definition,
38:03
our P-lag was amounted to require that the smallest of these numbers is non-negative. So this means this polynomial has a zero set, and the complement,
38:22
even the locus where the polynomial is positive, has several connected components, and requiring that all eigenvalues, all these numbers in the product are non-negative or positive means that you're in the connected component that contains the identity matrix.
38:47
And now it turns out that Lag harmonic functions are precisely those functions which are Lag-Peluri subharmonic, in such that this polynomial in the second
39:02
derivatives vanishes. And of course, in the general setting of upper semi-continuous functions, we have to take these equations in the viscosity sense. But now, here, we're really writing a differential operator in the classical sense.
39:26
Still a bit mysterious. Mr. Chairman, I'm over time now. Not yet? Yeah? Okay. Well, I think Blaine would be angry if I would have
39:51
stopped here. Yes? My dim memory of the viscosity solution is you somehow add epsilon
40:03
times the Laplacian to make it positive definite. I don't see that here at all. Ah, you mean you know another description of viscosity solutions. And I would bet that what you have in mind extends to this setting as well, but I
40:26
can't tell you. I will transfer this question to Blaine. I promised him to do so. Yes? Is there any notion of concavity here on MA? Okay, maybe the answer to your question comes a bit later.
40:45
So that's one more argument to let me continue. But please, at some point, you'll have to stop me, right? So the next slide,
41:05
I'll try to be a bit faster, which is a pity. Instead of playing with eigenvalues, we'd like to see some more intrinsic definition. I will say this in one sentence. Given a skew Hermitian quadratic form,
41:25
can define its absolute value, like the absolute value of a matrix, the square root of its square, the non-negative square root of its square. Now, mixing. So take absolute value of b times j. Now this is a skew symmetric matrix. So it defines a skew symmetric 2-form.
41:47
And now 2-forms can be viewed like, well, differential form, well, exterior forms, isomorphic to the Clifford algebra. And now you can let the Clifford algebra act on spinners
42:02
two to the n-dimensional representation. And so our quadratic form now defines an operator, an endomorphism of spinners. And now multiply by i, complex number i,
42:21
and add mu, this is half of the trace, times the identity. And take the determinant of this endomorphism. And this is exactly the Monjompere operator. So the derivation from this product of eigenvalues is rather straightforward.
42:41
But this is a more elegant way of viewing this polynomial. It's merely the determinant, but in some representation of dimension two to the n. Now, maybe I will skip this, because everything is straightforward and you have to believe me.
43:07
What does Blaine call a Gromov manifold? It's a symplectic manifold, together with the choice of almost complex structure, which is compatible in this sense, and taming in this sense.
43:26
So many of you remember the context in which Gromov introduces these kind of objects. And Blaine's claim is that everything I said up to now extends to this context. You merely need
43:40
replace what I called second derivatives by the Hessian with respect to the Levi-Civita connection of the metric you obtained by pairing the symplectic form and the complex structure. So what we get is a theory on Gromov manifolds. So that's not exactly symplectic geometry,
44:05
that's symplectic geometry enriched with the choice of an almost complex structure. Now, back to the Dirichlet problem. First, there is a homogeneous Dirichlet
44:21
problem that we can now state. We want to prescribe boundary values of a function together with the value of the Lagrangian operator on it. And this can be done on Lagrangian convex domains. And then comes the idea, which was completely new to me
44:50
until yesterday, that there are more branches. So you remember that in the definition of the
45:09
we call it an eigenvalue of the Monjomper operator applied to a given quadratic form. And so we have 2 to the n of these numbers called improperly eigenvalues.
45:26
And so instead of only one notion of Plurisa-Barbonicity that we had initially, we have 2 to the n of them. So the first one said that all these eigenvalues have to be
45:45
non-negative. The second one says that all these eigenvalues, but possibly the smallest one, are non-negative. This and so on. And it turns out that the homogeneous Dirichlet
46:06
functions are non-negative. Each of these notions of Plurisa-Barbonic functions. So even in the classical holomorphic setting, I was unaware of these notions,
46:22
which I find fascinating. And the final slide is just the beginning of a regularity theory. For such functions. So Riesz characteristic means start with the function
46:46
u, and now you want to understand its behavior at zero, and you want to analyze its regularity by producing some kind of tangent cone, as Jeff Chigurh explained to us when he does
47:07
this. So you form x times R a, a small positive number, and the statement that the Riesz characteristic is n is that 2 minus n u of x R, for any pseudo-Lagrangian-Plurisa-Barbonic function,
47:34
this has a limit. It has limits as R tends to zero. So tangent cones exist with this exponent here,
47:50
n. And now the second step in the regularity theory is to show uniqueness of tangent cones. In fact, it's a much more stronger statement hold sometimes, is that all cones, tangent cones,
48:07
which will be globally defined functions on Rn, will be multiples of a certain fundamental solution and its translates. And here is the fundamental solution. It's really familiar looking. Here we
48:24
are in R2n, and what you recognize is the expression of the fundamental solution of the Laplacian on Rn, but you extend it by the same expression to Cn. And now it's here that the slides end with a series of questions. So this regularity theory
48:52
has just begun with already pretty striking results. And here are the questions that Blaine wanted to set, which clearly generalize what we know
49:07
in the holomorphic setting. So sorry for having been so slow in the beginning and so fast in the end. And now you may try and ask questions.
49:37
So question. I'm relieved.
49:46
So there is one question that I will transfer to Blaine, is the alternate approach to viscosity and that you will describe to me and I'll ask him. But here is another question.
50:04
I will ask about if we can change this for any other complicated question, and if you don't have the maximum principle there. I was just wondering if you have a notion of maximum principle there as well.
50:23
So this beginning regularity theory works in a wider sense. So the starting point of Harvey and Lawson today, number 16, is conditions on second derivatives. So you specify a subset,
50:50
an open subset of the quadratic forms, and it may be convex or not, and it may be a cone or not,
51:02
but it always has this kind of monotonicity. And so stronger results holds when the set is convex, even stronger holds when the set has some symmetry invariant under some subgroup of the orthogonal group.
51:27
So you are right, there is room for even wider generalization than these p of g's that I described. You are right.
51:44
Is there a reference? You mentioned a manuscript. Is it a secret manuscript or something in the archive? So on the manuscript, it's written submitted, and I looked into the archive and I couldn't
52:04
find it. So that's the status. So the manuscript exists, it is submitted, but it is secret. But maybe you really want to have it? I won't know unless I look at it.
52:21
Then okay, I'll show you and then you decide. Well, I can write blind. Oh, well, yeah, sure. Of course, yeah. I have a question. Maybe it is contained in question two.
52:43
I used to work with Stein domain. That means compact domain, complex manifold with boundary, which are defined by pseudo subarachnoid function.
53:01
And there is dynamics on the boundary as well, important rib, because the boundary is contact manifold. And is there an analog of this picture in the case of lag category?
53:23
Is there some dynamics associated? So apparently, as you pointed out, this is probably what Lane has in mind with this question. There may be a substantial difference between the holomorphic case and this case.
53:48
There are two notions in the papers of lag harmonic and lag pluriharmonic, and apparently they differ. Or at least the g harmonic and g pluriharmonic are not always
54:03
the same. Maybe they coincide for lag. That's something I could not sort out from what I read up to now, because I did not read much. So there may be a pitfall there. So again, that's one more question I will transfer to Blaine,
54:21
and then he'll answer directly to you. I think we thank Jeff again.
Recommendations
Series of 36 media