The document discusses harmonic maps from the Riemann surface M=S1×R or CP1\{0,1,∞} into the complex projective space CPn. It presents the DPW method for constructing harmonic maps using loop groups. Specifically, it constructs equivariant harmonic maps in CPn from degree one potentials in the loop algebra Λgσ, relating these to whether the maps are isotropic, weakly conformal, or non-conformal. It then considers the system of ODEs and scalar ODE that must be solved to generate the harmonic maps using this method.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
We present a proof of the Generalized Riemann hypothesis (GRH) based on asymptotic expansions and operations on series. The advantage of our method is that it only uses undergraduate maths which makes it accessible to a wider audience.
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
1. Harmonic trinoids in complex projective spaces
Shimpei Kobayashi, Hirosaki University
12/12, 2008
2. Introduction
Harmonic maps into complex projective spaces
Preliminaries
Harmonic spheres
Harmonic tori
Equivariant harmonic maps in CPn
Isomorphisms between loop algebras
Potentials for equivariant harmonic maps
Harmonic trinoids in CPn
DPW method
System of ODEs and a scalar ODE
Hypergeometric equations
Unitarizability and interlace on the unit circle
Open problems
3. Let (M, g) and (N, h) be Riemannian manifolds and
Ψ : (M, g) → (N, h)
a C∞ map.
Define
E(Ψ) = |dΨ|2 dVg ,
M
where the norm is defined by g and h, and dV g is the volume form
of M.
4. Let (M, g) and (N, h) be Riemannian manifolds and
Ψ : (M, g) → (N, h)
a C∞ map.
Define
E(Ψ) = |dΨ|2 dVg ,
M
where the norm is defined by g and h, and dV g is the volume form
of M.
Consider the variation Ψt for Ψ.
def d
Ψ is harmonic ⇔ E(Ψt )|t=0 = 0 ⇔ τ (Ψ) = 0,
dt
where τ (Ψ) = trace dΨ is the tension field.
5. In particular, if dim M = 2, then the harmonicity can be written
as
Ψ ∂
∂ dΨ( ) = 0, (1)
∂¯
z ∂z
where z = x + iy and (x, y) is a conformal coordinate.
6. Harmonic spheres
If M = S2 , the followings (N, h) were studied in details:
Sn (RPn ) (Calabi, Chern)
CPn (D. Burns, Eells-Wood, Din-Zakrzewski, Glaser-Stora)
Gr2 (Cn ) (Chern-Wolfson, Burstall-Wood)
Grk (Cn ) (Wolfson, Wood)
These are based on
1) Holomorphic differential on S2 is zero
2) Techniques of Hermitian vector bundles.
7. Harmonic tori
If M = T2 , the followings (N, h) were studied in details :
S2 (Pinkall-Sterling)
S3 (Hitchin)
S4 (Pinkall-Ferus-Sterling-Pedit)
Sn , CPn (Burstall, McIntosh)
Gr2 (C4 ), HP3 (Udagawa)
Rank 1 compact symmetric spaces
(Burstall-Ferus-Pedit-Pinkall)
These are based on integrable system methods.
8. Goal of this talk
I would like to discuss harmonic maps from M = S 1 × R or
M = CP1 {0, 1, ∞} into N = CPn .
9. Goal of this talk
I would like to discuss harmonic maps from M = S 1 × R or
M = CP1 {0, 1, ∞} into N = CPn .
Consider a C∞ map Ψ from a Riemann surface M into a
symmetric space G/K:
∂ 1
Ψ dαk + 2 [αk ∧ αk ] = −[αp ∧ αp ] = 0,
∂ dΨ( )=0 ⇔
∂¯
z ∂z dαp + [αk ∧ αp ] = 0,
1
dαλ + 2 [αλ ∧ αλ ] = 0,
⇔
αλ = λ−1 αp + αk + λαp , λ ∈ S1 .
where α = F−1 dF is the Maurer-Cartan form of a lift
F : M → G, g = k ⊕ p and TMC = T M + T M.
10. Equivariant harmonic maps in k-symmetric spaces
Definition
A map Ψ : R2 → G/K is called R-equivariant if
Ψ(x, y) = exp(xA0 )Φ(y),
for some A0 ∈ g and Φ : R → G/K.
Theorem (Burstall-Kilian)
All equivariant primitive harmonic maps in k-symmetric spaces
G/K (with an order k-automorphism τ ) are constructed from
degree one potentials:
ξ = λ−1 ξ−1 + ξ0 + λξ1 ∈ Λgτ , (2)
where Λgτ = {ξ : S1 → g | ξ(e2πi/k λ) = τ ξ(λ)} is the loop
algebra of the Lie algebra g of G, ξj ∈ gj and ξj = ξ−j with the
eigenspace decomposition of gC = i∈Zk gi .
11. Equivariant harmonic maps in CPn
For CPn case, G = SU(n + 1) with the involution
σ = Ad diag [1, −1, . . . , −1], thus K = S(U(1) × U(n)) and
GC = SL(n + 1, C).
It is known that harmonic maps in CPn can be classified into
isotropic,
non-isotropic weakly conformal with isotropic dimension
r ∈ {1, . . . , n − 1},
non-conformal.
Problem: Which degree one potentials are corresponding to the
above cases?
12. Isomorphism
Lemma (Pacheco)
Let g be a Lie algebra, τ : g → g an automorphism of order k
and σ : g → g an involution.
Define Γ as a map between Λgτ and Λgσ
Γ(ξ)(λ) = s(λ)t(λ−2/k )ξ(λ2/k ) ∈ Λgσ for ξ ∈ Λgτ , (3)
where t : S1 → Aut g and s : S1 → Aut g are automorphism
such that t(e2πi/k ) = τ and s(−1) = σ respectively.
Then Γ is an isomorphism.
13. Isomorphism
Lemma (Pacheco)
Let g be a Lie algebra, τ : g → g an automorphism of order k
and σ : g → g an involution.
Define Γ as a map between Λgτ and Λgσ
Γ(ξ)(λ) = s(λ)t(λ−2/k )ξ(λ2/k ) ∈ Λgσ for ξ ∈ Λgτ , (3)
where t : S1 → Aut g and s : S1 → Aut g are automorphism
such that t(e2πi/k ) = τ and s(−1) = σ respectively.
Then Γ is an isomorphism.
Let t and s be t(λ) = Ad diag[1, λ, . . . , λ k−2 , λk−1 , . . . , λk−1 ]
and s(λ) = Ad diag[1, λ, . . . , λ] respectively. Then it is easy to
see t(e2πi/k ) = τ and s(−1) = σ. Define Γ as in (3), and let
t
ξ = λ−1 ξ−1 + ξ0 + λξ−1 ∈ Λsu(n + 1)τ
be the degree one potential.
14. Proposition
A harmonic map in CPn is R-equivariant if and only if it is
generated by the following degree one potential
η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)
where the order k of τ and the degree one potential ξ are given as
follows:
15. Proposition
A harmonic map in CPn is R-equivariant if and only if it is
generated by the following degree one potential
η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)
where the order k of τ and the degree one potential ξ are given as
follows:
(a) if it is isotropic:
k = n + 1 and ξ−1 is principal nilpotent.
16. Proposition
A harmonic map in CPn is R-equivariant if and only if it is
generated by the following degree one potential
η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)
where the order k of τ and the degree one potential ξ are given as
follows:
(a) if it is isotropic:
k = n + 1 and ξ−1 is principal nilpotent.
(b) if it is non-isotropic weakly conformal with the isotropic
dimension r ∈ {1, 2, · · · , n − 1}:
k = r + 2 ∈ {3, 4, · · · , n + 1} and ξ−1 is semisimple.
17. Proposition
A harmonic map in CPn is R-equivariant if and only if it is
generated by the following degree one potential
η = Γ(ξ) ∈ Λsu(n + 1)σ , (4)
where the order k of τ and the degree one potential ξ are given as
follows:
(a) if it is isotropic:
k = n + 1 and ξ−1 is principal nilpotent.
(b) if it is non-isotropic weakly conformal with the isotropic
dimension r ∈ {1, 2, · · · , n − 1}:
k = r + 2 ∈ {3, 4, · · · , n + 1} and ξ−1 is semisimple.
(c) if it is non-conformal:
k = 2 and ξ−1 is semisimple.
19. Loop groups
Definition
G : A compact simple Lie group, g : Lie algebra of G,
GC : The complexification of G, gC : Lie algebra of GC ,
σ : A involution of G, K : The fixed point set of σ
k : Lie algebra of K, g = k ⊕ p : Direct sum
B : The solvable part of an Iwasawa decomposition
KC = K · B, K ∩ B = e
20. Loop groups
ΛGσ := {H : S1 → G | σH(λ) = H(−λ)},
Λgσ := {h : S1 → g | σh(λ) = h(−λ)}
ΛgC σ := {h : S1 → gC | σh(λ) = h(−λ)}
ΛGC := {H : S1 → GC | σH(λ) = H(−λ)}
σ
H+ can be extend holomorphically
Λ+ GC :=
B σ H+ ∈ ΛGC |
σ to D1 and H+ (0) ∈ B
21. Loop groups
ΛGσ := {H : S1 → G | σH(λ) = H(−λ)},
Λgσ := {h : S1 → g | σh(λ) = h(−λ)}
ΛgC σ := {h : S1 → gC | σh(λ) = h(−λ)}
ΛGC := {H : S1 → GC | σH(λ) = H(−λ)}
σ
H+ can be extend holomorphically
Λ+ GC :=
B σ H+ ∈ ΛGC |
σ to D1 and H+ (0) ∈ B
We assume that the coefficients of all g ∈ Λg σ are in the Wiener
algebra
A= f(λ) = fn λn : C r → C ; |fn | < ∞ .
n∈Z n∈Z
The Wiener algebra is a Banach algebra relative to the norm
f = |fn |, and A consists of continuous functions.
22. DPW method
Step1 η(z, λ) = ∞ λk ξk (z) : ΛgC -valued 1-form on Σ ⊂ C,
k=−1 σ
where ξeven (z) ∈ Ω1,0 (kC ) and ξodd (z) ∈ Ω1,0 (pC ).
Step2 Solve ODE dC = Cη.
Step3 Iwasawa decomposition : C = FW + , F : Σ → ΛGσ and
W+ : Σ → Λ + GC .
B σ
Step4 Projection π ◦ F|λ∈S1 : Σ → G/K, where π : G → G/K.
Theorem (Dorfmeister-Pedit-Wu)
Multiplication ΛGσ × Λ+ GC → ΛGC is a diffeomorphism onto.
B σ σ
Theorem (Dorfmeister-Pedit-Wu, 1998)
Every harmonic map from a simply connected domain Σ into G/K
can be constructed in this way.
23. DPW method
Step1 η(z, λ) = ∞ λk ξk (z) : ΛgC -valued 1-form on Σ ⊂ C,
k=−1 σ
where ξeven (z) ∈ Ω1,0 (kC ) and ξodd (z) ∈ Ω1,0 (pC ).
Step2 Solve ODE dC = Cη.
Step3 Iwasawa decomposition : C = FW + , F : Σ → ΛGσ and
W+ : Σ → Λ + GC .
B σ
Step4 Projection π ◦ F|λ∈S1 : Σ → G/K, where π : G → G/K.
Theorem (Dorfmeister-Pedit-Wu)
Multiplication ΛGσ × Λ+ GC → ΛGC is a diffeomorphism onto.
B σ σ
Theorem (Dorfmeister-Pedit-Wu, 1998)
Every harmonic map from a simply connected domain Σ into G/K
can be constructed in this way.
From now on, CPn is represented as the symmetric space
U(n + 1)/U(1) × U(n) with the involution
σ = Ad diag [1, −1, . . . , −1].
24. System of ODEs and a scalar ODE
Consider
∞
ν, τj ∈ λ2k−3 g2k−3 g2k−3 is a holomorphic function on M ,
k=1
where i = 1, . . . , n and
dn+1 ν dn dn−1
− − ντ1 n−1 − · · · − ντn u = 0. (5)
dzn+1 ν dzn dz
25. System of ODEs and a scalar ODE
Consider
∞
ν, τj ∈ λ2k−3 g2k−3 g2k−3 is a holomorphic function on M ,
k=1
where i = 1, . . . , n and
dn+1 ν dn dn−1
− − ντ1 n−1 − · · · − ντn u = 0. (5)
dzn+1 ν dzn dz
Set
u1 , . . . un+1 : A fundamental solutions of (5),
(n)
u1 (n−1) (0)
ν u1 · · · u1
. . .
. .. .
.
C := . . . . ,
(n)
un+1 (n−1) (0)
ν
un+1 · · · un+1
(0) (k) dk u
where uj = uj and uj = dzk
, (k > 0).
26. Lemma
(1)
0 ν 0 ··· 0
τ1 0 1 · · · 0
−1
. . ..
. . .. .
.
η := C dC = . . . . . . (6)
. . ..
. . ..
. . . . 1
τn 0 · · · · · · 0
∞
(2) η = λk ξk is a holomorphic potential on M, where
k −1
ξeven ∈ Ω1,0 (kC ) and ξodd ∈ Ω1,0 (pC ).
Fact: Monodromy representations of (5) and (6) are the same.
27. Hypergeometric functions
n+1 Fn (α1 , . . . , αn+1 ; β1 , . . . , βn |z)
∞
(α1 )k · · · (αn+1 )k k
= z , (7)
k=0
(β1 )k · · · (βn )k k!
where α1 , . . . , αn+1 , β1 , . . . , βn ∈ C, (x)k is the Pochhammer
symbol or rising factorial
Γ(x + k)
(x)k = = x(x + 1) · · · (x + k − 1).
Γ(x)
n+1 Fn (α1 , . . . , αn+1 ; β1 , . . . , βn |z) is called the
hypergeometric function n+1 Fn .
2 F1 (α1 , α2 ; β1 |z) is the Gauß’s hypergeometric function.
28. Let D(α; β) = D(α1 . . . αn+1 ; β1 . . . βn+1 ) be the differential
operator
D(α; β) = (θ+β1 −1) . . . (θ+βn+1 −1)−z(θ+α1 ) . . . (θ+αn+1 )
d
for α1 , . . . , αn+1 , β1 , . . . , βn+1 ∈ C, where θ = z dz . The
hypergeometric equation is defined by
D(α; β)u = 0.
30. If βi are distinct mod Z, n + 1 independent solutions of
D(α; β)u = 0 are given by
z1−βi n+1 Fn (1+α1 −βi , . . . , 1+αn+1 −βi ; 1+β1 −βi , . ∨ ., 1+βn+1 −βi |z),
.
where i = 1, . . . , n + 1 and ∨ denotes omission of 1 + β i − βi .
V(α; β):The local solution space of D(α; β)u = 0 around z 0 .
G : The fundamental group π1 (CP1 {0, 1, ∞}, z0 ).
M(α, β) : G → GL(V(α; β)) : Monodromy representation
of D(α; β)u = 0.
31. Theorem (Beukers-Heckman, 1989)
Let M(α; β) be the Monodromy group of D(α; β)u = 0. Then
M(α; β) are simultaneously conjugated into U(n + 1).
iff
0 < α1 < β1 < α2 < β2 < . . . < αn+1 < βn+1 1
or
0 < β1 < α1 < β2 < α2 < . . . < βn+1 < αn+1 1 .
32. Remark
αj and βj are determined by solving the indicial equations, which
are n-th order algebraic equations.
There are several problems for an application to harmonic maps in
CPn .
αj and βj depend on the additional parameter λ ∈ C.
αj and βj need to be real and satisfy the inequality for almost
all λ ∈ S1 .
Products and sums of αj and βj are ν and τj as in the
holomorphic potential of (6).
33. The case n = 1 (Gauß’s hypergeometric equation)
Local exponents
z=0
z=∞ z=1
1−β α1 0
1
2 2
1 − β2
α2 γ= βj − αj − 1
1 1
Set
α1 = 1 − v 1 − v 2 − v 3 , α 2 = 1 − v 1 − v 2 + v 3 ,
and
β1 = 1 − 2v1 , β2 = 1,
where
1 1
vj = − 1 + wj (λ − λ−1 )2
2 2
34. Spherical triangle inequality
v1 + v 2 + v 3 < 1
v1 < v 2 + v 3
0 < α 1 < β1 < α2 < β2 1⇔ (8)
v2 < v 1 + v 3
v3 < v 1 + v 2
It is not difficult to show that the above inequality are satisfied for
some choices of wj . Moreover all problems can be solved
(Kilian-Kobayashi-Rossman-Schmitt, Dorfmeister-Wu).
Remark
Umehara-Yamada considered the similar inequality for CMC
H=1 in H3 . (No λ dependence!)
35. Examples of CMC trinoids in space forms
Figure: These figures are created by Nick Schmitt.
36. The case n > 1
Example
For the isotropic case, αj and βj do not depend on λ. Thus
there exist isotropic harmonic trinoids in CP n .
For n = 2, 3, the indicial equation can be solved explicitly.
We can show that there exist examples of harmonic trinoids in
CP2 and CP3 .
37. Open problem
What are behaviors around the punctures? Are they
asymptotically converge to equivariant ones?
Prove the existence of non-isotropic harmonic trinoids for
n 4.