Folding, tiling and tori: a Hamiltonian analysis, O.M. Lecian, 17 January 2018

Folding, tiling and tori: a Hamiltonian analysis
O.M. Lecian
Comenius University in Bratislava,
Faculty of Mathematics, Physics and Informatics,
KFTDF-Department of Theoretical Physics and Physics
Education,
and
Sapienza University of Rome, DICEA-
Department of Civil, Building and Environmental Engineering
17 January 2018
The Geometry and Algebra Seminars,
Sapienza University of Rome, SBAI
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis

Abstract
After introducing the modular group, the projective linear group
and its congruence subgroups are described; the Hecke groups are
defined, and their congruence subgroups realizations are
investigated. The differences between the congruence subgroups
for Gamma2 and Gamma(2) are outlined. Non-arithmetical groups
are considered, along with the possible subgroup structures. The
Picard and the Vinberg groups are examined in detail according to
the tools outlined. The folding (sub)group and the tiling
(sub)group structures are compared and specified. Definition for
geodesics trajectories are provided also after the Hamiltonian
analysis. The possible tori arising from this descriptions are defined.

Summary
• The modular group SL(2, C).
• The extended modular group PGL(2, C).
• Tiling vs folding.
• Non-arithmetical groups.
• The Hecke groups.
• The congruence subgroups of PGL(2, C): Γ0, Γ1, Γ2.
• The congruence subgroup Γ(2),
• Γ(2) tori,
• The Gutzwiller torus
• The Picard group
• The Vinberg groups
• Hamiltonain analysis:
• The Gauss-Kuzmin theorem for surds;
• Γ2 tori: complete tori and punctured tori.
• Non-modular algebraic structures for measures in C∗ algebras.
• PSL(2, Z) (Hamiltonian) tiling for generalized groups and congruence
subgroups of the extended modular group and for Hecke groups.
• Comparisons for ﬁnding congruences in non-arithmetical groups.

Orientation
T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, New York, USA (1976).

Deﬁnition
Jo/rgensen Groups with A, B transformations for a discrete
subgroup of SL2(C) ⇒
| tr2
(A) − 4 | + | tr(ABA−1
B−1
) − 2 |≥ 1
The modular group, Picard group and the 8-shape knot group
π1(R3/K) are Jo/rgensen groups.
H. Sato: The Picard, group, the Whitehead link and Jo/rgensen groups, in Progress in analysis : proceedings of the
3rd International ISAAC Congress,International Society for Analysis, Applications, and Computation, pp. 149-158,
Ed.’s: H.G.W. Begehr, R.P. Gilbert, M.W. Wong, World Scientiﬁc, New York, USA (2003).

Remark
Reﬂection groups on asymmetric domains are not Jo/rgensen
groups
Theorem
Jo/rgensen groups do not admit traslation subgroups
Remark They admit (also) traslation (sub-)grouppal extensions.
T. Jo/rgensen , A. Lascurain, T. Pignataro Translation extensions of the classical modular group, Complex
Variables, Theory and Application: An International 19, pp. 205-209 (1992).

Preserving orientation and
reversing orientation
Reﬂection congruence subgroups contain canonical reﬂections, whose op-
erators are unique (elements) which map sides in an orientation-preserving
manner and in an orientation-reversing manner.
R.S. Kulkarni, An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group, Am. Journ.
Math., 113, pp. 1053-1133 (1991).

The modular group SL(2, Z)
arithmetic subgroup for SL(2, R)
fundamental domain: sides
a1 : u = −1
2,
a2 : u = 1
2,
a3 : u2
+ v2
= 1.
generators of transformations:
T(z) = z + 1,
S(z) = −1
z ;
T2
= S3
= I
classification from
A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Springer-Verlag, New
York, USA (1985).

sides identiﬁcation
T : a1 → a2,
S : a3(u < 0) → a3(u > 0),

The modular group: domain
The domain of the modular group
sides identiﬁcations a1 → a2, a3(u < 0) → a3(u > 0)
induced by the transformations T and S:
T : a1 → a2, S : a3(u < 0) → a3(u > 0)

PGL2(Z)
a.k.a. the extended modular group
transformations
R1(z) = −¯z,
R2(z) = −¯z + 1,
R3(z) = 1
¯z
b1 : u = −1
2,
b2 : u = 0,
b3 : u2
+ v2
= 1.

The group PGL(2, C): domain
The domain of the extended modular group PGL(2, C).

no side-identiﬁcation is possible
comparison with the modular group:
T = R2R1, T−1
= R1R2,
S = R1R3, S−1
= S,
comparison: after C ֒→ R

The Θ Group
θ1 : u = −1,
θ2 : u = 1,
θ3 : u2
+ v2
= 1.
transformations
TΘ ≡ τ1(z) = z + 2,
SΘ ≡ τ2(z) = −
1
z
,

sides are identiﬁed as
TΘ : θ1 → θ2,
SΘ : θ3(−1 ≤ u ≤ 0) → θ3(0 ≤ u ≤ 1),

The Theta group
TΘ : θ1 ↔ θ2
SΘ : θ3(u < 0) ↔ θ3(u > 0)

Non-arithmetical groups Nω
fundamental domains: domain
u = 0, (10a)
u = − cos(
π
w
) ≡ uA, (10b)
v = 1 − u2, (10c)
cos α = −uA, with α ≡ π/ω is the angle between the considered
side and the goniometric circumference,
cos α = −uA.
Described as a group of three reﬂections generators of
transformations
T1 : z → −¯z, ,
T2 : z → −¯z − 2 ≡ uA,
T3 : z → −1
¯z ,

N(ω) ∈ PSL(2; R)
orientation preserving
T(z) = −1/z
Sω(z) = z + zA;
Sω ≡ TU, ⇒ S(z) = −
1
z + ua
uA = ua(ω) = 2 cos(π
ω ), ω in R/N.

Non-arithmetical groups: α < π
4
An example: α = π
3.2
< π
4
.

Non-arithmetical groups: α > π
4
An example: α = π
4.3
> π
4

The Hecke groups
H(λ) ∈ PSL(2; R)
orientation preserving
T(z) = −1/z
U(z) = z + λ;
S ≡ TU, ⇒ S(z) = −
1
z + λ
H(λ) discreteiﬀ λ = λq = 2 cos(π
q )
Z ֒→ R

Hq isomorphic to free product of
two ﬁnite permutation groups of order 2 and q, resp.:
T2 = Sq = I.
The even subgroup
He(λq) of Hq
is deﬁned for even values of q.
All Hecke groups are subgroups of
PSL(2; Z), Z ∈ Z[λq]

Reflections groups and Reflection subgroups
Reflections: characterized by z → f (¯z)
Traslations and mirror images: characterized by z → f (z)
Example:
traslation wrt degenerate geodesics: z → z − 1 reflection wrt
degenerate geodesics: z → −¯z − 1
inversion wrt non-degenerate geodesics: z → − z
z+1
reflection wrt non-degenerate geodesics: z → − ¯z
¯z+1
Theorem
Every subgroups of a free product (of groups) consists of a free
product of a free group and the elements conjugated in common to
the intersection of the considered free group.
A.G. Kurosh, Die Untergruppen der freien Produkte von beliebigen Gruppen, Mathematische Annalen, vol. 109,
pp. 647-660 (1934).

Admissible transformations
The identifications of any two sides of a (sub-)group domain is
admissible
iff
its the side-identification transformations consist of an independent
set of generators.

Comparison of tessellation
PGL(2, Z) generates a topological space by the extended modular
map for the extended modular tessellation.
Theorem
A special polygon is a fundamental domain for the subgroup
generated by the admissible side-pairing transformations. The
transformations form an independent set of generators for the
subgroup.
The converse holds.
R.S. Kulkarni, An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group, Am. Journ.
Math., 113, pp. 1053-1133 (1991).

Γ0(PGL(2, C)): The Γ0 congruence subgroup of PGL(2, C)
ζ1 : u = −1,
ζ2 : u = 0,
ζ3 : u2
+ v2
+ 2u = 0 − 1 ≤ u ≤ 1
2,
ζ4 : u2
+ v2
= 1 − 1
2 ≤ u ≤ 0
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = −1
¯z + 2 − 1 ≤ u ≤ −1
2
R4(z) = −1
¯z − 1
2 ≤ u ≤ 0
classification from
N.I. Koblitz, Introduction to Elliptic Curves and Modular Forms Springer Science and Busines Media, New York,
USA (1993).

sides identiﬁcations
ζ1 → ζ2,
ζ3 → ζ4

The Γ0 congruence subgroup
of (PGL(2, C))
ζ1 → ζ2
ζ3 → ζ4

σ1 : u = −1,
σ2 : u = 0,
σ3 : u2
+ v2
= 1 − 1 ≤ u ≤
1
2
,
σ4 : u2
+ v2
+ 2u = 0 −
1
2
≤ u ≤ 0
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = −1
¯z − 1 ≤ u ≤ −1
2,
R4(z) = −1
¯z + 2 − 1
2 ≤ u ≤ 0

sides identiﬁcation:
σ1 → σ2,
σ3 → σ4

Γ1(PGL(2, C)) The Γ1 congruence subgroup
of PGL(2, C)
σ1 → σ2
σ3 → σ4

ξ1 : u = −1,
ξ2 : u = 0,
ξ3 : u2
+ v2
+ u = 0,
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = − ¯z
2¯z+1

sides identiﬁcations
ξ1 → ξ2,
ξ3(−1 ≤ u ≤ −
1
2
) → ξ3(−
1
2
≤ u ≤ 0)

The Γ2 congruence subgroup
of (PGL(2, C))
ξ1 → ξ2
ξ3(−1 ≤ u ≤ −1
2 ) → ξ3(−1
2 ≤ u ≤ 0)

The Γ(2) subgroup for SL(2, Z)
̺1 : u = −1,
̺2 : u = 1,
̺3 : u2
− u + v2
= 0,
̺4 : u2
+ u + v2
= 0
transformations:
Γ1(z) = z + 2,
Γ2(z) = −
z
2z + 1
,

sides are identiﬁed as
Γ1 : ̺1 → ̺2,
Γ2 : ̺3 → ̺4,

The Γ(2) congruence subgroup
of (PGL(2, C))
̺1 → ̺2
̺3 ↔ ̺4

Γ(2) torus
ρ1 → ρ4
ρ2 ↔ ρ3

The Gutzwiller Γ(2) torus
sides identiﬁcations:
γk ↔ γk+4,
γk ↔ γk−4,
γK ↔ γK′ , K′
≡ 6.
M. C.Gutzwiller, Stochastic behavior in quantum scattering, Physica D: Nonlin. Phen., 7, pp. 341-355 (1983).

The Picard Group
transformations
P1 : z → z + 1,
P2 : z → z + i,
P3 : z → −
1
z
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
S.L. Kleiman, ”The Picard scheme”, Fundamental algebraic geometry, Math. Surveys Monogr., 123, pp. 235321,
American Mathematical Society, Providence, USA (2005); [arXiv:math/0504020].

The Picard group
The Picard group: domain

P1 : π1 → π2,
P2 : π3(−1
2 < u0) → π3(< u0 < 1
2),
π4 : (−1
2 < u0) → π4(0 < u < 1
2),
P3 : π5(−1
2 < u0) → π5(0 < u0 < 1
2)

The Picard group: sides
identiﬁcation
Sides identiﬁcation for the Picard group.
Picture from: 0305048v2

The ’symmetrized’ Picard group
transformations
P1 : z → z + 2,
P2 : z → z + 2i,
P3 : z → −
1
z
domain:
π1 : u0 = −
1
2
, −
1
2
< u1 <
1
2
,
π2 : u0 =
1
2
, −
1
2
< u1 <
1
2
,
π3 : u1 = −1
2 , −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,

The u1 ’symmetrized’ Picard
group
The ’symmetrized’ domain of the Picard group wrt u1.

The Γ0 (Pic) group
domain
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 = 1, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
π6 : u2
0 + u2
1 + (v − 1)2
= 1,

The Γ0 (Pic) group
P1 : π1 → π2,
P2 : π3(−
1
2
< u0) → π3(< u0 <
1
2
), π4 : (−
1
2
< u0) → π4(0 < u <
1
2
),
P3 : π5(−
1
2
< u0) → π5(0 < u0 <
1
2
)

The Γ0 Picard group
The Γ0 Picard group Γ0(Pic) wrt u1.

The Vinberg group
example 1; a ∈ Rn, a = (1, 0), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 = 2 cos
π
m
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
Wolf prize in mathematics Vol 2, I.I. Piatetskii’-S’apiro, Regions of the type of the upper half plane in the theory of
functions of several complex varibles, p190, Ref. [16], Selected Works pp. 487-512, Ed.’s S.-S. Chern, F.
Hirzebruch; World Scientiﬁc, New York, Usa (2001).

sides identiﬁcations possible
P1 : π1 → π2,
P3 : π5(−
1
2
< u0) → π5(0 < u0 <
1
2
)

The Vinberg group
The Vinberg group with −1
2 ≤ u0 ≤ 1
2 and 0 ≤ u1 ≤
√
2
3 .

The Vinberg group
2 ≤ u0 ≤ 1
2 and 0 ≤ u1 ≤
√
5
3 .

The Vinberg group
example 1′; a = (1, 0), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 = 2 cos
π
m
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =,
1
2
−
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,

The Vinberg group
2 ≤ u0 ≤
√
2
3 and 0 ≤ u1 ≤ 1
2 .

The Vinberg group
2 ≤ u0 ≤
√
5
3 and 0 ≤ u1 ≤ 1
2 .

sides identiﬁcation possible
P2 :π3(−1
2 < u0) → π3(< u0 < 1
2),
π4 : (−1
2 < u0) → π4(0 < u < 1
2),

The Vinberg group
example 2; a = (0, 1), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 = 2 cos
π
ω
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,

Comparison of the Picard
group and the Vinberg group
The Vinberg group −1
2 ≤ u0 ≤
√
5
3 , 0 ≤ u1 ≤ 1
2 and the Picard group.
The u0-direction positivemost sides of the Picard group are delimited by
the black (solid) arc of circumpherence nad by the dashed lines.

Historical motivations for PSL(2, Z)
PSL(2, Z)
• analyzed as conjugate to a congruence subgroup of the
modular group for natural extensions of the symbolic
dynamics;
D. Mayer, F. Stroemberg, Symbolic dynamics for the geodesic ﬂow on Hecke surfaces, Journ. of Mod.
Dyn. 2, pp. 581-627 (2008) [arXiv:0801.3951].
• Hamiltonian formulation of chaotic systems in generalized
triangles.
D. Fried, Symbolic dynamics for triangle groups, Inventiones mathematicae, 125, Issue 3, pp 487521
(1996).

Historical motivations for SL(2, Z)
Comparison of the free diffeomorphism group and Γ(2(SL(2, Z))).
C. Series, The geometry of Markoff numbers, Math. Intell., 7, pp. 2029 (1985).
The Free diffeomorphism Group on the Torus
Study the free group for vanishing Hamiltonian potential.
M. R. Bridson and K. Vogtmann, On the geometry of the automorphism group of a free group, Bull. London
Math. Soc., 27 (1995), pp. 544552. M. R. Bridson and K. Vogtmann, Homomorphisms from automorphism groups
of free groups, Bull. London Math. Soc., 35 (2003), pp. 785792. M. R. Bridson, K. Vogtmann Automorphism
groups of free groups, surface groups and free abelian groups, in Problems on mapping class groups and related
topics, Proc. Symp. Pure and Applied Math. (B. Farb, ed.) (2005).

Tessellation groups
The tessellation groups can be compared as corresponding to the
subgrouppal structures of the geodesic flow invariant under the
free diffeomorphism group.
The folding group corresponds to the folding of trajectories
(solution) to a Hamiltonian system whose potential is consistent
with a congruence subgroup of PGL(2, C) and the composition of
operators for the symbolic dynamics description.
This is equivalent to classifying the folding group for the solutions
of a Hamiltonian problem of a free particle, eventually ruled by a
infinite-wall potential.

Folding groups
The interval [0, 1] is classified according to the Gauss-Kuzmin theorem.
S. J. Miller, R. Takloo-Bighash, An Invitation to Modern Number Theory, PUP (2006).
A Hamiltonian system whose potential is consistent with the
congruence subgroup Γ2 for PGL(2, C), i.e. Γ2(2, C), identifies
geodesics (invariant also under the free diffeoemorphisms group)
specified as containing at least one point in the interval 1 < x < 1
(surds).
Γ2(2, C) Surds are classified according to the Gauss-Kuzmin
theorem.
Γ2(2, C) tori are classified according to the surds defined in the
associated Hamiltonian problem.

Motivations for the Γ2 Gutzwiller Torus
The Γ2 Gutzwiller Torus is a subgroup of SL(2, Z) (instead of
PGL(2, Z)), the generalized polygonal (non-triangular) domain allows for
a description as quotient of the plane after a coordinate identiﬁcation.
Several representations of the group domains are equivalent; among
which
G1 : γ1 → γ4,
G2 : γ2 → γ3,

A domain for the Γ2(PGL(2, Z)) torus
A domain for the Γ2 torus with sides identiﬁcations
ςj ↔ ςj+4
ςι ↔ ςι+4
ςj ↔ ςj−4
ςι ↔ ςι−4
ΣJ → ΣJ′ , J′
≡ 6.

Folding for Γ2
An example of folding for Γ2: the right-most part of the pink domain and
the left-most part of the yellow domain.

Modular folding for Γ2
Tiling for Γ2:
each quadrilateral tile delimited by ΣJ , ΣJ+1, ςj and ςι.

Γ2 tori
A Γ2 torus (on the left); and a Γ2 punctured torus (on the right).
Tori are classiﬁed according to the surds of the associated Hamiltonian
problem by Gauss-Kuzmin theorem.

Surds have the properties to uniquely deﬁne the composition of
operators in the symbolic dynamics codes.
The initial conditions uniquely
• the folding of singular geodesics tiles a punctured torus;
• the folding of non-singular geodesics tile a complete torus.

Outlook 1
Spaces equipped with measures
Algebraic modular structures are employed also for the definition of
measures for (abstract) C* algebras in abstract spaces (without
boundaries) by means of Gelfand triples and evolutionary Gelfand
triples. Feinsilver, P. J., Schott, R.: Algebraic structures and operator calculus, Kluwer (1993).
A. J. Kurdila, M. Zabarankin, Convex Functional Analysis, Birkhuser Verlag, Basel (2005). J. Wloka, Partial
Differential Equations. Cambridge University Press, Cambridge (1987); E. Zeidler, Nonlinear Functional Analysis
and Its Applications: Linear Monotone Operators, Springer, New York (1990).
R. Haag, Local quantum fields: Fields, Particles, Algebras, Springer, Heidelberg (1996).

Outlook 2
Non-modular algebraic structures: orientable manifolds From
these constructions, it is possible to describe algebraic
non-arithmetical (sub-)grouppal structures more general than the
algebraic modular structures.
From the deﬁnition of oriented (grouppal) domains, such
structures are of advantage on
• orientable manifolds;
• oriented manifolds;
• the deﬁnition of the algebraic structures for measures on such
manifolds.
Wolfgang Schwarz; Thomas Maxsein; Paul Smith An example for Gelfand’s theory of commutative Banach algebras
Mathematica Slovaca, Vol. 41 (1991), No. 3, 299–310.
I: Canguel, D. Singerman, . Normal subgroups of Hecke groups and regular maps. Mathematical Proceedings of
the Cambridge Philosophical Society, 123, pp. 59-74 (1998).

Outlook 3
Oriented manifolds: beyond the algebraic modular structures
I. Ivrissimtzis, D. Singermanb, Regular maps and principal congruence subgroups of Hecke groups, Europ. Journ.
Combinatorics, 26 , pp. 437-456 (2005).
The analysis of the subgroups allows to deﬁne (sub-)group(pal)
structures
(more general than modular structures)
for the measures for the associated (C∗) operator algebra.
Reﬂection groups on non-symmetric domains
The use of the measure for topological spaces allows for the
analysis of the corresponding structures on oriented manifolds.
M. Amini, C* Algebras of generalized Hecke pairs, Math. Slovaca 61, pp. 645652 (2011).

Discussion
T. Hsu, Identifying congruence subgroups of the modular group, April 1996Proceedings of the American
Mathematical Society 124(5).
Enumeration;
M.-L. Lang, C.-H. Lim, S.-P. Tan, An Algorithm for Determining if a Subgroup of the Modular Group is
Congruence, Journ. London Math. Soc., 51, 491502 (1995).
Existence of the (polygonal) group domain(s) and its shape;
T. Hamilton, D. Loeﬄer, Congruence testing for odd subgroups of the modular group, LMS J. Comput. Math. 17
(1) (2014) 206208 C 2014
For SL2(Z):
Wohlfahrt’s Theorem;
odd Hecke subgroups.
B. Demir, Oezden Koruog’lu, R. Sahin, On Normal Subgroups of Generalized Hecke Groups, Analele Universitatii
Constanta - Seria Matematica 24, pp. 169-184 (2016).

Outlook: non-arithmetical
subgroups
A congruence for a non-arithemtical group with (0 ≤ π/w ≤ π/4).

A congruence for a non-arithemtical group with π/4 ≤ π/w ≤ π/2.

Deﬁnition of non-arithmetical tori
For uA < −1/2, (0 ≤ π/w ≤ π/4),
u04 = 2uA(uA+1)
1+2uA
, r4 =
1+2uA+2u2
A
1+2uA
,
u05 =
2u2
A
1+2uA
, r5 ≡ r4 =
1+2uA+2u2
A
1+2uA
,
u06 = uA, r6 = 1 + uA
For uA > −1/2, π/4 ≤ π/w ≤ π/2,
u = 0,
u = −2 cos α, v = 2uuA − u2
A
according to the sides for v < vA and the goniometric circumpherence
v2
= (− 1
2uA
)2
− (u − u0)2
, u0 =
2u2
A+1
2uA
,
v2
= (− 1
2uA
)2
− (u − u′
0)2
, u′
0 = 1
2uA
and, for for v > vA, (10b) and v2
= 1 − (u − 2uA)2
.

Bibliography
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Acknowledgments
This work was partially supported by The National Scholarship
Programme of the Slovak Republic (NS’P) SAIA, and partially by
DIAEE- Sapienza University of Rome.
OML is grateful to Prof. L. Accardi for outlining the progresses in
these research directions, to Prof. V. Balek and Prof. P. Zlatos’
for discussion about the use of Hamiltonian systems, to Prof. R.
Conti for following the focuses of the calculations, and to Prof. R.
Jajcay for stressing the relevance of the geometrical description of
manifolds.
OML is grateful to Comenius University in Bratislava, Faculty of
Mathematics, Physics and Informatics, Department of Theoretical
Physics and Didactics of Physics (KTFDF), Bratislava, and to
Sapienza University of Rome- SBAI- Department for Basic
Sciences and Applications for Engineering, for warmest hospitality.
Orchidea Maria Lecian gently thanks Prof. M. Testa, Prof. R.
Ruﬃni and Prof. G. Immirzi for kindly reading the seminar slides.

Folding, tiling and tori: a Hamiltonian analysis, O.M. Lecian, 17 January 2018

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