text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 59 of the relations on the gluing parameters only give toric singularities. That is, all R 2,1 corners are polyhedral. The moduli space is shown in Figure 9. R Figure 9. 2,1 The moduli space of quilted disks with boundary and seam markings admits natural forgetful maps forgetting both boundary and seam markings generalizing ∈ { } ∈ { } ≥ those in (24): For indices i 1, . . . , d and j 1, . . . , e maps (for d, e 1) R → R − R → R − (55) f : d,e d 1,e, fj : d,e d,e 1 i given by forgetting the i-th boundary resp. j-th seam marking and collapsing unstable components. As in (25), the forgetful morphisms f may be identiﬁed with i projection from part of the boundary of the universal curve while the morphism fj may be identiﬁed with the part of the universal seam between two markings, with the additional caveat that each nodal point on the seam is potentially replaced with two points as in Figure 10 (so the map to the universal seam is two-to-one in this case.) We leave it to the reader to check that these maps are continuous with respect R d,e to the topology on constructed above and (using the gluing construction) a topological ﬁbration with interval ﬁbers. By induction and the fact that the moduli R R R 1,0 d,e d,e space is a point, is a topological disk. In particular, is compact and Hausdorﬀ. Proposition 6.7. For any positive integers d, e, there exists an isomorphism of R K decomposed spaces from d,e to d,e. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: of

text: the

text: relations

text: on

text: the

text: gluing

text: parameters

text: only

text: give

text: toric

text: singularities.

text: That

text: is,

text: all

text: R

text: corners

text: are

text: polyhedral.

text: The

text: moduli

text: space

text: is

text: shown

text: in

text: Figure

text: R

text: Figure

text: The

text: moduli

text: space

text: of

text: quilted

text: disks

text: with

text: boundary

text: and

text: seam

text: markings

text: admits

text: natural

text: forgetful

text: maps

text: forgetting

text: both

text: boundary

text: and

text: seam

text: markings

text: generalizing

text: those

text: in

text: For

text: indices

text: i

text: d

text: and

text: j

text: e

text: maps

text: (for

text: d,

text: e

text: R

text: R

text: R

text: R

text: f

text: d,e

text: d

text: fj

text: d,e

text: d,e

text: i

text: given

text: by

text: forgetting

text: the

text: i-th

text: boundary

text: resp.

text: j-th

text: seam

text: marking

text: and

text: collapsing

text: unstable

text: components.

text: As

text: in

text: the

text: forgetful

text: morphisms

text: f

text: may

text: be

text: identiﬁed

text: with

text: i

text: projection

text: from

text: part

text: of

text: the

text: boundary

text: of

text: the

text: universal

text: curve

text: while

text: the

text: morphism

text: fj

text: may

text: be

text: identiﬁed

text: with

text: the

text: part

text: of

text: the

text: universal

text: seam

text: between

text: two

text: markings,

text: with

text: the

text: additional

text: caveat

text: that

text: each

text: nodal

text: point

text: on

text: the

text: seam

text: is

text: potentially

text: replaced

text: with

text: two

text: points

text: as

text: in

text: Figure

text: (so

text: the

text: map

text: to

text: the

text: universal

text: seam

text: is

text: two-to-one

text: in

text: this

text: case.)

text: We

text: leave

text: it

text: to

text: the

text: reader

text: to

text: check

text: that

text: these

text: maps

text: are

text: continuous

text: with

text: respect

text: R

text: d,e

text: to

text: the

text: topology

text: on

text: constructed

text: above

text: and

text: (using

text: the

text: gluing

text: construction)

text: a

text: topological

text: ﬁbration

text: with

text: interval

text: ﬁbers.

text: By

text: induction

text: and

text: the

text: fact

text: that

text: the

text: moduli

text: R

text: R

text: R

text: d,e

text: d,e

text: space

text: is

text: a

text: point,

text: is

text: a

text: topological

text: disk.

text: In

text: particular,

text: is

text: compact

text: and

text: Hausdorﬀ.

text: Proposition

text: For

text: any

text: positive

text: integers

text: d,

text: e,

text: there

text: exists

text: an

text: isomorphism

text: of

text: R

text: K

text: decomposed

text: spaces

text: from

text: d,e

text: to

text: d,e.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 60 S. MA’U, K. WEHRHEIM, AND C. WOODWARD z 1 w4 w2 w5 w1 z 0 Figure 10. The part of the seam between two markings ∈ { } Proof. In the base cases (d, e) (1, 0), (0, 1) both moduli spaces are points and so K the result holds. By the combinatorial description of d,e and induction, it suﬃces R d,e to show that the compactiﬁed moduli space is the cone on its boundary. But R d,e since is a topological disk, it is homeomorphic to the cone on its boundary, a (cid:3) sphere. In order to deﬁne the natural transformations, we count quilted surfaces with strip-like ends for a family constructed from the space of quilted disks with interior and exterior markings constructed in the previous section. For analytical reasons, this requires replacing the family of quilted disks in the previous section with one in which degeneration is given by neck-stretching: Proposition 6.8. (Existence of families of quilted surfaces with strip-like ends over the bimultiplihedra) For any positive integers d, e, there exists a collection of S R d,e d,e families of quilted surfaces with strip-like ends over , with the property S R that the restriction d,e of the family to a stratum d,e isomorphic to a product Γ Γ of multiplihedra, biassociahedra, and associahedra is a product of the corresponding families of surfaces and quilted surfaces with strip-like ends, and collar neighborhoods S of d,e are given by gluing along strip-like ends. Γ Proof. By induction using Theorem 1.3, starting from the case of three-marked disk where we choose a genus zero surface with strip-like ends, using the already (cid:3) constructed families of surfaces with strip-like ends in Proposition 4.7. In this section we construct pseudoholomorphic quilts with interior seam condi- tions on various Lagrangian correspondences. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: z

text: z

text: Figure

text: The

label: caption

text: part

text: of

text: the

text: seam

text: between

text: two

text: markings

text: Proof.

text: In

text: the

text: base

text: cases

text: (d,

text: e)

text: both

text: moduli

text: spaces

text: are

text: points

text: and

text: so

text: K

text: the

text: result

text: holds.

text: By

text: the

text: combinatorial

text: description

text: of

text: d,e

text: and

text: induction,

text: it

text: suﬃces

text: R

text: d,e

text: to

text: show

text: that

text: the

text: compactiﬁed

text: moduli

text: space

text: is

text: the

text: cone

text: on

text: its

text: boundary.

text: But

text: R

text: d,e

text: since

text: is

text: a

text: topological

text: disk,

text: it

text: is

text: homeomorphic

text: to

text: the

text: cone

text: on

text: its

text: boundary,

text: a

text: sphere.

text: In

text: order

text: to

text: deﬁne

text: the

text: natural

text: transformations,

text: we

text: count

text: quilted

text: surfaces

text: with

text: strip-like

text: ends

text: for

text: a

text: family

text: constructed

text: from

text: the

text: space

text: of

text: quilted

text: disks

text: with

text: interior

text: and

text: exterior

text: markings

text: constructed

text: in

text: the

text: previous

text: section.

text: For

text: analytical

text: reasons,

text: this

text: requires

text: replacing

text: the

text: family

text: of

text: quilted

text: disks

text: in

text: the

text: previous

text: section

text: with

text: one

text: in

text: which

text: degeneration

text: is

text: given

text: by

text: neck-stretching:

text: Proposition

text: (Existence

text: of

text: families

text: of

text: quilted

text: surfaces

text: with

text: strip-like

text: ends

text: over

text: the

text: bimultiplihedra)

text: For

text: any

text: positive

text: integers

text: d,

text: e,

text: there

text: exists

text: a

text: collection

text: of

text: S

text: R

text: d,e

text: d,e

text: families

text: of

text: quilted

text: surfaces

text: with

text: strip-like

text: ends

text: over

text: with

text: the

text: property

text: S

text: R

text: that

text: the

text: restriction

text: d,e

text: of

text: the

text: family

text: to

text: a

text: stratum

text: d,e

text: isomorphic

text: to

text: a

text: product

text: Γ

text: Γ

text: of

text: multiplihedra,

text: biassociahedra,

text: and

text: associahedra

text: is

text: a

text: product

text: of

text: the

text: corresponding

text: families

text: of

text: surfaces

text: and

text: quilted

text: surfaces

text: with

text: strip-like

text: ends,

text: and

text: collar

text: neighborhoods

text: S

text: of

text: d,e

text: are

text: given

text: by

text: gluing

text: along

text: strip-like

text: ends.

text: Γ

text: Proof.

text: By

text: induction

text: using

text: Theorem

text: starting

text: from

text: the

text: case

text: of

text: three-marked

text: disk

text: where

text: we

text: choose

text: a

text: genus

text: zero

text: surface

text: with

text: strip-like

text: ends,

text: using

text: the

text: already

text: constructed

text: families

text: of

text: surfaces

text: with

text: strip-like

text: ends

text: in

text: Proposition

text: In

text: this

text: section

text: we

text: construct

text: pseudoholomorphic

text: quilts

text: with

text: interior

text: seam

text: condi-

text: tions

text: on

text: various

text: Lagrangian

text: correspondences.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 62 S. MA’U, K. WEHRHEIM, AND C. WOODWARD Proof. We show the A∞ functor axiom (78). The axiom follows from a signed count of the boundary components of the one-dimensional component of the moduli space M d,e in Proposition 6.9. The boundary components are of three combinatorial types: (a) (Quilted sphere bubbles) Facets where some subset of the markings w , . . . , w 1 e on the interior circle have bubbled oﬀ onto a quilted sphere with values in M , M ; 0 1 (b) (Quilted disk bubbles) Facets corresponding to partitions of the interior and exterior markings, corresponding to quilted disk bubbles; (c) (Disk bubbles) Facets where some subset of the markings z , . . . , z on the 1 d boundary have bubbled oﬀ onto a quilted sphere with values in M ; 0 (d) (Trajectory bubbling) Bubbling oﬀ trajectories at the interior or exterior markings. See Figure 11 for the case of two interior markings; the facet on the left represents the lim(cid:88)it when the two interior marked points come together. Counting boundaries of the ﬁrst type gives an expression ±T − (56) e j+1(α , . . . , α , 1 i i,j (cid:104) (cid:105) (cid:104) (cid:105) µj −× (αi+1, . . . , αi+j), αi+j+1, . . . , αe)( x1 , . . . , xd ). (cid:88) Fuk(M0 M1) The second type of boundary component contributes a sum of terms of the form ± T (57) µmFuk (M1)(Φ(L001))i1(. . .) . . . (Φ(L001))ij1−1(. . .) ij1,k1(. . .) T (Φ(L101))ij1+1(. . .) . . . (Φ(L101)ij2−1(. . .) ij2,k2(. . .) T . . . ijs,ks(. . .)Φ(Le )ijs+1(. . .) . . . (Φ(Le )im(. . .)) 01 01 (cid:104) (cid:105) where each . . . is an expression in the α and x ’s, r is the number of bubbles, s is i j the number of bubbles with interior markings, i represents the number of exterior l markings on the bubbles, k represents the number of interior markings on bubbles l with interior markings, and j , . . . , j are the indices of the bubbles with interior 1 s markings. The third and fourth type of boundary are similar to those considered (cid:81) before and will not be discussed. Combining this with (82) and (81) proves the A∞ fun(cid:80)ctor axiom up(cid:80)to sign. It remains to check the signs. The sig(cid:80)n for the R × R → R inclusion of a facet equal to the image of an embedding di,ei f,0 d,e i|(cid:80)s (|−1) mi=1(di−1+ei)i+ i<j(di−1)(ei). The degree of T j(α1, . . .i , αj) is 1 − ji=1(1 − α ). The signs appearing in (82) and in the higher compositions are given by sums i | | − | | − ( xj 1)( yl 1) where xj, yl are intersection points corresponding to the i<k,j,l i k i k inner and outer markings respectively on on the i-th resp. k-th bubble. The terms | || | of the form xj yl are accounted for by Koszul signs. Combining with the two i♥ k (cid:3) occurrences of , this gives the signs claimed in (78). ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: Proof.

text: We

text: show

text: the

text: A∞

text: functor

text: axiom

text: The

text: axiom

text: follows

text: from

text: a

text: signed

text: count

text: of

text: the

text: boundary

text: components

text: of

text: the

text: one-dimensional

text: component

text: of

text: the

text: moduli

text: space

text: M

text: d,e

text: in

text: Proposition

text: The

text: boundary

text: components

text: are

text: of

text: three

text: combinatorial

text: types:

text: (a)

label: list

text: (Quilted

text: sphere

text: bubbles)

text: Facets

text: where

text: some

text: subset

text: of

text: the

text: markings

text: w

text: w

text: e

text: on

text: the

text: interior

text: circle

text: have

text: bubbled

text: oﬀ

text: onto

text: a

text: quilted

text: sphere

text: with

text: values

text: in

text: M

text: M

text: (b)

text: (Quilted

text: disk

text: bubbles)

text: Facets

text: corresponding

text: to

text: partitions

text: of

text: the

text: interior

text: and

text: exterior

text: markings,

text: corresponding

text: to

text: quilted

text: disk

text: bubbles;

text: (c)

text: (Disk

text: bubbles)

text: Facets

text: where

text: some

text: subset

text: of

text: the

text: markings

text: z

text: z

text: on

text: the

text: d

text: boundary

text: have

text: bubbled

text: oﬀ

text: onto

text: a

text: quilted

text: sphere

text: with

text: values

text: in

text: M

text: (d)

text: (Trajectory

text: bubbling)

text: Bubbling

text: oﬀ

text: trajectories

text: at

text: the

text: interior

text: or

text: exterior

text: markings.

text: See

text: Figure

text: for

text: the

text: case

text: of

text: two

text: interior

text: markings;

text: the

text: facet

text: on

text: the

text: left

text: represents

text: the

text: when

text: the

text: two

text: interior

text: marked

text: points

text: come

text: together.

text: Counting

text: boundaries

text: of

text: the

text: ﬁrst

text: type

text: gives

text: an

text: expression

text: ±T

text: e

text: α

text: i

text: i,j

text: µj

text: αi+j),

text: αe)(

text: xd

text: The

text: second

text: type

text: of

text: boundary

text: component

text: contributes

text: a

text: sum

text: of

text: terms

text: of

text: the

text: form

text: T

text: µmFuk

text: T

text: T

text: ijs,ks(.

text: .)Φ(Le

text: (Φ(Le

text: )im(.

text: where

text: each

text: is

text: an

text: expression

text: in

text: the

text: α

text: and

text: x

text: ’s,

text: r

text: is

text: the

text: number

text: of

text: bubbles,

text: s

text: is

text: i

text: j

text: the

text: number

text: of

text: bubbles

text: with

text: interior

text: markings,

text: i

text: represents

text: the

text: number

text: of

text: exterior

text: l

text: markings

text: on

text: the

text: bubbles,

text: k

text: represents

text: the

text: number

text: of

text: interior

text: markings

text: on

text: bubbles

text: l

text: with

text: interior

text: markings,

text: and

text: j

text: j

text: are

text: the

text: indices

text: of

text: the

text: bubbles

text: with

text: interior

text: s

text: markings.

text: The

text: third

text: and

text: fourth

text: type

text: of

text: boundary

text: are

text: similar

text: to

text: those

text: considered

text: before

text: and

text: will

text: not

text: be

text: discussed.

text: Combining

text: this

text: with

text: and

text: proves

text: the

text: A∞

text: axiom

text: sign.

text: It

text: remains

text: to

text: check

text: the

text: signs.

text: The

text: for

text: the

text: R

text: R

text: R

text: inclusion

text: of

text: a

text: facet

text: equal

text: to

text: the

text: image

text: of

text: an

text: embedding

text: di,ei

text: d,e

text: The

text: degree

text: of

text: T

text: .i

text: αj)

text: is

text: α

text: The

text: signs

text: appearing

text: in

text: and

text: in

text: the

text: higher

text: compositions

text: are

text: given

text: by

text: sums

text: i

text: xj

text: yl

text: where

text: xj,

text: yl

text: are

text: intersection

text: points

text: corresponding

text: to

text: the

text: i<k,j,l

text: i

text: k

text: i

text: k

text: inner

text: and

text: outer

text: markings

text: respectively

text: on

text: on

text: the

text: i-th

text: resp.

text: k-th

text: bubble.

text: The

text: terms

text: of

text: the

text: form

text: xj

text: yl

text: are

text: accounted

text: for

text: by

text: Koszul

text: signs.

text: Combining

text: with

text: the

text: two

text: i♥

text: k

text: occurrences

text: of

text: this

text: gives

text: the

text: signs

text: claimed

text: in

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 64 S. MA’U, K. WEHRHEIM, AND C. WOODWARD Theorem 6.14. (Independence of the functors up to quasiisomorphism) Let M , M 0 1 be symplectic backgrounds with the same monotonicity constant, and let L be an 01 object in Fuk (M , M ). The functor Φ(L ) constructed in Section is indepen- 0 1 01 dent up to quasiisomorphism of all choices (the choice of family of quilts, that is, S → R d,e d,e holomorphic structures on the ﬁbers of , and the perturbation data.) (cid:48) Proof. Suppose two such choices are given, with corresponding functors Φ(L ), Φ (L ). ∈ 01 01 The Floer cocycle α CF (L , L ) corresponding to the identity in HF (L , L ) 01 01 (cid:48) 01 01 deﬁnes a natural transformation β from Φ(L ) to Φ (L ). Its transpose deﬁnes a (cid:48) 01 01 natural transformation from Φ (L ) to Φ(L ). The composition of the two natural 01 01 transformations is given by the product of α with β under the composition map µ2 in Fuk (M , M ). By Theorem 1.1, the composition of natural transformations is 0 1 (cid:3) the identity transformation. Proposition 6.15. (Functor for the diagonal correspondence) Let M be a compact ⊂ − × monotone symplectic background. Suppose that M is spin and ∆ M M is the diagonal equipped with the relative spin structure corresponding to a spin structure on M. Then Φ(∆) is quasiisomorphic to the identity functor from Fuk (M) to Fuk (M). ∅ Proof. The correspondence ∆ is quasiisomorphic to in Fuk (M, M), with isomor- phism given by the cohomological unit in CF (∆). By Proposition 5.15 and Theorem (cid:3) 1.1 Φ(∆) is quasiisomorphic to the identity functor in Func(Fuk (M), Fuk (M)). 7. Algebraic and geometric composition In this section we study the composition of A∞ functors for Lagrangian correspon- dences. In particular, we prove Theorem 1.2 on the homotopy equivalence of the A∞ functor for a geometric composition and the A∞ composition of the corresponding A∞ functors. The proof of the composition result 1.2 depends on generalizations of the mul- ≥ K tiplihedra which feature multiple interior circles. For d 1, let d,0,0 denote the cell complex whose cells correspond to parenthesized expressions in formal variables a , . . . , a and operations h , h with h always following h . More precisely, these 1 d 1 2 2 1 expressions correspond to bicolored trees which are trees equipped with two types of colored vertices: Deﬁnition 7.1. (Bicolored trees) A bicolored, rooted tree with d leaves consists of data (Edge(Γ), Vert(Γ), Edge∞(Γ), Vert(1)(Γ), Vert(2)(Γ)) where: ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: Theorem

text: (Independence

text: of

text: the

text: functors

text: up

text: to

text: quasiisomorphism)

text: Let

text: M

text: M

text: be

text: symplectic

text: backgrounds

text: with

text: the

text: same

text: monotonicity

text: constant,

text: and

text: let

text: L

text: be

text: an

text: object

text: in

text: Fuk

text: (M

text: M

text: The

text: functor

text: Φ(L

text: constructed

text: in

text: Section

text: is

text: indepen-

text: dent

text: up

text: to

text: quasiisomorphism

text: of

text: all

text: choices

text: (the

text: choice

text: of

text: family

text: of

text: quilts,

text: that

text: is,

text: S

text: R

text: d,e

text: d,e

text: holomorphic

text: structures

text: on

text: the

text: ﬁbers

text: of

text: and

text: the

text: perturbation

text: data.)

text: Proof.

text: Suppose

text: two

text: such

text: choices

text: are

text: given,

text: with

text: corresponding

text: functors

text: Φ(L

text: Φ

text: (L

text: The

text: Floer

text: cocycle

text: α

text: CF

text: (L

text: L

text: corresponding

text: to

text: the

text: identity

text: in

text: HF

text: (L

text: L

text: deﬁnes

text: a

text: natural

text: transformation

text: β

text: from

text: Φ(L

text: to

text: Φ

text: (L

text: Its

text: transpose

text: deﬁnes

text: a

text: natural

text: transformation

text: from

text: Φ

text: (L

text: to

text: Φ(L

text: The

text: composition

text: of

text: the

text: two

text: natural

text: transformations

text: is

text: given

text: by

text: the

text: product

text: of

text: α

text: with

text: β

text: under

text: the

text: composition

text: map

text: µ2

text: in

text: Fuk

text: (M

text: M

text: By

text: Theorem

text: the

text: composition

text: of

text: natural

text: transformations

text: is

text: the

text: identity

text: transformation.

text: Proposition

text: (Functor

text: for

text: the

text: diagonal

text: correspondence)

text: Let

text: M

text: be

text: a

text: compact

text: monotone

text: symplectic

text: background.

text: Suppose

text: that

text: M

text: is

text: spin

text: and

text: M

text: M

text: is

text: the

text: diagonal

text: equipped

text: with

text: the

text: relative

text: spin

text: structure

text: corresponding

text: to

text: a

text: spin

text: structure

text: on

text: M.

text: Then

text: Φ(∆)

text: is

text: quasiisomorphic

text: to

text: the

text: identity

text: functor

text: from

text: Fuk

text: (M)

text: to

text: Fuk

text: (M).

text: Proof.

text: The

text: correspondence

text: is

text: quasiisomorphic

text: to

text: in

text: Fuk

text: (M,

text: M),

text: with

text: isomor-

text: phism

text: given

text: by

text: the

text: cohomological

text: unit

text: in

text: CF

text: By

text: Proposition

text: and

text: Theorem

text: Φ(∆)

text: is

text: quasiisomorphic

text: to

text: the

text: identity

text: functor

text: in

text: Func(Fuk

text: (M),

text: Fuk

text: (M)).

text: Algebraic

text: and

text: geometric

text: composition

text: In

text: this

text: section

text: we

text: study

text: the

text: composition

text: of

text: A∞

text: functors

text: for

text: Lagrangian

text: correspon-

text: dences.

text: In

text: particular,

text: we

text: prove

text: Theorem

text: on

text: the

text: homotopy

text: equivalence

text: of

text: the

text: A∞

text: functor

text: for

text: a

text: geometric

text: composition

text: and

text: the

text: A∞

text: composition

text: of

text: the

text: corresponding

text: A∞

text: functors.

text: The

text: proof

text: of

text: the

text: composition

text: result

text: depends

text: on

text: generalizations

text: of

text: the

text: mul-

text: K

text: tiplihedra

text: which

text: feature

text: multiple

text: interior

text: circles.

text: For

text: d

text: let

text: denote

text: the

text: cell

text: complex

text: whose

text: cells

text: correspond

text: to

text: parenthesized

text: expressions

text: in

text: formal

text: variables

text: a

text: a

text: and

text: operations

text: h

text: h

text: with

text: h

text: always

text: following

text: h

text: More

text: precisely,

text: these

text: d

text: expressions

text: correspond

text: to

text: bicolored

text: trees

text: which

text: are

text: trees

text: equipped

text: with

text: two

text: types

text: of

text: colored

text: vertices:

text: Deﬁnition

text: (Bicolored

text: trees)

text: A

text: bicolored,

text: rooted

text: tree

text: with

text: d

text: leaves

text: consists

text: of

text: data

text: (Edge(Γ),

text: Vert(Γ),

text: Edge∞(Γ),

text: where:

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 67 ∈ (b) exactly two inner circles appear in the component D corresponding to v ∩ v Vert(Γ) if and only if v lies in (Vert(1)(Γ) Vert(2)(Γ)); ∈ (c) for any biquilted disk component D , v Vert(2)(Γ) the ratio v ∈ ∞ radius(C )/ radius(C ) [1, ) 2 1 of radii of the two inner circles C , C is independent of the choice of com- 1 2 ponent and satisfying the root marking property that the root marking z is, if it lies on 0 a quilted component, the intersection of the inner circle(s) with the boundary and the monotonicity property that on any non-self-crossing path of components from the root marking z to the component containing a marking z , there is exactly one 0 i component with an inner circle and one component with an outer circle. A nodal biquilted disk is stable if the corresponding bicolored tree is stable, that is, each disk with at least one resp. no interior circles has at least two resp. three special points. R d,0,0 Denote by the set of isomorphism classes of stable nodal biquilted disks. The topology on the moduli space of stable nodal biquilted disks is deﬁned in a similar way as that for the moduli space of stable nodal quilted disks with interior and seam markings in Section . A sequence of biquilted marked disks is said to converge to a stable limit if, for each component of the limit, there exist a family of reparametrizations satisfying certain properties that we will not detail here. The R R 1,0,0 2,0,0 moduli space is a topological interval while the moduli space is shown in Figure 13. R Figure 13. Vertices of 2,0,0 The local structure of the moduli space of stable nodal biquilted disks may be described in terms of gluing parameters taking values in the positive real part of a toric variety. Deﬁnition 7.4. (Balanced gluing parameters) Let Γ = (Edge ∞(Γ), Vert(Γ), Edge∞(Γ), Vert(1)(Γ), Vert(2)(Γ)) < ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: (b)

text: exactly

text: two

text: inner

text: circles

text: appear

text: in

text: the

text: component

text: D

text: corresponding

text: to

text: v

text: v

text: Vert(Γ)

text: if

text: and

text: only

text: if

text: v

text: lies

text: in

text: (c)

text: for

text: any

text: biquilted

text: disk

text: component

text: D

text: v

text: the

text: ratio

text: v

text: radius(C

text: radius(C

text: of

text: radii

text: of

text: the

text: two

text: inner

text: circles

text: C

text: C

text: is

text: independent

text: of

text: the

text: choice

text: of

text: com-

text: ponent

text: and

text: satisfying

text: the

text: root

text: marking

text: property

text: that

text: the

text: root

text: marking

text: z

text: is,

text: if

text: it

text: lies

text: on

text: a

text: quilted

text: component,

text: the

text: intersection

text: of

text: the

text: inner

text: circle(s)

text: with

text: the

text: boundary

text: and

text: the

text: monotonicity

text: property

text: that

text: on

text: any

text: non-self-crossing

text: path

text: of

text: components

text: from

text: the

text: root

text: marking

text: z

text: to

text: the

text: component

text: containing

text: a

text: marking

text: z

text: there

text: is

text: exactly

text: one

text: i

text: component

text: with

text: an

text: inner

text: circle

text: and

text: one

text: component

text: with

text: an

text: outer

text: circle.

text: A

text: nodal

text: biquilted

text: disk

text: is

text: stable

text: if

text: the

text: corresponding

text: bicolored

text: tree

text: is

text: stable,

text: that

text: is,

text: each

text: disk

text: with

text: at

text: least

text: one

text: resp.

text: no

text: interior

text: circles

text: has

text: at

text: least

text: two

text: resp.

text: three

text: special

text: points.

text: R

text: Denote

text: by

text: the

text: set

text: of

text: isomorphism

text: classes

text: of

text: stable

text: nodal

text: biquilted

text: disks.

text: The

text: topology

text: on

text: the

text: moduli

text: space

text: of

text: stable

text: nodal

text: biquilted

text: disks

text: is

text: deﬁned

text: in

text: a

text: similar

text: way

text: as

text: that

text: for

text: the

text: moduli

text: space

text: of

text: stable

text: nodal

text: quilted

text: disks

text: with

text: interior

text: and

text: seam

text: markings

text: in

text: Section

text: A

text: sequence

text: of

text: biquilted

text: marked

text: disks

text: is

text: said

text: to

text: converge

text: to

text: a

text: stable

text: limit

text: if,

text: for

text: each

text: component

text: of

text: the

text: limit,

text: there

text: exist

text: a

text: family

text: of

text: reparametrizations

text: satisfying

text: certain

text: properties

text: that

text: we

text: will

text: not

text: detail

text: here.

text: The

text: R

text: R

text: moduli

text: space

text: is

text: a

text: topological

text: interval

text: while

text: the

text: moduli

text: space

text: is

text: shown

text: in

text: Figure

text: R

text: Figure

text: Vertices

text: of

text: The

text: local

text: structure

text: of

text: the

text: moduli

text: space

text: of

text: stable

text: nodal

text: biquilted

text: disks

text: may

text: be

text: described

text: in

text: terms

text: of

text: gluing

text: parameters

text: taking

text: values

text: in

text: the

text: positive

text: real

text: part

text: of

text: a

text: toric

text: variety.

text: Deﬁnition

text: (Balanced

text: gluing

text: parameters)

text: Let

text: Γ

text: (Edge

text: ∞(Γ),

text: Vert(Γ),

text: Edge∞(Γ),

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 71 R Figure 14. Facets of 2,0,0 (cid:89) spaces for the vertices, R R pre = . Γ v ∈ v Vert(Γ) Let R → ∞ (cid:55)→ ρ : pre [0, ]k, r := (r ) ∈ ρ (r) := (ρ(r )) ∈ ∩ Γ Γ v v Vert(Γ) Γ v v Vert(1)(Γ) Vert(2)(Γ) be the map derived from the ratios of the radii of the circles for each biquilted ∞ component as in (58), if there are biquilted components, or taking the value , if there are no biquilted components. Figure 15. A free biquilted disk In terms of the map ρ deﬁned in Deﬁnition 7.10 we have Γ R − (60) = ρ 1(∆) Γ Γ { | ∈ R} ⊂ R where ∆ = (x, . . . , x) x k is the thin diagonal. The combinatorial type of a free biquilted disk is a free bicolored tree which is deﬁned in the same way as (cid:48) Deﬁnition 7.1, but without item (bii). If Γ is a combinatorial type of free biquilted ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: R

text: Figure

text: Facets

text: of

text: spaces

text: for

text: the

text: vertices,

text: R

text: R

text: pre

text: Γ

text: v

text: v

text: Vert(Γ)

text: Let

text: R

text: ρ

text: pre

text: ]k,

text: r

text: (r

text: ρ

text: (r)

text: (ρ(r

text: Γ

text: Γ

text: v

text: v

text: Vert(Γ)

text: Γ

text: v

text: v

text: be

text: the

text: map

text: derived

text: from

text: the

text: ratios

text: of

text: the

text: radii

text: of

text: the

text: circles

text: for

text: each

text: biquilted

text: component

text: as

text: in

text: if

text: there

text: are

text: biquilted

text: components,

text: or

text: taking

text: the

text: value

text: if

text: there

text: are

text: no

text: biquilted

text: components.

text: Figure

text: A

text: free

text: biquilted

text: disk

text: In

text: terms

text: of

text: the

text: map

text: ρ

text: deﬁned

text: in

text: Deﬁnition

text: we

text: have

text: Γ

text: R

text: ρ

text: Γ

text: Γ

text: R}

text: R

text: where

text: (x,

text: x)

text: x

text: k

text: is

text: the

text: thin

text: diagonal.

text: The

text: combinatorial

text: type

text: of

text: a

text: free

text: biquilted

text: disk

text: is

text: a

text: free

text: bicolored

text: tree

text: which

text: is

text: deﬁned

text: in

text: the

text: same

text: way

text: as

text: Deﬁnition

text: but

text: without

text: item

text: (bii).

text: If

text: Γ

text: is

text: a

text: combinatorial

text: type

text: of

text: free

text: biquilted

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 84 S. MA’U, K. WEHRHEIM, AND C. WOODWARD T is well-deﬁned. Furthermore the “diﬀerential” of is the limit T T ≤ (µ1 F F )d = lim (µ1 F F d0)d. Hom( 1, 2) d0→∞ Hom( 1, 2) T T ◦ So = ( d) ≥ is a homotopy from Φ(L ) Φ(L ) to Φ(L L ). d 0 |1T2| |H|01 − 01 ζ 12 It remains to check the signs. Since = = 1, the signs in the formula R d,0,0 (83) vanish. Consider the signs of the inclusions of strata into : An embedding R × R → R f e,0,0 d,0,0 corresponding to an unquilted bubble containin(cid:80)g the markings − i + 1, . . . , i + f has sign ( 1)if+i, cf. (32). For the facets induced by embeddings R × × R × R → R − − − ( i1,0 . . . im,0) e,0 d,0,0 gluing acts on signs by 1 + ( 1) mj=1(m j)(ij 1) R × × c.f. Lemma 5.9. For the facets induced by embeddings ( i1,0,0 ∞ . . . ∞ (cid:80) [0, ] [0, ] R × R → R im,0,0) m d,0,0 (where the real number is the ratio of the radii of the two − − interior circles) the gluing map has sign 1 + ( 1) mj=1(m j)ij. For the facet given by R → R the embedding d,0 d,0,0 the gluing map is orientation preserving. The signs for the embeddings of the facets combine with the Koszul signs to give the signs in (cid:3) the formulas (83) and (81). In this section we prove Theorem 1.2 relating the composition of functors with the functor for the composition of correspondences. First we show that the A∞ functors are quasi-isomorphic. Let L , L be Lagrangian correspondences as above with the 01 12 ◦ property that the geometric composition L := L L is smooth and embedded − × 02 01 12 by projection into M M as in (1). 0 2 Proposition 7.21. (Algebraic versus geometric composition) Let M , M , M be ⊂0 1− ×2 symplectic backgrounds with the same monotonicity constant and L M M ⊂ − × 01 0 1 and L M M admissible Lagrangian correspondences with brane structure. 12 1 2 ◦ −× Suppose that L = L L is transverse and embedded, and admissible in M M . 02 01 12 0 2 Then for any ζ > 0 the functors Φ(L L ) and Φ(L ) are quasi-isomorphic in 01 ζ 12 02 Func(Fuk (M ), Fuk (M )). 0 2 ◦ Proof. Theorem 7.20 shows that Φ(L ) Φ(L ), Φ(L L ) are homotopic, in par- 12 01 01 ζ 12 ticular, quasiisomorphic. To show that Φ(L L ), Φ(L ) are quasi-isomorphic re- 01 ζ 12 02 ∈ call that in [42] the second two authors constructed cocycles φ CF (L L , L ) ∈ 01 ζ 12 02 and ψ CF (L , L L ) with the property that 02 01 ζ 12 (75) ◦ ∈ ◦ ∈ [ψ] [φ] = 1 HF (L , L ), [ψ] [φ] = 1 HF (L L , L L ). L02 02 02 L01 ζL12 01 12 01 ζ 12 An alternative argument for the existence of the cocycle is given in Lekili-Lipyanskiy [18]. Let T (φ), T (ψ) denote the corresponding natural natural transformations. It ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: T

text: is

text: well-deﬁned.

text: Furthermore

text: the

text: “diﬀerential”

text: of

text: is

text: the

text: limit

text: T

text: T

text: F

text: F

text: )d

text: lim

text: F

text: F

text: Hom(

text: d0→∞

text: Hom(

text: T

text: T

text: So

text: d)

text: is

text: a

text: homotopy

text: from

text: Φ(L

text: Φ(L

text: to

text: Φ(L

text: L

text: d

text: ζ

text: It

text: remains

text: to

text: check

text: the

text: signs.

text: Since

text: the

text: signs

text: in

text: the

text: formula

text: R

text: vanish.

text: Consider

text: the

text: signs

text: of

text: the

text: inclusions

text: of

text: strata

text: into

text: An

text: embedding

text: R

text: R

text: R

text: f

text: corresponding

text: to

text: an

text: unquilted

text: bubble

text: the

text: markings

text: i

text: i

text: f

text: has

text: sign

text: cf.

text: For

text: the

text: facets

text: induced

text: by

text: embeddings

text: R

text: R

text: R

text: R

text: gluing

text: acts

text: on

text: signs

text: by

text: j)(ij

text: R

text: c.f.

text: Lemma

text: For

text: the

text: facets

text: induced

text: by

text: embeddings

text: R

text: R

text: R

text: m

text: (where

text: the

text: real

text: number

text: is

text: the

text: ratio

text: of

text: the

text: radii

text: of

text: the

text: two

text: interior

text: circles)

text: the

text: gluing

text: map

text: has

text: sign

text: j)ij.

text: For

text: the

text: facet

text: given

text: by

text: R

text: R

text: the

text: embedding

text: the

text: gluing

text: map

text: is

text: orientation

text: preserving.

text: The

text: signs

text: for

text: the

text: embeddings

text: of

text: the

text: facets

text: combine

text: with

text: the

text: Koszul

text: signs

text: to

text: give

text: the

text: signs

text: in

text: the

text: formulas

text: and

text: In

text: this

text: section

text: we

text: prove

text: Theorem

text: relating

text: the

text: composition

text: of

text: functors

text: with

text: the

text: functor

text: for

text: the

text: composition

text: of

text: correspondences.

text: First

text: we

text: show

text: that

text: the

text: A∞

text: functors

text: are

text: quasi-isomorphic.

text: Let

text: L

text: L

text: be

text: Lagrangian

text: correspondences

text: as

text: above

text: with

text: the

text: property

text: that

text: the

text: geometric

text: composition

text: L

text: L

text: L

text: is

text: smooth

text: and

text: embedded

text: by

text: projection

text: into

text: M

text: M

text: as

text: in

text: Proposition

text: (Algebraic

text: versus

text: geometric

text: composition)

text: Let

text: M

text: M

text: M

text: be

text: ⊂0

text: 1−

text: ×2

text: symplectic

text: backgrounds

text: with

text: the

text: same

text: monotonicity

text: constant

text: and

text: L

text: M

text: M

text: and

text: L

text: M

text: M

text: admissible

text: Lagrangian

text: correspondences

text: with

text: brane

text: structure.

text: Suppose

text: that

text: L

text: L

text: L

text: is

text: transverse

text: and

text: embedded,

text: and

text: admissible

text: in

text: M

text: M

text: Then

text: for

text: any

text: ζ

text: the

text: functors

text: Φ(L

text: L

text: and

text: Φ(L

text: are

text: quasi-isomorphic

text: in

text: ζ

text: Func(Fuk

text: (M

text: Fuk

text: (M

text: Proof.

text: Theorem

text: shows

text: that

text: Φ(L

text: Φ(L

text: Φ(L

text: L

text: are

text: homotopic,

text: in

text: par-

text: ζ

text: ticular,

text: quasiisomorphic.

text: To

text: show

text: that

text: Φ(L

text: L

text: Φ(L

text: are

text: quasi-isomorphic

text: re-

text: ζ

text: call

text: that

text: in

text: the

text: second

text: two

text: authors

text: constructed

text: cocycles

text: φ

text: CF

text: (L

text: L

text: L

text: ζ

text: and

text: ψ

text: CF

text: (L

text: L

text: L

text: with

text: the

text: property

text: that

text: ζ

text: [ψ]

text: [φ]

text: HF

text: (L

text: L

text: [ψ]

text: [φ]

text: HF

text: (L

text: L

text: L

text: L

text: ζ

text: An

text: alternative

text: argument

text: for

text: the

text: existence

text: of

text: the

text: cocycle

text: is

text: given

text: in

text: Lekili-Lipyanskiy

text: Let

text: T

text: (φ),

text: T

text: (ψ)

text: denote

text: the

text: corresponding

text: natural

text: natural

text: transformations.

text: It

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 85 follows from (75) that ◦ ∈ ◦ ∈ T (φ) T (ψ) Aut(Φ(L L )), T (ψ) T (φ) Aut(Φ(L )) 01 ζ 12 02 are cohomologous to the identity natural transformations. The proposition follows (cid:3) by combining (75), Theorem 1.1, and Theorem 7.20. Proof of Theorem 1.2. The statement of the Theorem is a parametrized version of S d,0,0 the main result of [43]. We construct a family of quilted surfaces over the R d,0,0 bimultiplihedron for which the strip with boundary conditions L , L has ∞ 01 12 varying width between 0 and ; this diﬀers from the algebraic composition theorem where the width was bounded below by a non-zero constant ρ . Such a family can 0 S d,0 be obtained from by inserting a strip of width ζ for ζ small and varying, and S R d,0,0 d,0,0 extended to a family over by the previous procedure. The argument will be the same as that for the algebraic composition in Theorem 7.20, but the facet of R d,0,0 where the seams come together has ﬁbers given by quilted disks with a single seam. The corresponding boundary stratum in the moduli space of pseudoholomor- ◦ phic quilts now corresponds to the terms for the geometric composition Φ(L L ) 01 12 in the statement of Theorem 1.2. To show that the moduli space of pseudoholomorphic quilts over this new family has the same properties as before requires the arguments of [43]: near any pseu- doholomorphic quilt with seam in L there is a unique nearby pseudoholomorphic 02 quilt with small width ζ of the strip in between the seams labelled L and L , by 01 12 a parametrized version of the implicit function theorem given in [43]. The operation of replacing the seam with a strip of width ζ deﬁnes a thickening map R × → R T : d,0 [0, δ) d,0,0. r → Given a quilt u : S M with seam I mapping to L , we denote by S(ζ) the quilt R×02 → × obtained by replacing the seam I with a strip S = [0, ζ]. Let γ : I L L ζ → 01 M1 12 be the unique lift of the restriction of u to I. Let u(ζ) : S(ζ) M be the map equal to u on the components except S , and u(ζ)(s, t) = π (γ(s)) on S . Given a section ξ ∗ ζ 1 ζ of u(ζ) T M with Lagrangian boundary and seam conditions, a suitable exponential → e (ξ) : S M with the Lagrangian boundary and seam conditions is deﬁned in [43, u (cid:48) ∈ S → 20]. Pseudoholomorphic quilts near u correspond to pairs r d,0,0, v : S M (cid:48) r with r = T (σ, ζ), v = e (ξ). These are zeroes of the map r u F R × ∗ → ∗ (76) : T d,0 Ω0(Sd,0 , u(ζ) T M) Ω0,1(Sd,0 , u(ζ) T M), u,ζ r Tr(σ,ζ) Tr(σ(cid:55)→,ζ) − (σ, ξ) Φ 1 ∂ e (ξ) u(ζ) Tr(σ,ζ),J,K u − where Φ 1 denotes almost complex parallel transport. In a suitable Sobolev com- u(ζ) F pletion, the map is Fredholm and satisﬁes uniform quadratic and error esti- u,ζ mates, and has a derivative with uniformly bounded right inverse. As in [43] the argument requires splitting oﬀ two more strips of width ζ near the seam, and using the folding construction in [43, Section 3]. The Sobolev completions are described ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: follows

text: from

text: that

text: T

text: (φ)

text: T

text: (ψ)

text: Aut(Φ(L

text: L

text: T

text: (ψ)

text: T

text: (φ)

text: Aut(Φ(L

text: ζ

text: are

text: cohomologous

text: to

text: the

text: identity

text: natural

text: transformations.

text: The

text: proposition

text: follows

text: by

text: combining

text: Theorem

text: and

text: Theorem

text: Proof

text: of

text: Theorem

text: The

text: statement

text: of

text: the

text: Theorem

text: is

text: a

text: parametrized

text: version

text: of

text: S

text: the

text: main

text: result

text: of

text: We

text: construct

text: a

text: family

text: of

text: quilted

text: surfaces

text: over

text: the

text: R

text: bimultiplihedron

text: for

text: which

text: the

text: strip

text: with

text: boundary

text: conditions

text: L

text: L

text: has

text: varying

text: width

text: between

text: and

text: this

text: diﬀers

text: from

text: the

text: algebraic

text: composition

text: theorem

text: where

text: the

text: width

text: was

text: bounded

text: below

text: by

text: a

text: non-zero

text: constant

text: ρ

text: Such

text: a

text: family

text: can

text: S

text: be

text: obtained

text: from

text: by

text: inserting

text: a

text: strip

text: of

text: width

text: ζ

text: for

text: ζ

text: small

text: and

text: varying,

text: and

text: S

text: R

text: extended

text: to

text: a

text: family

text: over

text: by

text: the

text: previous

text: procedure.

text: The

text: argument

text: will

text: be

text: the

text: same

text: as

text: that

text: for

text: the

text: algebraic

text: composition

text: in

text: Theorem

text: but

text: the

text: facet

text: of

text: R

text: where

text: the

text: seams

text: come

text: together

text: has

text: ﬁbers

text: given

text: by

text: quilted

text: disks

text: with

text: a

text: single

text: seam.

text: The

text: corresponding

text: boundary

text: stratum

text: in

text: the

text: moduli

text: space

text: of

text: pseudoholomor-

text: phic

text: quilts

text: now

text: corresponds

text: to

text: the

text: terms

text: for

text: the

text: geometric

text: composition

text: Φ(L

text: L

text: in

text: the

text: statement

text: of

text: Theorem

text: To

text: show

text: that

text: the

text: moduli

text: space

text: of

text: pseudoholomorphic

text: quilts

text: over

text: this

text: new

text: family

text: has

text: the

text: same

text: properties

text: as

text: before

text: requires

text: the

text: arguments

text: of

text: near

text: any

text: pseu-

text: doholomorphic

text: quilt

text: with

text: seam

text: in

text: L

text: there

text: is

text: a

text: unique

text: nearby

text: pseudoholomorphic

text: quilt

text: with

text: small

text: width

text: ζ

text: of

text: the

text: strip

text: in

text: between

text: the

text: seams

text: labelled

text: L

text: and

text: L

text: by

text: a

text: parametrized

text: version

text: of

text: the

text: implicit

text: function

text: theorem

text: given

text: in

text: The

text: operation

text: of

text: replacing

text: the

text: seam

text: with

text: a

text: strip

text: of

text: width

text: ζ

text: deﬁnes

text: a

text: thickening

text: map

text: R

text: R

text: T

text: δ)

text: r

text: Given

text: a

text: quilt

text: u

text: S

text: M

text: with

text: seam

text: I

text: mapping

text: to

text: L

text: we

text: denote

text: by

text: S(ζ)

text: the

text: quilt

text: obtained

text: by

text: replacing

text: the

text: seam

text: I

text: with

text: a

text: strip

text: S

text: ζ].

text: Let

text: γ

text: I

text: L

text: L

text: ζ

text: be

text: the

text: unique

text: lift

text: of

text: the

text: restriction

text: of

text: u

text: to

text: I.

text: Let

text: u(ζ)

text: S(ζ)

text: M

text: be

text: the

text: map

text: equal

text: to

text: u

text: on

text: the

text: components

text: except

text: S

text: and

text: u(ζ)(s,

text: t)

text: π

text: (γ(s))

text: on

text: S

text: Given

text: a

text: section

text: ξ

text: ζ

text: ζ

text: of

text: u(ζ)

text: T

text: M

text: with

text: Lagrangian

text: boundary

text: and

text: seam

text: conditions,

text: a

text: suitable

text: exponential

text: e

text: (ξ)

text: S

text: M

text: with

text: the

text: Lagrangian

text: boundary

text: and

text: seam

text: conditions

text: is

text: deﬁned

text: in

text: u

text: S

text: Pseudoholomorphic

text: quilts

text: near

text: u

text: correspond

text: to

text: pairs

text: r

text: v

text: S

text: M

text: r

text: with

text: r

text: T

text: (σ,

text: ζ),

text: v

text: e

text: (ξ).

text: These

text: are

text: zeroes

text: of

text: the

text: map

text: r

text: u

text: F

text: R

text: T

text: u(ζ)

text: T

text: M)

text: u(ζ)

text: T

text: M),

text: u,ζ

text: r

text: Tr(σ,ζ)

text: (σ,

text: ξ)

text: Φ

text: e

text: (ξ)

text: u(ζ)

text: Tr(σ,ζ),J,K

text: u

text: where

text: Φ

text: denotes

text: almost

text: complex

text: parallel

text: transport.

text: In

text: a

text: suitable

text: Sobolev

text: com-

text: u(ζ)

text: F

text: pletion,

text: the

text: map

text: is

text: Fredholm

text: and

text: satisﬁes

text: uniform

text: quadratic

text: and

text: error

text: esti-

text: u,ζ

text: mates,

text: and

text: has

text: a

text: derivative

text: with

text: uniformly

text: bounded

text: right

text: inverse.

text: As

text: in

text: the

text: argument

text: requires

text: splitting

text: oﬀ

text: two

text: more

text: strips

text: of

text: width

text: ζ

text: near

text: the

text: seam,

text: and

text: using

text: the

text: folding

text: construction

text: in

text: Section

text: The

text: Sobolev

text: completions

text: are

text: described

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 87 Deﬁnition 8.1. (A∞ categories, functors etc.) Z C (a) (A∞ categories) A -graded A∞ category consists of the following data: N C (cid:76) (i) A class of objects Obj( ); ∈ C Z (ii) for each pair C , C Obj( ), a -graded abelian group of mor- 1 2 N phisms Hom≥C(C1, C2) = i∈ZN HomiC(C1, C2);∈ C (iii) for each d 0 and (d + 1)-tuple C , . . . , C Obj( ), a multilinear 0 d composition map ⊗ ⊗ → − µdC := µdC : HomC(C , C ) . . . HomC(C − , C ) HomC(C , C )[2 d] ,C0,...,Cd 0 1 d 1 d 0 d (cid:88) satisfying the A∞(cid:80)-associativity equations − | | − (77) 0 = ( 1)n+ ni=1 ai µdC m+1(a1, . . . , an, ≤ n+m d µmC (a , . . . , a ), a , . . . , a ) n+1 n+m n+m+1 d for any tuple of homogeneous elements a , . . . , a . The signs are the shifted 1 d Koszul signs, that is, the Koszul signs for the shifted grading in which the structure maps have degree one as in Kontsevich-Soibelman [15]. The ele- ∈ ment µ0C (1) Hom(C , C )[2] is the curvature of the object C . An A∞ ,C0 0 0 0 category is ﬂat if µ0C (1) vanishes for every object C . We remark that we ,C0 0 do not assume that our A∞ categories have units. C C F C (b) (A∞ functor) Let , be ﬂat A∞ categories. An A∞ functor from to C 0 1 0 consists of the following data: 1 F C → C (i) a map : Obj( ) Obj( ); and ≥ 0 1 ∈ C (ii) for any d 1 and d + 1-tuple C , . . . , C Obj( ), a map 0 d 0 F × → F F − d : Hom(C , C ) . . . Hom(C − , C ) Hom( (C ), (C ))[1 d] 0 1 d 1 d 0 d (cid:88) such that th(cid:80)e following holds: − | |F − (78) ( (cid:88)1)i+ ij=(cid:88)1 aj d j+1(a1, . . . , ai, µjC (ai+1, . . . , ai+j), ai+j+1, . . . , ad) = ≤ 0 i+j d F F ≥ µmC1( i1(a1, . . . , ai1), . . . , im(ai1+...+im−1+1, . . . , ad)). m 1 i1+...+im=d F F (c) (Composition of A∞ fu(cid:88)nctors)(cid:88)The composition of A∞ functors , is 1 2 deﬁned by composition of maps on the level of objects, and F ◦ F F F (79) ( )d(a , . . . , a ) = m( i1(a , . . . , a ), 1 2 1 d ≥ 1 2 1 i1 m 1 i1+...+im=d F . . . , 2im(ai1+...+im−1+1, . . . , ad)) on the level of morphisms. F C → C (d) (Cohomology functor) Any A∞ functor : between ﬂat A∞ cat- C C 1F 2 C → C egories , deﬁnes an ordinary functor H( ) : H( ) H( ) acting 1 2 1 2

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: Deﬁnition

text: (A∞

text: categories,

text: functors

text: etc.)

text: Z

text: C

text: (a)

text: (A∞

text: categories)

text: A

text: -graded

text: A∞

text: category

text: consists

text: of

text: the

text: following

text: data:

text: N

text: C

text: (i)

text: A

text: class

text: of

text: objects

text: Obj(

text: C

text: Z

text: (ii)

text: for

text: each

text: pair

text: C

text: C

text: Obj(

text: a

text: -graded

text: abelian

text: group

text: of

text: mor-

text: N

text: phisms

text: i∈ZN

text: C

text: (iii)

text: for

text: each

text: d

text: and

text: (d

text: C

text: C

text: Obj(

text: a

text: multilinear

text: d

text: composition

text: map

text: µdC

text: µdC

text: HomC(C

text: C

text: HomC(C

text: C

text: HomC(C

text: C

text: d]

text: d

text: d

text: d

text: satisfying

text: the

text: equations

text: ai

text: µdC

text: an,

text: n+m

text: d

text: µmC

text: (a

text: a

text: a

text: a

text: n+m

text: d

text: for

text: any

text: tuple

text: of

text: homogeneous

text: elements

text: a

text: a

text: The

text: signs

text: are

text: the

text: shifted

text: d

text: Koszul

text: signs,

text: that

text: is,

text: the

text: Koszul

text: signs

text: for

text: the

text: shifted

text: grading

text: in

text: which

text: the

text: structure

text: maps

text: have

text: degree

text: one

text: as

text: in

text: Kontsevich-Soibelman

text: The

text: ele-

text: ment

text: Hom(C

text: C

text: is

text: the

text: curvature

text: of

text: the

text: object

text: C

text: An

text: A∞

text: category

text: is

text: ﬂat

text: if

text: vanishes

text: for

text: every

text: object

text: C

text: We

text: remark

text: that

text: we

text: do

text: not

text: assume

text: that

text: our

text: A∞

text: categories

text: have

text: units.

text: C

text: C

text: F

text: C

text: (b)

text: (A∞

text: functor)

text: Let

text: be

text: ﬂat

text: A∞

text: categories.

text: An

text: A∞

text: functor

text: from

text: to

text: C

text: consists

text: of

text: the

text: following

text: data:

text: F

text: C

text: C

text: (i)

text: a

text: map

text: Obj(

text: Obj(

text: and

text: C

text: (ii)

text: for

text: any

text: d

text: and

text: d

text: C

text: C

text: Obj(

text: a

text: map

text: d

text: F

text: F

text: F

text: d

text: Hom(C

text: C

text: Hom(C

text: C

text: Hom(

text: (C

text: (C

text: d]

text: d

text: d

text: d

text: such

text: that

text: following

text: holds:

text: |F

text: aj

text: d

text: ai,

text: µjC

text: ai+j),

text: ad)

text: i+j

text: d

text: F

text: F

text: ad)).

text: m

text: F

text: F

text: (c)

text: (Composition

text: of

text: A∞

text: composition

text: of

text: A∞

text: functors

text: is

text: deﬁned

text: by

text: composition

text: of

text: maps

text: on

text: the

text: level

text: of

text: objects,

text: and

text: F

text: F

text: F

text: F

text: )d(a

text: a

text: m(

text: a

text: d

text: m

text: F

text: ad))

text: on

text: the

text: level

text: of

text: morphisms.

text: F

text: C

text: C

text: (d)

text: (Cohomology

text: functor)

text: Any

text: A∞

text: functor

text: between

text: ﬂat

text: A∞

text: cat-

text: C

text: C

text: C

text: C

text: egories

text: deﬁnes

text: an

text: ordinary

text: functor

text: H(

text: H(

text: H(

text: acting


text: same N2 second order covariance moments as the top textures. The third column shows the vorticity ﬁeld of a two-dimensional turbulent ﬂuid. The Gaussian realization is thus a Kolmogorov type model, which does not restore the ﬁlament geometry. The third row gives reconstructions from scattering coeﬃcients, ≤ limited to order m 2. The scattering vector is computed at the maximum scale 2J = N, with wavelets having K = 8 diﬀerent orientations. It is thus completely invariant to translations. The dimension of Φ x (cid:28) J is about (K log N)2/2 N2. Scattering moments restore better texture geometries than the Gaussian 2 models obtained with N2 covariance moments. This geometry is mostly captured by second order scattering coeﬃcients, providing scale interaction terms. Indeed, ﬁrst order scattering moments can only reconstruct images which are similar to realizations of Gaussian processes. First and second order scattering moments also provide good models of ergodic audio textures [8]. The fourth image has very sparse wavelet coeﬃcients. In this case the image is nearly perfectly restored by its scattering coeﬃcients, up to a random translation. The reconstruction is centered for comparison. Section explains that if wavelet coeﬃcients are sparse then a rectiﬁer or an absolute value contractions ρ | − (cid:48) | | − (cid:48)| (cid:48) does not contract as much distances with other signals. Indeed, ρ(α) ρ(α ) = α α if α = 0 or α = 0. Inverting a scattering transform is a non-linear inverse problem, which requires to recover a lost phase information. Sparsity has an important role on such phase recovery problems [32]. Translating randomly the last motorcycle image deﬁnes a non-ergodic stationary process, whose wavelet coeﬃcients are not as sparse. As a result, the reconstruction from a random initialization is very diﬀerent, and does not preserve patterns which are important for most classiﬁcation tasks. This is not surprising since there is much less scattering coeﬃcients than image pixels. If we reduce 2J so that the number of scattering coeﬃcients reaches the number of pixels then the reconstruction is of good quality, but there is little variance reduction. Concentrating on the translation group is not so eﬀective to reduce variance when the process is not translation ergodic. Applying wavelet ﬁlters can destroy important structures which are not sparse over wavelets. Next section addresses both issues. Impressive texture synthesis results have been obtained with deep convolutional networks trained on image data bases [14], but with much more output coeﬃcients. Numerical reconstructions [13] also show that one can also recover complex patterns, such as birds, airplanes, cars, dogs, ships, if the network is trained to recognize the corresponding image classes. The network keeps some form of memory of important classiﬁcation patterns. § 7 Multiscale Hierarchical Convolutional Networks Scattering transforms on the translation group are restricted deep convolutional network architectures, which suﬀer from variance issues and loss of information. We shall explain why channel combinations provide the ﬂexibility needed to avoid some of these limitations. We analyze a general class of convolutional network architectures by extending the tools previously introduced. Contractions and invariants to translations are replaced by contractions along groups of local symmetries adapted to f, which are deﬁned by parallel transports in each network layer. The network is structured by factorizing groups of symmetries, as depth increases. It implies that all linear operators can be written as generalized convolutions across multiple channels. To preserve the classiﬁcation margin, wavelets must also be replaced by adapted ﬁlter weights, which separate discriminative patterns in multiple network ﬁbers. Separation margin Network layers x = ρW x − are computed with operators ρW which contract and j j j 1 j separate components of x . We shall see that W also needs to prepare x for the next transformation W , j j j j+1 so consecutive operators W and W are strongly dependant. Each W is a contractive linear operator, (cid:107) (cid:107) ≤ (cid:107) (cid:107) j j+1 j W z z to reduce the space volume, and avoid instabilities when cascading such operators [31]. A j layer x − must separate f so that we can write f(x) = f − (x − ) for some function f − (z). To simplify j 1 j 1 j 1 j 1 explanations, we concentrate on classiﬁcation, where separation is an (cid:15) > 0 margin condition: 13 ##LTLine##

text: same

text: second

text: order

text: covariance

text: moments

text: as

text: the

text: top

text: textures.

text: The

text: third

text: column

text: shows

text: the

text: vorticity

text: ﬁeld

text: of

text: a

text: two-dimensional

text: turbulent

text: ﬂuid.

text: The

text: Gaussian

text: realization

text: is

text: thus

text: a

text: Kolmogorov

text: type

text: model,

text: which

text: does

text: not

text: restore

text: the

text: ﬁlament

text: geometry.

text: The

text: third

text: row

text: gives

text: reconstructions

text: from

text: scattering

text: coeﬃcients,

text: limited

text: to

text: order

text: m

text: The

text: scattering

text: vector

text: is

text: computed

text: at

text: the

text: maximum

text: scale

text: N,

text: with

text: wavelets

text: having

text: K

text: diﬀerent

text: orientations.

text: It

text: is

text: thus

text: completely

text: invariant

text: to

text: translations.

text: The

text: dimension

text: of

text: Φ

text: x

text: J

text: is

text: about

text: (K

text: log

text: Scattering

text: moments

text: restore

text: better

text: texture

text: geometries

text: than

text: the

text: Gaussian

text: models

text: obtained

text: with

text: covariance

text: moments.

text: This

text: geometry

text: is

text: mostly

text: captured

text: by

text: second

text: order

text: scattering

text: coeﬃcients,

text: providing

text: scale

text: interaction

text: terms.

text: Indeed,

text: ﬁrst

text: order

text: scattering

text: moments

text: can

text: only

text: reconstruct

text: images

text: which

text: are

text: similar

text: to

text: realizations

text: of

text: Gaussian

text: processes.

text: First

text: and

text: second

text: order

text: scattering

text: moments

text: also

text: provide

text: good

text: models

text: of

text: ergodic

text: audio

text: textures

text: The

text: fourth

text: image

text: has

text: very

text: sparse

text: wavelet

text: coeﬃcients.

text: In

text: this

text: case

text: the

text: image

text: is

text: nearly

text: perfectly

text: restored

text: by

text: its

text: scattering

text: coeﬃcients,

text: up

text: to

text: a

text: random

text: translation.

text: The

text: reconstruction

text: is

text: centered

text: for

text: comparison.

text: Section

text: explains

text: that

text: if

text: wavelet

text: coeﬃcients

text: are

text: sparse

text: then

text: a

text: rectiﬁer

text: or

text: an

text: absolute

text: value

text: contractions

text: ρ

text: does

text: not

text: contract

text: as

text: much

text: distances

text: with

text: other

text: signals.

text: Indeed,

text: ρ(α)

text: ρ(α

text: α

text: α

text: if

text: α

text: or

text: α

text: Inverting

text: a

text: scattering

text: transform

text: is

text: a

text: non-linear

text: inverse

text: problem,

text: which

text: requires

text: to

text: recover

text: a

text: lost

text: phase

text: information.

text: Sparsity

text: has

text: an

text: important

text: role

text: on

text: such

text: phase

text: recovery

text: problems

text: Translating

text: randomly

text: the

text: last

text: motorcycle

text: image

text: deﬁnes

text: a

text: non-ergodic

text: stationary

text: process,

text: whose

text: wavelet

text: coeﬃcients

text: are

text: not

text: as

text: sparse.

text: As

text: a

text: result,

text: the

text: reconstruction

text: from

text: a

text: random

text: initialization

text: is

text: very

text: diﬀerent,

text: and

text: does

text: not

text: preserve

text: patterns

text: which

text: are

text: important

text: for

text: most

text: classiﬁcation

text: tasks.

text: This

text: is

text: not

text: surprising

text: since

text: there

text: is

text: much

text: less

text: scattering

text: coeﬃcients

text: than

text: image

text: pixels.

text: If

text: we

text: reduce

text: so

text: that

text: the

text: number

text: of

text: scattering

text: coeﬃcients

text: reaches

text: the

text: number

text: of

text: pixels

text: then

text: the

text: reconstruction

text: is

text: of

text: good

text: quality,

text: but

text: there

text: is

text: little

text: variance

text: reduction.

text: Concentrating

text: on

text: the

text: translation

text: group

text: is

text: not

text: so

text: eﬀective

text: to

text: reduce

text: variance

text: when

text: the

text: process

text: is

text: not

text: translation

text: ergodic.

text: Applying

text: wavelet

text: ﬁlters

text: can

text: destroy

text: important

text: structures

text: which

text: are

text: not

text: sparse

text: over

text: wavelets.

text: Next

text: section

text: addresses

text: both

text: issues.

text: Impressive

text: texture

text: synthesis

text: results

text: have

text: been

text: obtained

text: with

text: deep

text: convolutional

text: networks

text: trained

text: on

text: image

text: data

text: bases

text: but

text: with

text: much

text: more

text: output

text: coeﬃcients.

text: Numerical

text: reconstructions

text: also

text: show

text: that

text: one

text: can

text: also

text: recover

text: complex

text: patterns,

text: such

text: as

text: birds,

text: airplanes,

text: cars,

text: dogs,

text: ships,

text: if

text: the

text: network

text: is

text: trained

text: to

text: recognize

text: the

text: corresponding

text: image

text: classes.

text: The

text: network

text: keeps

text: some

text: form

text: of

text: memory

text: of

text: important

text: classiﬁcation

text: patterns.

text: Multiscale

text: Hierarchical

text: Convolutional

text: Networks

text: Scattering

text: transforms

text: on

text: the

text: translation

text: group

text: are

text: restricted

text: deep

text: convolutional

text: network

text: architectures,

text: which

text: suﬀer

text: from

text: variance

text: issues

text: and

text: loss

text: of

text: information.

text: We

text: shall

text: explain

text: why

text: channel

text: combinations

text: provide

text: the

text: ﬂexibility

text: needed

text: to

text: avoid

text: some

text: of

text: these

text: limitations.

text: We

text: analyze

text: a

text: general

text: class

text: of

text: convolutional

text: network

text: architectures

text: by

text: extending

text: the

text: tools

text: previously

text: introduced.

text: Contractions

text: and

text: invariants

text: to

text: translations

text: are

text: replaced

text: by

text: contractions

text: along

text: groups

text: of

text: local

text: symmetries

text: adapted

text: to

text: f,

text: which

text: are

text: deﬁned

text: by

text: parallel

text: transports

text: in

text: each

text: network

text: layer.

text: The

text: network

text: is

text: structured

text: by

text: factorizing

text: groups

text: of

text: symmetries,

text: as

text: depth

text: increases.

text: It

text: implies

text: that

text: all

text: linear

text: operators

text: can

text: be

text: written

text: as

text: generalized

text: convolutions

text: across

text: multiple

text: channels.

text: To

text: preserve

text: the

text: classiﬁcation

text: margin,

text: wavelets

text: must

text: also

text: be

text: replaced

text: by

text: adapted

text: ﬁlter

text: weights,

text: which

text: separate

text: discriminative

text: patterns

text: in

text: multiple

text: network

text: ﬁbers.

text: Separation

text: margin

text: Network

text: layers

text: x

text: ρW

text: x

text: are

text: computed

text: with

text: operators

text: ρW

text: which

text: contract

text: and

text: j

text: j

text: j

text: j

text: separate

text: components

text: of

text: x

text: We

text: shall

text: see

text: that

text: W

text: also

text: needs

text: to

text: prepare

text: x

text: for

text: the

text: next

text: transformation

text: W

text: j

text: j

text: j

text: so

text: consecutive

text: operators

text: W

text: and

text: W

text: are

text: strongly

text: dependant.

text: Each

text: W

text: is

text: a

text: contractive

text: linear

text: operator,

text: j

text: j

text: W

text: z

text: z

text: to

text: reduce

text: the

text: space

text: volume,

text: and

text: avoid

text: instabilities

text: when

text: cascading

text: such

text: operators

text: A

text: j

text: layer

text: x

text: must

text: separate

text: f

text: so

text: that

text: we

text: can

text: write

text: f(x)

text: f

text: (x

text: for

text: some

text: function

text: f

text: (z).

text: To

text: simplify

text: j

text: j

text: j

text: j

text: explanations,

text: we

text: concentrate

text: on

text: classiﬁcation,

text: where

text: separation

text: is

text: an

text: margin

text: condition:

text: ##LTLine##


text: [19] Krizhevsky A, Sutskever I, Hinton G. 2012 ImageNet classiﬁcation with deep convolutional neural net- works, In Proc. of NIPS, p. 1090-1098, 2012. 1, 8 [20] Le Cun Y, Boser B, Denker J, Henderson D, Howard R, Hubbard W, Jackelt L. 1990 Handwritten digit recognition with a back-propagation network, In Proc. of NIPS. 3. 1, 8 [21] Le Cun Y, Bengio Y, Hinton G. 2015 Deep learning, Nature, 521. 1, 2, 8, 9, 17 [22] Leung MK, Xiong HY, Lee LJ, Frey BJ. 2014 Deep learning of the tissue regulated splicing code, Bioin- formatics, 30. 1 [23] Lowe DG. 2004 SIFT: Scale Invariant Feature Transform, J. of Computer Vision, 60. 8 [24] Mallat S. 2012, Group Invariant Scattering, Comm. in Pure and Applied Mathematics, 65. 2, 6, 7, 10, 11 [25] Mesgarani M, Slaney M, Shamma S. 2006 Discrimination of speech from nonspeech based on multiscale spectro-temporal modulations IEEE Trans. Audio, Speech, Lang. Process., 14. 8, 16 [26] J. Petitot, 2008 Neurog´em´etrie de la vision, E´ditions de l’E´cole Polytechnique. 14 [27] Oyallon E, Mallat S. 2015 Deep roto-translation scattering for object classiﬁcation, Proc. of CVPR. 16 [28] Radford A, Metz L, Chintala S. 2016 Unsupervised representation learning with deep convolutional generative adversarial networks, ICLR 2016. 5 [29] Sifre L, Mallat S. 2013 Rotation, Scaling and Deformation Invariant Scattering for Texture Discrimi- nation, In Proc. of CVPR. 11, 15, 16 [30] Sutskever I, Vinyals O, Le QV. 2015 Sequence to sequence learning with neural networks In Proc. of NIPS, 27. 1 [31] Szegedy C, Erhan D, Zaremba W, Sutskever I, Goodfellow I, Bruna J, Fergus R. 2014, Intriguing properties of neural networks, In Proc. of ICLR. 9, 13 [32] Waldspurger I. 2015 Wavelet transform modulus: phase retrieval and scattering, Ph.D Ecole Normale Sup`erieure. 11, 13 19

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text: Krizhevsky

text: A,

text: Sutskever

text: I,

text: Hinton

text: G.

text: ImageNet

text: classiﬁcation

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text: neural

text: net-

text: works,

text: In

text: Proc.

text: of

text: NIPS,

text: p.

text: Le

text: Cun

text: Y,

text: Boser

text: B,

text: Denker

text: J,

text: Henderson

text: D,

text: Howard

text: R,

text: Hubbard

text: W,

text: Jackelt

text: L.

text: Handwritten

text: digit

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text: a

text: back-propagation

text: network,

text: In

text: Proc.

text: of

text: NIPS.

text: Le

text: Cun

text: Y,

text: Bengio

text: Y,

text: Hinton

text: G.

text: Deep

text: learning,

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text: Leung

text: MK,

text: Xiong

text: HY,

text: Lee

text: LJ,

text: Frey

text: BJ.

text: Deep

text: learning

text: of

text: the

text: tissue

text: regulated

text: splicing

text: code,

text: Bioin-

text: formatics,

text: Lowe

text: DG.

text: SIFT:

text: Scale

text: Invariant

text: Feature

text: Transform,

text: J.

text: of

text: Computer

text: Vision,

text: Mallat

text: S.

text: Group

text: Invariant

text: Scattering,

text: Comm.

text: in

text: Pure

text: and

text: Applied

text: Mathematics,

text: Mesgarani

text: M,

text: Slaney

text: M,

text: Shamma

text: S.

text: Discrimination

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text: speech

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text: nonspeech

text: based

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text: multiscale

text: spectro-temporal

text: modulations

text: IEEE

text: Trans.

text: Audio,

text: Speech,

text: Lang.

text: Process.,

text: J.

text: Petitot,

text: Neurog´em´etrie

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text: vision,

text: E´ditions

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text: l’E´cole

text: Polytechnique.

text: Oyallon

text: E,

text: Mallat

text: S.

text: Deep

text: roto-translation

text: scattering

text: for

text: object

text: classiﬁcation,

text: Proc.

text: of

text: CVPR.

text: Radford

text: A,

text: Metz

text: L,

text: Chintala

text: S.

text: Unsupervised

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text: learning

text: with

text: deep

text: convolutional

text: generative

text: adversarial

text: networks,

text: ICLR

text: Sifre

text: L,

text: Mallat

text: S.

text: Rotation,

text: Scaling

text: and

text: Deformation

text: Invariant

text: Scattering

text: for

text: Texture

text: Discrimi-

text: nation,

text: In

text: Proc.

text: of

text: CVPR.

text: Sutskever

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text: Vinyals

text: O,

text: Le

text: QV.

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text: to

text: sequence

text: learning

text: with

text: neural

text: networks

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text: Proc.

text: of

text: NIPS,

text: Szegedy

text: C,

text: Erhan

text: D,

text: Zaremba

text: W,

text: Sutskever

text: I,

text: Goodfellow

text: I,

text: Bruna

text: J,

text: Fergus

text: R.

text: Intriguing

text: properties

text: of

text: neural

text: networks,

text: In

text: Proc.

text: of

text: ICLR.

text: Waldspurger

text: I.

text: Wavelet

text: transform

text: modulus:

text: phase

text: retrieval

text: and

text: scattering,

text: Ph.D

text: Ecole

text: Normale

text: Sup`erieure.


text: In conclusion, generally a very good agreement is obtained calculated using a natural logarithm of the ﬁrst random number between Eq. (3) and Eq. (4) for a wide range of parameters that and the inverse of the sum of all rates of jumps that are possible reﬂect realistic experimental conditions. However, it is worth in the system. In the next step it is decided which type of jump mentioning that since a set of conditions needs to be fulﬁlled will be carried out. For this purpose all rates are normalised to for the Zeeman projection to provide an acceptable approxi- sum up to ’1’, and arranged in a cumulative sum array. An ele- mation [16], there are also parameter regimes where the error is ment in this array is identiﬁed which is higher in value than the likely to be greater than shown here. In particular, the dynamics second random number. The operation corresponding to that in model spin systems with very small average distances (below rate in the array element is carried out. Accordingly, operations 2Å) or long transverse relaxation time constants is not well ap- with higher rate values are statistically more likely to occur than proximated by an adiabatic elimination procedure to project the events with low rates. Once the operation or jump is carried out, dynamics onto the Zeeman subspace. In this case coherences the eﬀective rates for the ﬂip ﬂop jump between a nucleus and are not negligible for the evolution of the spin system and need the electron (Eq. (7)) are re-calculated based on the current sys- to be taken into account by including states outside of the Zee- tem state. The steps outlined are repeated until a desired time man subspace into the calculations. is reached in the trajectory. Many trajectories are computed 3.2. Kinetic Monte Carlo simulations and averaged to approximate the polarization dynamics of the system. The method does not require vasts amounts of mem- The dynamics of spin systems containing a large number N ory since only an N-element binary array is required that stores of coupled spins cannot be calculated by conventional propa- the current orientation of the z-component of each individual gation using Eq. (4) because the dimensions of the required spin. The KMC method was implemented using MATLAB on state space scales with 2N. However, because of the classical a 3GHz 20 core desktop computer. 1 nature of Eq. (4), the dynamics can be well approximated using a kinetic Monte Carlo (KMC) method [26]. It provides a CPU eﬃcient way to ﬁnd numerical solutions for the evolution of A comparison of the polarization dynamics was carried out the density operator in Eq. (4). Like the static variants, KMC for the model spin system described previously with a = 5 Å relies on averaging over many trajectories that correspond to inter-spin separation using the same parameter set as previously possible evolutions of the system. The Zeeman-projected mas- listed. ter equation (4) is already in a form suitable for this method. The eﬀective rates in Eqs. (5 - 8) along with their correspond- ing Lindblad operators can be directly used. The formalism is schematically described in Fig. 4. Figure 4: Flow diagram of the kinetic Monte Carlo algorithm The initial state for a trajectory is generated based on the ther- mal equilibrium polarization values of the electron (the nuclear spin polarization was assumed to be zero). Figure 5: Comparison between Zeeman master equation and KMC Due to the Markovian nature of the dynamics in the many spin system only knowledge of the current state is suﬃcient to prop- agate the system in time. The Lindblad rates (Eqs. (5) - (8)) are ﬁrst calculated based on the system parameters. The rate for the ﬂip-ﬂop interaction between an electron and a nuclei (Eq. (7)) is state-dependent and hence is initiated using the thermal equilibrium state. Next, two random numbers r and r are gen- 1 2 erated. The time required for the next event to take place is 1The full code can be made available by contacting the authors 5 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: In

text: conclusion,

text: generally

text: a

text: very

text: good

text: agreement

text: is

text: obtained

text: calculated

text: using

text: a

text: natural

text: logarithm

text: of

text: the

text: ﬁrst

text: random

text: number

text: between

text: Eq.

text: and

text: Eq.

text: for

text: a

text: wide

text: range

text: of

text: parameters

text: that

text: and

text: the

text: inverse

text: of

text: the

text: sum

text: of

text: all

text: rates

text: of

text: jumps

text: that

text: are

text: possible

text: reﬂect

text: realistic

text: experimental

text: conditions.

text: However,

text: it

text: is

text: worth

text: in

text: the

text: system.

text: In

text: the

text: next

text: step

text: it

text: is

text: decided

text: which

text: type

text: of

text: jump

text: mentioning

text: that

text: since

text: a

text: set

text: of

text: conditions

text: needs

text: to

text: be

text: fulﬁlled

text: will

text: be

text: carried

text: out.

text: For

text: this

text: purpose

text: all

text: rates

text: are

text: normalised

text: to

text: for

text: the

text: Zeeman

text: projection

text: to

text: provide

text: an

text: acceptable

text: approxi-

text: sum

text: up

text: to

text: ’1’,

text: and

text: arranged

text: in

text: a

text: cumulative

text: sum

text: array.

text: An

text: ele-

text: mation

text: there

text: are

text: also

text: parameter

text: regimes

text: where

text: the

text: error

text: is

text: ment

text: in

text: this

text: array

text: is

text: identiﬁed

text: which

text: is

text: higher

text: in

text: value

text: than

text: the

text: likely

text: to

text: be

text: greater

text: than

text: shown

text: here.

text: In

text: particular,

text: the

text: dynamics

text: second

text: random

text: number.

text: The

text: operation

text: corresponding

text: to

text: that

text: in

text: model

text: spin

text: systems

text: with

text: very

text: small

text: average

text: distances

text: (below

text: rate

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text: the

text: array

text: element

text: is

text: carried

text: out.

text: Accordingly,

text: operations

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text: long

text: transverse

text: relaxation

text: time

text: constants

text: is

text: not

text: well

text: ap-

text: with

text: higher

text: rate

text: values

text: are

text: statistically

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text: likely

text: to

text: occur

text: than

text: proximated

text: by

text: an

text: adiabatic

text: elimination

text: procedure

text: to

text: project

text: the

text: events

text: with

text: low

text: rates.

text: Once

text: the

text: operation

text: or

text: jump

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text: carried

text: out,

text: dynamics

text: onto

text: the

text: Zeeman

text: subspace.

text: In

text: this

text: case

text: coherences

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text: rates

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text: the

text: ﬂip

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text: a

text: nucleus

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text: negligible

text: for

text: the

text: evolution

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text: spin

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text: are

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text: based

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text: to

text: be

text: taken

text: into

text: account

text: by

text: including

text: states

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text: tem

text: state.

text: The

text: steps

text: outlined

text: are

text: repeated

text: until

text: a

text: desired

text: time

text: man

text: subspace

text: into

text: the

text: calculations.

text: is

text: reached

text: in

text: the

text: trajectory.

text: Many

text: trajectories

text: are

text: computed

text: Kinetic

text: Monte

text: Carlo

text: simulations

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text: averaged

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text: approximate

text: the

text: polarization

text: dynamics

text: of

text: the

text: system.

text: The

text: method

text: does

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text: require

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text: mem-

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text: dynamics

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text: spin

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text: containing

text: a

text: large

text: number

text: N

text: ory

text: since

text: only

text: an

text: N-element

text: binary

text: array

text: is

text: required

text: that

text: stores

text: of

text: coupled

text: spins

text: cannot

text: be

text: calculated

text: by

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text: the

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text: the

text: z-component

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text: Eq.

text: because

text: the

text: dimensions

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text: the

text: required

text: spin.

text: The

text: KMC

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text: implemented

text: using

text: MATLAB

text: on

text: state

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text: scales

text: with

text: However,

text: because

text: of

text: the

text: classical

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text: core

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text: computer.

text: nature

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text: Eq.

text: the

text: dynamics

text: can

text: be

text: well

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text: using

text: a

text: kinetic

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text: Carlo

text: (KMC)

text: method

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text: provides

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text: eﬃcient

text: way

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text: ﬁnd

text: numerical

text: solutions

text: for

text: the

text: evolution

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text: comparison

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text: the

text: polarization

text: dynamics

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text: carried

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text: the

text: density

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text: Like

text: the

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text: KMC

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text: the

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text: the

text: system.

text: The

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text: listed.

text: ter

text: equation

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text: already

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text: a

text: form

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text: this

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text: The

text: eﬀective

text: rates

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text: Eqs.

text: along

text: with

text: their

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text: ing

text: Lindblad

text: operators

text: can

text: be

text: directly

text: used.

text: The

text: formalism

text: is

text: schematically

text: described

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text: Fig.

text: Figure

text: Flow

text: diagram

text: of

text: the

text: kinetic

text: Monte

text: Carlo

text: algorithm

text: The

text: initial

text: state

text: for

text: a

text: trajectory

text: is

text: generated

text: based

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text: the

text: ther-

text: mal

text: equilibrium

text: polarization

text: values

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text: the

text: electron

text: (the

text: nuclear

text: spin

text: polarization

text: was

text: assumed

text: to

text: be

text: zero).

text: Figure

text: Comparison

text: between

text: Zeeman

text: master

text: equation

text: and

text: KMC

text: Due

text: to

text: the

text: Markovian

text: nature

text: of

text: the

text: dynamics

text: in

text: the

text: many

text: spin

text: system

text: only

text: knowledge

text: of

text: the

text: current

text: state

text: is

text: suﬃcient

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text: agate

text: the

text: system

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text: time.

text: The

text: Lindblad

text: rates

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text: are

text: ﬁrst

text: calculated

text: based

text: on

text: the

text: system

text: parameters.

text: The

text: rate

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text: ﬂip-ﬂop

text: interaction

text: between

text: an

text: electron

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text: is

text: state-dependent

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text: hence

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text: initiated

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text: the

text: thermal

text: equilibrium

text: state.

text: Next,

text: two

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text: erated.

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text: ##LTFigure##

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text: ##LTFigure##

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text: ##LTLine##


text: 1 An eﬀective numerical method to solve a class of nonlinear singular boundary value problems using improved diﬀerential transform method ∗ † Lie-jun Xie , Cai-lian Zhou and Song Xu Department of Mathematics, Faculty of Science, Ningbo University Ningbo Zhejiang, 315211 P. R. China In this work, an eﬀective numerical method is developed to solve a class of singular boundary value problems arising in various physical models by using the improved diﬀer- ential transform method (IDTM). The IDTM applies the Adomian polynomials to handle the diﬀerential transforms of the nonlinearities arising in the given diﬀerential equation. The relation between the Adomian polynomials of those nonlinear functions and the coef- ﬁcients of unknown truncated series solution is given by a simple formula, through which one can easily deduce the approximate solution which takes the form of a convergent series. An upper bound for the estimation of approximate error is presented. Several physical problems are discussed as illustrative examples to testify the validity and appli- cability of the proposed method. Comparisons are made between the present method and the other existing methods. Keywords: Singular boundary value problem; diﬀerential transform method; Adomian polynomials; improved diﬀerential transform method; approximate series solutions 1. Introduction Singular boundary value problems (SBVPs) is an important class of boundary value problems, and arises frequently in the modeling of many actual problems related to physics and engineering areas such as in the study of electro hydrodynamics, theory of thermal explosions, boundary layer theory, the study of astrophysics, three layer beam, electro- magnetic waves or gravity driven ﬂows, inelastic ﬂows, the theory of elastic stability and so on. In general, SBVPs is diﬃcult to solve analytically. Therefore, various numerical techniques have been proposed to treat it by many researchers. However, the solution of SBVPs is numerically challenging due to the singularity behavior at the origin. In this work, we are interested again in the following SBVPs arising frequently in applied science and engineering: (cid:48)(cid:48) α (cid:48) ≤ ≥ u (x) + u (x) = f(x, u), 0 < x 1, α 1, (1) x subject to the boundary value conditions (cid:48) u (0) = 0 (2) ∗ Corresponding author. † E-mail address: xieliejun@nbu.edu.cn, xieliejun@hotmail.com. ##LTLine## ##LTLine##

text: An

label: title

text: eﬀective

text: numerical

text: method

text: to

text: solve

text: a

text: class

text: of

text: nonlinear

text: singular

text: boundary

text: value

text: problems

text: using

text: improved

text: diﬀerential

text: transform

text: method

label: author

text: Lie-jun

text: Xie

text: Cai-lian

text: Zhou

text: and

text: Song

text: Xu

text: Department

text: of

text: Mathematics,

text: Faculty

text: of

text: Science,

text: Ningbo

text: University

text: Ningbo

text: Zhejiang,

text: P.

text: R.

text: China

text: In

label: abstract

text: this

text: work,

text: an

text: eﬀective

text: numerical

text: method

text: is

text: developed

text: to

text: solve

text: a

text: class

text: of

text: singular

text: boundary

text: value

text: problems

text: arising

text: in

text: various

text: physical

text: models

text: by

text: using

text: the

text: improved

text: diﬀer-

text: ential

text: transform

text: method

text: (IDTM).

text: The

text: IDTM

text: applies

text: the

text: Adomian

text: polynomials

text: to

text: handle

text: the

text: diﬀerential

text: transforms

text: of

text: the

text: nonlinearities

text: arising

text: in

text: the

text: given

text: diﬀerential

text: equation.

text: The

text: relation

text: between

text: the

text: Adomian

text: polynomials

text: of

text: those

text: nonlinear

text: functions

text: and

text: the

text: coef-

text: ﬁcients

text: of

text: unknown

text: truncated

text: series

text: solution

text: is

text: given

text: by

text: a

text: simple

text: formula,

text: through

text: which

text: one

text: can

text: easily

text: deduce

text: the

text: approximate

text: solution

text: which

text: takes

text: the

text: form

text: of

text: a

text: convergent

text: series.

text: An

text: upper

text: bound

text: for

text: the

text: estimation

text: of

text: approximate

text: error

text: is

text: presented.

text: Several

text: physical

text: problems

text: are

text: discussed

text: as

text: illustrative

text: examples

text: to

text: testify

text: the

text: validity

text: and

text: appli-

text: cability

text: of

text: the

text: proposed

text: method.

text: Comparisons

text: are

text: made

text: between

text: the

text: present

text: method

text: and

text: the

text: other

text: existing

text: methods.

text: Keywords:

text: Singular

text: boundary

text: value

text: problem;

text: diﬀerential

text: transform

text: method;

text: Adomian

text: polynomials;

text: improved

text: diﬀerential

text: transform

text: method;

text: approximate

text: series

text: solutions

text: Introduction

text: Singular

text: boundary

text: value

text: problems

text: (SBVPs)

text: is

text: an

text: important

text: class

text: of

text: boundary

text: value

text: problems,

text: and

text: arises

text: frequently

text: in

text: the

text: modeling

text: of

text: many

text: actual

text: problems

text: related

text: to

text: physics

text: and

text: engineering

text: areas

text: such

text: as

text: in

text: the

text: study

text: of

text: electro

text: hydrodynamics,

text: theory

text: of

text: thermal

text: explosions,

text: boundary

text: layer

text: theory,

text: the

text: study

text: of

text: astrophysics,

text: three

text: layer

text: beam,

text: electro-

text: magnetic

text: waves

text: or

text: gravity

text: driven

text: ﬂows,

text: inelastic

text: ﬂows,

text: the

text: theory

text: of

text: elastic

text: stability

text: and

text: so

text: on.

text: In

text: general,

text: SBVPs

text: is

text: diﬃcult

text: to

text: solve

text: analytically.

text: Therefore,

text: various

text: numerical

text: techniques

text: have

text: been

text: proposed

text: to

text: treat

text: it

text: by

text: many

text: researchers.

text: However,

text: the

text: solution

text: of

text: SBVPs

text: is

text: numerically

text: challenging

text: due

text: to

text: the

text: singularity

text: behavior

text: at

text: the

text: origin.

text: In

text: this

text: work,

text: we

text: are

text: interested

text: again

text: in

text: the

text: following

text: SBVPs

text: arising

text: frequently

text: in

text: applied

text: science

text: and

text: engineering:

text: α

text: u

text: (x)

text: u

text: (x)

text: f(x,

text: u),

text: x

text: α

text: x

text: subject

text: to

text: the

text: boundary

text: value

text: conditions

text: u

text: Corresponding

text: author.

text: E-mail

text: address:

text: xieliejun@nbu.edu.cn,

text: xieliejun@hotmail.com.

text: ##LTLine##

text: ##LTLine##