An introduction to embedding
problems in symplectic
geometry
Dusa McDuff
Department of Mathematics, Barnard College, Columbia University

dusa@math.columbia.edu

Talk to Women in Mathematics Conference, Japan
September 7, 2022

Plan of lecture:1.

A symplectic structure is a rather elusive geometric structure that can be
put on an even dimensional space. After a brief introduction I will explainsome recent developments in the symplectic embedding problem.
I(I): Very brief introduction to symplectic geometry

I(II) The symplectic embedding problem

I(III) The Fibonacci staircase

What is symplectic geometry?

2.

Euclidean (or Riemannian) geometry makes measurements using a symmetricbilinear form, such as
x · y = x1 y1 + · · · xn yn ,x, y ∈ Rn .

Symplectic geometry makes measurements using a skew symmetric bilinear
form, such asω0 = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n on R2n .
More generally, a symplectic form ω on a (necessarily even dimensional)
manifold is a differential 2-form ω that locally looks like ω0 with respect tosuitable local coordinates.
Thus in 2-dimensions, ω is an area form, and symplectic geometry is the studyof area preserving diffeomorphisms. These have many interesting properties(e.g. more fixed points than an arbitrary diffeomorphism). Arnol’d realised thatin higher dimensions these properties do not hold for volume preserving
diffeomorphisms, but he conjectured that they do hold for symplectic
diffeomorphisms. His conjectures have inspired much work.

The standard symplectic form ω0 in R4
ω0 := dx1 ∧ dy1 + dx2 ∧ dy2 ,a sum of area forms
S is a piece of surface thatyou project in two different ways
and then add the areas.

Since ω0 is closed, i.e. dω0 = 0,
we get flabby measurements:
by Stokes’ theorem, the area
Rof a surface S can be written as an integral S ω0 ; that does not change as S
moves as long as the boundary remains fixed. In physics, the pair xi , yi
represents the position and velocity of a particle in one direction – so a particle
moving in 3-space gives 6 coordinates. The symplectic form in R6 gives animportant (but geometrically somewhat obscure) measurement of the mutual
entanglement of position and velocity.

3.

Many symplectomorphisms:

4.

In general, a symplectic structure on a 2n-dimensional manifold M is a closed,
nondegenerate 2-form ω.
Every function H : M → R generates a flow φHt , t ∈ R (1-parameter group of∗
motions of space) that preserves ω: (φH)(ω)= ω. Such transformations aret
called symplectomorphisms.
The flow is a solution of Hamilton’s differential equations — generated by thevector field XH given by ω(XH , ·) = dH(·) – so that in R2n we have∂x= ∂H, ∂y = − ∂H.
∂t∂y ∂t∂x
Example: if H = 12 (x 2 + y 2 ) on (R2 , dx ∧ dy ), we finddH = xdx + ydy =⇒ XH = y ∂x − x∂ygiving a clockwise rotation, preserving the circles H = const.
But there are many, much more twisty symplectomorphisms Because there areso many, symplectic geometry is very flexible.
But it also displays interesting rigidity

Symplectic embeddings

5.

In 1985 Gromov asked:
what can be said about the image φ(B 2n ) of a ball under a symplectomorphism
φ of R2n ?
In 2-dimensions nothing interesting happens:(Moser - 1965) If a closed disc D ⊂ R2 is diffeomorphic to a closed region U of∼=the same total area, there is an area preserving diffeomorphism φ : D → U.
But higher dimensions are very interesting. Consider the ball
B 2n (a) = (z1 , . . . , zn ) : π(|z1 |2 + · · · + |zn |2 ) ≤ a ⊂ Cn = R2nand the cylinder Z (A) = (z1 , . . . , zn ) : π|z1 |2 ≤ A ⊂ R2n .
Nonsqueezing Theorem (Gromov: 1985) There is a symplectic embeddings
B 2n (a) ,→ Z (A) if and only if a ≤ A.
The volume preserving map (z1 , . . . , zn ) 7→ λz1 , λ1 z2 , z3 , . . . , zn ),qA
λ :=squeezes the ball into the cylinder.
aSymplectic capacities:

6.

Although the Nonsqueezing Theorem may seem just like a curiosity, it turns
out to be a cornerstone of the modern theory.
Gromov, Ekeland–Hofer (1980s): Given an open U ⊂ R2n define thesymplectic capacity byc(U) = sup{a : B 2n (a) embeds symplectically in U}.
Ic(U) is a symplectic invariant;

Iit is essentially 2-dimensional, eg the cylinder Z (A) is a set of infinite
volume with finite capacityI

any orientation preserving diffeomorphism φ that preserves c (i.e.
c(φ(U)) = c(U) for all U) is an (anti)-symplectomorphism, i.e.
φ∗ (ω) = ± ω. So: it characterizes symplectomorphisms

I

What other obstructions are there to symplectic embeddings?

Embedding 4-dimensional Ellipsoids
27.
2

Let E (a, b) be the ellipsoid {(z1 , z2 ) : π |z1a| + |z2b|≤ 1}.

¼(jz1j2=a + jz2j2=b) · 1

the ellipsoid Ea;b

Hofer conjectured around 2010 that intE (a, b) embeds symplectically inintE (c, d) if and only if N (a, b) ≤ N (c, d). Here N (a, b) is the set of allnumbers ka + `b, k, ` ≥ 0, arranged with multiplicities in increasing order.So,N (2, 2) = (0, 2, 2 , 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 8, . . . ), and|{z} | {z } | {z } | {z }N (1, 4) = (0, 1, 2 , 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, . . . Thus N (1, 4) ≤ N (2, 2) because|{z} | {z } | {z } | {z }the first sequence is termwise no larger than the second.
These numbers are the actions of the ECH generators: ECH = embedded contact
homology — a 4-dimensional Floer-type homology theory related to gauge theory;
whose generators are unions of closed orbits of the boundary Hamiltonian flow.
This conjecture now proved by McDuff (2012). An illustration of what it means:

The “ellipsoid into ball” embedding capacity8.

For x ≥ 1 define c(x) := inf µ : E (1, x) embeds sympl. in B 4 (µ) .
This function was calculated by McDuff–Schlenk (2012).
3(b2;b)(¿4;¿2)
2

1

¿4
1

2

3

4

5

67(17=6)289√

For x < τ 4 ≈ 6.7 (where τ = 1+2 5 ) there is an infinite staircase (withnumerics based on the Fibonacci numbers),
2√I for x ≥ 8 1 = 17 , c(x) = x – no obstruction except for volume366III1τ 4 < x < 8 36
is a transitional region;

there are rather few results or plausible guesses as to behavior in dim > 4.
The obvious analog of Hofer’s conjecture is false (Guth) sinces
E (1, S, S),→E (3 + ε, 3 + ε, S 3 ) for all S

Connection with number theory via Toric models 13.
Using coordinates (t = π|z|2 , θ = arg z) and forgetting θ gives a map
C2 → R2 ,(z1 , z2 ) 7→ (t1 , t2 ) := (π|z1 |2 , π|z2 |2 ).
2
2It takes the ellipsoid E (a, b) = π |z1a| + π |z2b| ≤ 1 to the triangle
T (a, b) :=4(t1 , t2 ) ∈ R2+ : 0 ≤ t1 , t2 , ta1 + tb2 ≤ 1 .

T(3;4)
0

3

The ball B(1) maps to the standard triangle T (1, 1), with various affine
equivalent images given by integral changes of basis of C2 (eg use (z1 , z1 + z2 )).
Thus T (1, 3) can be cut into three standard triangles; cf. diagram on right.
Hence E (1, 3) contains three disjoint balls B(1) t B(1) t B(1).

General Triangle decompositions

9In fact any triangle T (a, b) with a, b ∈ Z can be decomposed into
standard 4s of different sizes, (a standard triangle is linearly equiv to someT (c, c))

On the top: we build T (5, 3) from T (3, 3) t T (2, 2) t T (1, 1) t T (1, 1).
s

13p3q1ss2rss

1 1332pqrr

On bottom: we build a rectangle from squares. Combinatorially these
decomp are same, but the second is much easier to understand. To get thetriangles from the rectangles, remove the top right point and collapse top and rightsides to points in affine way.

Continued fractions and decompositions

15

The combinatorics of this decomposition of a triangle into standard
triangles (or of a rectangle into squares), is the same as that governing
the weight expansion W ( pq ) of a rational fraction qp . e.g.
9, 9 , 7, 2, 2, 2, 1, 1 )W ( 259 ) = (|{z}| {z } |{z}corresponds to25
1
1= [2; 1, 3, 2] = 2 +=2+= 2 + 79 .
1
91 + 3+ 1
1 + 27
2The entries of the continued fraction are the multiplicities of the weights

Embedding ellipsoids vs disjoint balls.

10Key Points: Suppose that pq has normalized weight expansionw ( pq ) = (w1 , w2 , . . . , wn ) (where w1 = 1). ThenI

E (1, pq ) embeds into B 4 (µ) if and only if one can embed the disjoint unionof balls tni=1 B(wi ) into B 4 (µ).

Itni=1 B(wi ) embeds into the ball B 4 (µ) if and only if there is a symplectic
form on the n-fold blow up of CP 2 (µ) in the cohomology class
α := µ PD(L) − w1 PD(E1 ) − · · · − wn PD(En ) (PD(Ei ) is the Poincaré dual
to the exceptional divisor given by the ith blow up)

I(Li-Li, 1990s) this happens if and only if2- α
R > 0 (volume constraint)- E α > 0 for all exceptional divisors in the blow up.

So we now have a geometric/algebraic problem — what are the homologyclasses of the relevant exceptional divisors? In the case when we map
into a ball, their coefficients come from Fibonacci numbers.

Fibonacci staircase revisited

113(b2;b)(¿4;¿2)21¿41234567(17=6)289

We found that the exceptional classes in blow ups of CP 2 that gave thesharpest
P obstruction to embedding E (p, q) had the formdL − mi Ei ,where (mi ) = W (p/q) and d, p, q are odd placed Fibonaccinumbers. These are called perfect classes.
I Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
I For example at p/q = 2/1 with W (2) = (1×2 ) the obstruction isL − E12 = L − E1 − E2 (blow up of a line through 2 generic points.
I at p/q = 5/1 with W (5) = (1×5 ) the obstruction is 2L − E1···5 (blow upof a conic through 5 generic points.
I at p/q = 13/2, we have W (13/2) = (2×6 , 1×6 ) and the obstructive classis 5L − 2E1···6 − E78 .
I Recently I (and various collaborators) have been studying the staircasesthat exist for embeddings ellipsoids into Hirzebruch surfaces (one point
blow up of CP 2 ). We found a very interesting fractal pattern.
IA very few references:

12.

Maria Bertozzi, Tara Holm, Emily Maw, Grace Mwakyoma, Ana Rita Pires,
Morgan Weiler: Infinite staircases for Hirzebruch surfaces, (arXiv:2010.08567),
WISCon proceedings, ICERM workshop, Springer
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones
Mathematicae, 82 (1985), 307–47.
D. McDuff, Symplectic Structures - a new approach to geometry, Notices
A.M.S. 45 (1998), 952–960.
D. McDuff, Symplectic embeddings and continued fractions: a survey, Jpn. J.
Math. 4 (2009), 121–139.
D. McDuff, ”What is Symplectic Geometry?”, in Hobbs, Catherine; Paycha,Sylvie, European Women in Mathematics. Proceedings of the 13th General
Meeting, World Scientific, pp. 33–51, ISBN 9789814277686, (2010)