Clayton Shonkwiler PRO
Mathematician and artist
/usna21
this talk!
Symplectic geometry can sound scary, but it turns out to be useful in some surprising contexts.
—Dusa McDuff & Dietmar Salamon
Introduction to Symplectic Topology
—Ana Cannas da Silva
A nice space, where multivariable calculus makes sense. Think spheres, tori, projective spaces, …
A rule for assigning a number to pairs of tangent vectors which transforms like (signed) area of the parallelogram they span.
Must be closed (think: divergence-free) and non-degenerate (doesn’t annihilate any vector).
Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)
Non-Example: \((\mathbb{R}^3, dx \wedge dy)\)
For any vector \(v\), \(dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0\), so \(dx \wedge dy\) is degenerate.
Non-Example: \((\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)\)
Because of the coefficient \(y_1\), this is not closed (“not divergence-free”).
\((S^2,d\theta\wedge dz)\)
\((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)
\((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)
\((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)
\((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)
\((\mathbb{C}^{m \times n}, \omega)\) with \(\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)\).
\((T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)\)
phase space
position
momentum
\(\omega^{\wedge n} = \omega \wedge \dots \wedge \omega\) is a volume form on \(M\), and induces a measure
called Liouville measure on \(M\).
In particular, if \(M\) is compact, this can be normalized to give a probability measure.
Theorem [Archimedes]
\(\mathrm{d}\theta \wedge \mathrm{d}z\) is the standard area form on the unit sphere \(S^2\).
KoenB [ ] from Wikimedia Commons
“Evolution of a physical system follows the symplectic gradient”
If \(H: M \to \mathbb{R}\) is smooth, then there exists a unique vector field \(X_H\) so that \({dH = \iota_{X_H}\omega}\), i.e.,
(\(X_H\) is called the Hamiltonian vector field for \(H\), or sometimes the symplectic gradient of \(H\))
Example. \(H: (S^2, d\theta\wedge dz) \to \mathbb{R}\) given by \(H(\theta,z) = z\).
\(dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)\), so \(X_H = \frac{\partial}{\partial \theta}\).
\(H\) is constant on orbits of \(X_H\):
\(\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0\)
Suppose we have a particle of mass \(m\) with position \(\vec{q} = (q_1,q_2,q_3)\).
Suppose the particle is subject to a potential \(V(\vec{q})\).
Newton’s Second Law: \(-\nabla V(\vec{q}) = m \frac{d^2 \vec{q}}{dt^2}\).
Momenta \(p_i := m \frac{dq_i}{dt}\).
\((\vec{q},\vec{p})\in T^*\mathbb{R}^3 \cong \mathbb{R}^6\); canonical symplectic form \(\omega = dq_1 \wedge dp_1 + dq_2 \wedge dp_2 + dq_3 \wedge dp_3\).
Energy function (or Hamiltonian)
\(H(\vec{q},\vec{p}) := \frac{1}{2m}|\vec{p}|^2 + V(\vec{q})\).
\(dH = \sum\left(\frac{\partial H}{\partial p_i} dp_i + \frac{\partial H}{\partial q_i}dq_i\right)\),
\((\vec{q}(t),\vec{p}(t))\) is an integral curve for \(X_H\) if and only if
\(\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{1}{m} p_i\)
\(\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial V}{\partial q_i}.\)
This is Newton’s Second Law written as a first-order system!
\(X_H = \sum\left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)\).
“Every continuous symmetry has a corresponding conserved quantity”
A circle action on \((M,\omega)\) determines a vector field \(X\) by
\(S^1=U(1)\) acts on \((S^2,d\theta \wedge dz)\) by
So \(X = \frac{\partial}{\partial \theta}\).
Definition. A circle action on \((M,\omega)\) is Hamiltonian if there exists a map
so that \(d\mu = \iota_{X}\omega = \omega(X,\cdot)\), where \(X\) is the vector field generated by the circle action. In other words, \(X = X_\mu\).
\(X = \frac{\partial}{\partial \theta}\)
\(\mu(\theta,z) = z\)
\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = (d\theta \wedge dz)\left(\frac{\partial}{\partial \theta},\cdot \right) = dz \)
Theorem [Atiyah and Guillemin–Sternberg]
If \((M,\omega)\) is symplectic and \(\mathbb{T}=(S^1)^k\) acts in a Hamiltonian fashion, the conserved quantities are recorded by \(\mu:M \to \mathbb{R}^k\) with:
Theorem [Duistermaat–Heckman]
If \(\dim(M)=2n\) and \(\dim(\mathbb{T})=k\), then the pushforward measure on the moment polytope \(\mu(M)\) is absolutely continuous w.r.t. Lebesgue measure, with continuous, piecewise-polynomial density of degree \(\leq n-k\).
In particular, if \(n=k\), then the pushforward measure is a constant multiple of Lebesgue measure.
Theorem (Archimedes)
Let \(\mu: S^2 \to \mathbb{R}\) be given by \(\mu(x,y,z) = z\). Pushing forward the uniform measure on \(S^2\) to the image \([-1,1]\) gives Lebesgue measure.
Proposition. If we rotate both factors of \(S^2 \times S^2\) simultaneously around the \(z\)-axis, the moment map is \(((x_1,y_1,z_1),(x_2,y_2,z_2))\mapsto z_1 + z_2\), and the pushforward measure on \([-2,2]\) has density given by the triangle function.
The \(\mathbb{T}^2 = U(1) \times U(1)\) action on \(\mathbb{CP}^2\) given by
has moment map
In solution, linear polymers become crumpled:
Protonated P2VP
Roiter–Minko, J. Am. Chem. Soc. 127 (2005), 15688-15689
[CC BY-SA 3.0], from Wikimedia Commons
Modern polymer physics is based on the analogy between a polymer chain and a random walk.
Continuous symmetry \(\Rightarrow\) conserved quantity
The space of equilateral \(n\)-gons has lots of symmetries...
Theorem [with Cantarella]
The standard volume form on polygon space is
\(\mathrm{d}\theta_1 \wedge \dots \wedge \mathrm{d}\theta_{n-3} \wedge \mathrm{d} r_1 \wedge \dots \wedge \mathrm{d}r_{n-3}\)
Corollary [with Cantarella, Duplantier, Uehara]
The first efficient, provably correct algorithm for sampling equilateral random polygons in \(\mathbb{R}^3\).
Signal: \(v \in \mathbb{C}^d\)
Design: \(f_1, \dots , f_n \in \mathbb{C}^d\)
Measurements: \(\langle f_1, v \rangle, \dots , \langle f_n, v \rangle \).
Parseval’s Theorem
If \(f_1, \dots , f_d \in \mathbb{C}^d\) is an orthonormal basis,
\(\|v\|^2 = \sum |\langle f_i, v \rangle |^2 = \|F^* v\|^2\)
for any \(v \in \mathbb{C}^d\).
Even better,
\(v = \sum \langle f_i, v \rangle f_i = FF^* v \).
If \(F = [f_1 \, f_2 \dots f_n]\), the measurement vector is \(F^*v\).
\(\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}\)
Parseval frame
Fact: For any \(n \geq d\), there exist Parseval frames \(f_1, \dots , f_n \in \mathbb{C}^d\) so that \(\|f_i\|=\|f_j\|\) for all \( i,j\); these are the equal-norm Parseval frames (ENPs).
This is fragile! What if a measurement gets lost?
Some nice group actions on \(\mathbb{C}^{d \times n}\):
Parseval frames
\(\mu_{U(d)}^{-1}(I_{d\times d})\)
equal-norm frames
\(\mu_{U(1)^N}^{-1}\left(-\frac{1}{2}r,\dots , -\frac{1}{2}r\right)\)
ENPs: \(\mu_{U(d)}^{-1}(I_{d \times d}) \cap \mu_{U(1)^n}^{-1}\left(-\frac{d}{2n}, \dots , -\frac{d}{2n}\right)\)
Theorem [with Needham]
There is a continuous interpolation between any two frames \(f_1, \dots , f_n\) and \(g_1, \dots , g_n\) with \(FF^* = GG^*\) and \(\|f_i\| = \|g_i\|\) preserving these conditions.
It is often desirable to require every minor of the \(d \times N\) matrix \(F\) to be invertible. Such an \(F\) is said to have full spark.
Theorem [with Needham]
Fix the spectrum of \(FF^\ast\) and fix \(\|f_1\|,\dots,\|f_N\|\). There are three possibilities:
Funding: National Science Foundation (DMS–2107700) and Simons Foundation
Symplectic geometry and connectivity of spaces of frames
Tom Needham and Clayton Shonkwiler
Advances in Computational Mathematics 47 (2021), no. 1, 5
The symplectic geometry of closed equilateral random walks in 3-space
Jason Cantarella and Clayton Shonkwiler
Annals of Applied Probability 26 (2016), no. 1, 549–596
A fast direct sampling algorithm for equilateral closed polygons
Jason Cantarella, Bernard Duplantier, Clayton Shonkwiler, and Erica Uehara
Journal of Physics A: Mathematical and Theoretical 49 (2016), no. 27, 275202
Toric symplectic geometry and full spark frames
Tom Needham and Clayton Shonkwiler
By Clayton Shonkwiler