Tony’s Take September 2021

This month’s topics:

5D topology in photonic metamaterials

Magnetic monopoles (e.g. a positive pole with no negative) have been studied seriously as theoretical objects since 1931 when Paul Dirac proved their link to the quantization of electric charge. They are of great mathematical interest because when restricted to a small sphere about the monopole, the magnetic vector potential forces the set of phases of charged particles on the sphere to have the structure of a circle bundle with (first) Chern number $C_1=\pm 1$, the sign depending on orientations. This bundle, discovered by Heinz Hopf (also in 1931) and now called the “Hopf Bundle,” is a foundational object in algebraic and differential topology. Its total space (the union of its fibers) turns out to be a 3-dimensional sphere.

Hopf bundle
This still from a video by Nils Johnson, used with permission, shows part of the Hopf Bundle. The 3-spherical total space is represented as ${\bf R}^3$ with a point at infinity. A point on the sphere is color-matched with the corresponding fiber —in the monopole, this is the circle of possible phases of charged particles at that point. Towards the bottom of the sphere, the fibers are converging to the single circle at the core of the innermost torus; towards the top, to the straight vertical line representing the circle through infinity. Note that any two fibers are simply linked, corresponding to $C_1=\pm 1$.

In 1978 the physicist C. N. Yang published a “generalization of [the magnetic] monopole to $SU_2$ gauge fields.” The idea was to substitute for the 1-dimensional circle group relating phases the 3-dimensional symmetry group $SU_2$. (The complex analogue of the circle, this is is the group of rotations of complex 2-dimensional space.) The “Yang monopole,” as it came to be called, lives in 5-dimensional space.

This year Mirage news service and Science Daily picked up a press release (PDF) from the University of Hong Kong, “HKU Physicists and collaborators co-observe a higher-dimensional topological state with metamaterials,” dated August 26. The research in question was published in Science on July 30; the authors are a team of eight led by Shuang Zhang of the HKU Physics and Electric and Electronic Engineering Departments. As they report in their abstract, “We constructed a system possessing Yang monopoles and Weyl surfaces based on metamaterials with engineered electromagnetic properties, leading to the observation of several intriguing bulk and surface phenomena, such as linking of Weyl surfaces and surface Weyl arcs, via selected three-dimensional subspaces.”

The “Weyl surfaces” ${\mathcal M}_1$ and ${\mathcal M}_2$ are linked two-dimensional surfaces in 5-space associated with the Yang monopole. In 3-dimensional sections of the configuration, they appear as linked 1 spheres (circles; top and bottom center), as a 2-sphere linked with a 0-sphere (2 points, one here at infinity; bottom left) and as a 0-sphere linked with a 2-sphere (closing up at infinity; bottom right). From an image by Shaojie Ma, used with permission.

An analogy with the magnetic monopole: “This nonzero linking number between the projected ${\mathcal M}_1$ and ${\mathcal M}_2$ reveals the nontrivial $C_2$ of the Weyl surfaces in the 5D space.” Just as circle bundles over a surface are characterized by their first Chern number $C_1$, principal $SU_2$-bundles over a 4-dimensional surface are characterized by their second Chern number $C_2$. The Yang monopole has $C_2=\pm 1$ depending on orientations, corresponding to the simple linking observed between the Weyl surfaces.

Dynamics of topological defects on cell membranes

“Topological braiding and virtual particles on the cell membrane” appeared in PNAS on August 20, 2021. As the authors, a team of seven from MIT, Harvard, and the Flatiron Institute, explain at the start, “Combining direct experimental observations with mathematical modeling and chemical perturbations, we investigate the dynamics of spiral wave defects on the surfaces of starfish egg cells.” They continue: “To investigate the braiding dynamics of biochemical spiral waves in living cells, we compared here experimental observations of Rho-GTP activation waves on starfish oocyte membranes with predictions of a generic continuum theory. Rho-GTP is a highly conserved signaling protein pivotal in regulating cellular division and mechanics across a wide variety of eukaryotic species.”

oocyte evolution
“Time evolution of chemical Rho signaling wave patterns on the starfish oocyte from a homogeneous initial state to a quasi-steady state exhibiting turbulent spiral patterns (Scale bar: 40 $\mu$m.)” The full movie shows the spirals rotating inward (clockwise or counterclockwise) as they grow and interact. Image from PNAS118 e2104191118, used according to PNAS License to Publish.

The authors report: “Topological defects in the phase field are singular points with winding number +1 or −1 corresponding to counterclockwise or clockwise rotating centers of propagating spiral waves. These phase defects are created and annihilated in pairs, conserving the total topological charge. By tracking the 2 + 1-dimensional world lines of both defect types, we observed complex creation, annihilation, and braiding dynamics, similar to those in Bose−Einstein condensates. … In addition to short-lived loops which dominate at high activity, low-activity states exhibit a large number of long-lived defect world lines that undergo spontaneous braiding dynamics. Space−time braiding of spiral cores is indicative of chaotic dynamics of the Rho-GTP signaling patterns.”

Braiding of defect world lines. Blue lines correspond to clockwise rotating spiral cores; red lines to counterclockwise. Space-bars: 10$\mu$m, time 90s. Image courtesy of Jinghui Liu.

“As a topological measure of complexity in dynamical systems, braiding analysis has the advantage that it is well grounded in group theory. Mathematically, a sequence of braiding history between particles can be treated as a series of sequentially multiplied generators, where each generator denotes the direction of ‘crossing’ between one particle and its neighbors projected onto a reference line at an instantaneous time. Analyzing such product of generators as a function of time then gives a measurement of complexity growth in the system.” Specifically, they represent each of the generators by an $(n-1)\times(n-1)$ matrix and take $\Sigma_n(t)$ as the product of the matrices corresponding to the crossings up to time $t$. “The matrix product $\Sigma_n(t)$ records the braiding history of particles and therefore contains information about the system dynamics. One important piece of information is the magnitude of its largest eigenvalue, $E_n(t)$, often termed as the braiding factor. In random matrix theory, the exponential growth rate of $E_n(t)$ at long time limit approximates the Lyapunov exponent of a chaotic system, which has also been verified in numerical experiments. Such exponential growth rate is therefore termed as the braiding exponent,
For our two-dimensional defect trajectories, the braiding factor calculated from taking the average of all reference line projections … displayed consistently positive braiding exponents $\lambda(n)$.”

Braiding of defect world lines forces mixing of the surrounding surface elements (time increases vertically). Image courtesy of Jan Totz.

Ingrid Daubechies in the N. Y. Times Magazine

Siobhan Roberts contributed a long profile of the mathematician Ingrid Daubechies to the September 19, 2021 edition of the New York Times Magazine. “A professor at Duke University, in Durham, N.C., Daubechies’ métier is figuring out optimal ways to represent and analyze images and information. The great mathematical discovery of her early career, made in 1987 when she was 33, was the ‘Daubechies wavelet.’ Her work, together with further wavelet developments, was instrumental to the invention of image-compression algorithms, like the JPEG2000, that pervade the digital age.”

What are wavelets? Roberts tries to give us some idea. First, waves. “Daubechies says […] ‘You can build almost anything by combining, in clever ways, waves of different wavelengths.’ This idea dates back two centuries: In 1822, the French physicist and mathematician Joseph Fourier […] proposed that all periodic functions — all periodic phenomena — could be understood as sums of sine and cosine waves. […] But this approach had its limitations: It couldn’t efficiently handle signals with abrupt changes, like spoken language or pictures with sharp edges and sudden transitions in luminosity.”

Then wavelets, a 20th-century innovation. “Sometimes Daubechies gives a fancifully impractical musical metaphor to describe the difference. For Fourier analysis, she envisions a room full of thousands of idealized tuning forks, each sustaining a uniquely assigned note indefinitely. […] Wavelets, by contrast, are a more sophisticated symphony orchestra of tuning forks that each ring for a shorter time. They can, in a manner of speaking, read and convey all the information contained in the musical score: information about tempo and note duration, and about even more granular nuances of musicality, like […] the attack at the start of a note, or the purity of tone held for bars at a time. ‘With wavelets you can decompose all that in an efficient way,’ Daubechies says.”

Mathematically speaking, Daubechies wavelets are a family of functions db1, db2, etc. that play a role (as suggested above) similar in some ways to the families $\sin x, \sin 2x$, etc. and $1, \cos x, \cos 2x$, etc., which can be combined, with the right coefficients, to represent any function of period $2\pi$. Here are the first few:

first four Daubechies wavelets
The first four Daubechies wavelets (the first one is actually the original wavelet invented by Alfréd Haar in 1909). Note that each of them is zero outside of a finite interval and that the number of relative maxima (roughly analogous to a frequency) increases with the index. Also note that while db1 is discontinuous, db2 is continuous but ragged and from then on the graphs get smoother as the index increases.

Each of the wavelets generates a bi-infinite family of its own through horizontal scaling by powers of 2 and shifting by integer lengths. All these functions together are orthogonal (the product of any two integrates to zero), just like the sines and cosines, and a judicious finite linear combination of them can give an excellent and efficient approximation of of one-dimensional signals as diverse as seismograph records or speech (at this level of explanation, the signal needs to have first been adjusted to have average value zero). Their extension to two dimensions has become essential in compressing photographic images and movies so that they can be stored and transmitted efficiently.

Roberts continues, “Daubechies is most famous as a pioneer of wavelets, but more broadly, her scientific contributions over the last three decades have rippled out in all directions from the field of ‘signal processing.’ […] Jordan Ellenberg, a mathematician at the University of Wisconsin-Madison […], points out that signal processing ‘makes up a huge proportion of applied math now, since so much of applied math is about the geometry of information as opposed to the geometry of motion and force.'” Roberts mentions in particular Daubechies’s recent participation in the restoration of The Adoration of the Mystic Lamb (the Ghent Altarpiece—closed and open), “a 15th-century polyptych attributed to Hubert and Jan van Eyck, arguably among the most important paintings in history.”

Much of the profile covers Daubechies’s life and personality. One striking detail: Daubechies had her first baby in 1988. “It was an unsettling and disorienting period, because she lost her ability to do research-level mathematics for several months postpartum. ‘Mathematical ideas wouldn’t come,’ she says. That frightened her. She told no one, not even her husband, until gradually her creative motivation returned. On occasion, she has since warned younger female mathematicians about the baby-brain effect, and they have been grateful for the tip.”

Plans for the future? “Machine learning’s success […] is something that Daubechies believes mathematicians and mathematically inclined scientists should attend to more. ‘Machine learning works very well, and we don’t know why it works so well,’ she says. ‘I consider that a challenge for mathematicians, to understand it better.’ […] Usually, the argument is that beautiful, pure mathematics eventually — in a year, in a century — produces compelling applications. Daubechies believes that the cycle also turns in the opposite direction, that successful applications can lead to beautiful, pure mathematics. Machine learning is a promising example. ‘You can’t argue with success,’ she says. ‘I believe if something works, there is a reason. We have to find the reason.'”