2020-2021 Department Colloquia

Spring 2021

Stacking van der Waals atomic layers: quest for new quantum materials

April 27, 2021

Philip Kim, Harvard University
Host: Brian Swingle

Abstract: Modern electronics heavily rely on the technology to confine electrons in the interface layers of semiconductors. In recent years, scientists discovered that various atomically thin van der Waals (vdW) layered materials can be isolated. In these atomically thin materials, quantum physics allows electrons to move only in an effective 2-dimensional (2D) space. By stacking these 2D quantum materials, one can also create atomic-scale heterostructures with a wide variety of electronic and optical properties. We demonstrate the enhanced electronic and optoelectronic performances in the vdW heterostructures, suggesting that these a few atom thick interfaces may provide a fundamental platform to realize novel physical phenomena. In this talk, we will discuss several research efforts to realize unusual quasiparticle pairing mesoscopic devices based on stacked vdW interfaces between 2-dimensional materials.

Electric Multipole Insulators

April 13, 2021

Taylor Hughes, University of Illinois, Urbana-Champaign
Host: Brian Swingle

Abstract: In this talk I will present a general framework to distinguish different classes of charge insulators based on whether or not they insulate or conduct higher multipole moments (dipole, quadrupole, etc.). This formalism applies to generic many-body systems that support multipolar conservation laws. Applications of this work provide a key link between recently discovered higher order topological phases and fracton phases of matter.

Astrophysical Black Holes

April 6, 2021

Ramesh Narayan, Harvard University
Host: Brian Swingle

Abstract: A black hole is an object which is so compact, and whose gravitational pull is so strong, that not even light can escape from its interior. The concept of a black hole is very bizarre and one feels that something in physics ought to prevent such objects from forming. But on the contrary, the universe actually contains countless numbers of black holes; some weigh only a few or few tens of solar masses, while others weigh millions to billions of solar masses. Even though light cannot escape from inside a black hole, gas flowing into the hole often emits intense electromagnetic radiation before falling in. Astronomers study this radiation to infer a variety of information about the underlying systems. In favorable cases, using millimeter-wave interferometers, they are able even to take close-up pictures of black holes. This talk will describe the kinds of black holes that have been discovered in the universe, their observational manifestations, and what they teach us about physics in the regime of strong gravity.

Universality and quantum criticality far from equilibrium

March 30, 2021

Romain Vasseur, University of Massachusetts, Amherst
Host: Brian Swingle

Abstract: Interacting many-body quantum systems often exhibit chaotic dynamics that rapidly ``scramble’’ quantum information and lead to highly entangled states whose local properties are thermal and classical. At long times, most many-body quantum systems reach thermal equilibrium on their own. In this talk, I will discuss recently discovered exceptions to this rule, and focus on systems in which thermalization can be prevented either by many-body localization, integrability or by local projective measurements. I will discuss the new types of non-equilibrium phases and phase transitions that can arise in such systems, with an emphasis on the emergence of universality far from thermal equilibrium.

Scaling down the laws of thermodynamics

March 23, 2021

Chris Jarzynski, University of Maryland
Host: Brian Swingle

Abstract: Thermodynamics provides a robust conceptual framework and set of laws that govern the exchange of energy and matter. Although these laws were originally articulated for macroscopic objects, nanoscale systems also exhibit “thermodynamic¬-like” behavior – for instance, biomolecular motors convert chemical fuel into mechanical work, and single molecules exhibit hysteresis when manipulated using optical tweezers. To what extent can the laws of thermodynamics be scaled down to apply to individual microscopic systems, and what new features emerge at the nanoscale? I will describe some of the challenges and recent progress – both theoretical and experimental – associated with addressing these questions. Along the way, my talk will touch on non-equilibrium fluctuations, “violations” of the second law, the thermodynamic arrow of time, nanoscale feedback control, strong system-environment coupling, and quantum thermodynamics.

The LHC’s Next Frontier: Searching for Pairs of Higgs Bosons to Understand the Standard Model and Beyond

March 8, 2021

Maximilian Swiatlowski, TRIUMF
Host: Gabriella Sciolla

Abstract: The discovery of the Higgs boson at the Large Hadron Collider completed the Standard Model, but many fundamental open questions remain. One question is particularly simple: how could the Big Bang produce the matter-dominated universe we observe without anti-matter, which should have been produced in equal parts? As the LHC switches into its High-Luminosity phase, the huge datasets the ATLAS experiment will collect can provide answers to this question, and others, by measuring the extremely rare production of pairs of Higgs bosons. Though difficult to observe, these signatures can directly measure the shape of the Higgs potential: deviations from the Standard Model's expectations could allow us to understand not just the history of the early universe that created the matter/anti-matter asymmetry, but questions like the future stability of the universe. This colloquium will focus on the challenges to detecting Higgs boson pairs, and how to interpret them to understand the shape of the Higgs potential and consequences for physics beyond the Standard Model. I will also describe how ATLAS is utilizing new developments in detector upgrades and machine learning to improve these measurements and to make the most of the data in the face of unprecedented experimental conditions.

Dismantling the Nuclear Doomsday Machine: Science, Technology and Policy Challenges

March 2, 2021

Zia Mian, Princeton University
Host: Brian Swingle

Abstract: Eighty years ago physicists first worked out what it would take to build a simple nuclear weapon and the immediate effects of its use. Five years later, theory and experiment became devastating facts as the United States built, tested, and then used the first nuclear weapons to destroy cities. Today, there are nine nuclear armed states and over 13,000 nuclear weapons in the world. The nuclear-armed states are developing or modernizing their arsenals, easing constraints on when these weapons might be used, and pursue policies that risk accidental nuclear war. The hard-won international arrangements intended to halt, reverse and eventually eliminate nuclear weapons programs are unraveling. One potentially hopeful development, the 2017 United Nations Treaty on the Prohibition of Nuclear Weapons, has elicited opposition rather than support from these states. This talk will look at what role scientists have played in the past and can play today in helping address the challenge of reducing and eliminating the threat from nuclear weapons in the United States and globally, it also will introduce the new Physicists Coalition for Nuclear Threat Reduction.

Fracton-elasticity duality

February 23, 2021

Leo Radzihovsky, University of Colorado
Host: Brian Swingle

Abstract: I will discuss a recent discovery that elasticity of a two-dimensional quantum crystal is dual to an unusual gauge theory, thereby providing a physical realization of gapless “fractonic” quantum order. Restricted-mobility of topological defects in a crystal map onto fractonic charges of the gauge theory. This duality leads to predictions of fractonic phases and quantum Higgs-like phase transitions to their descendants, that are duals of the quantum commensurate crystal, supersolid, smectic, and hexatic liquid crystals. Extensions of this duality to generalized elasticity theories provide a route to discovery of new fractonic models and their potential experimental realizations.

Tension in the Hubble Constant: Is There a Crisis in Cosmology?

February 16, 2021

Wendy Freedman, University of Chicago
Host: Brian Swingle

Abstract: One of the most exciting questions in cosmology today is whether there is new physics that is missing from our current standard Lambda Cold Dark Matter (LCDM) model. A current discrepancy in the measurement of the Hubble constant could be signaling a new physical property of the universe or, more mundanely, unrecognized measurement uncertainties. I will discuss two of our most precise methods for measuring distances in the local universe: Cepheids and the Tip of the Red Giant Branch (TRGB). I will present new results from the Carnegie-Chicago Hubble Program (CCHP), the goal of which is to independently measure a value of the Hubble constant to a precision and accuracy of 2%. Using the Hubble Space Telescope Advanced Camera for Surveys, we are using the TRGB to calibrate Type Ia supernovae. Our value of the Hubble constant, Ho = 69.6 +/- 0.8 (statistical) +/- 1.7 (systematic) km/sec/Mpc, falls midway between the value obtained from the Planck Cosmic Microwave Background analysis, and that obtained using Cepheids. I will address the uncertainties, discuss the current tension in Ho, and whether there is need for additional physics beyond the standard LCDM model.

Berko Symposium

February 9, 2021

Ian Hunter, Brandeis University
Host: John Wardle

"Experimental Equivariant Dynamics in a Network of Reaction-Diffusion Oscillators"

Abstract: Does form follow function in natural coupled-oscillator networks? Symmetry controls both the steady-state and the transient spatiotemporal patterns that form in mathematically ideal networks to a remarkable degree. But what happens in the real-world networks, with imperfections in their nodes and connections? To address these questions, we developed a model experimental reaction-diffusion network formed by an oscillatory chemical reaction confined to a square symmetric ring of 4 diffusively coupled microfluidic reactors. We compared experimental dynamics to theory assuming perfect symmetry and theory incorporating slight heterogeneity. We observed that even slight heterogeneity selectively modifies and eliminates some patterns, while preserving others. This work demonstrates that a surprising degree of the natural network’s dynamics are constrained by symmetry in spite of the breakdown of the assumptions of perfect symmetry and raises the question of why heterogeneity destabilizes some symmetry predicted states, but not others.

 Zach Schillaci, Brandeis University
Host: John Wardle

"Searching for Hints of New Physics in Four-lepton Events with the ATLAS Detector"

Abstract: The Run 2 operation of the Large Hadron Collider (2015-2018) provided an enormous dataset of √s = 13 TeV proton-proton collisions, corresponding to an integrated luminosity of 139 fb−1 delivered to the ATLAS detector. This large dataset allows for the study of very rare Standard Model processes and, with that, the possibility of discovering new physics if deviations from state-of-the-art theoretical predictions are observed. The four-lepton final state exhibits a complex structure with contributions from several Standard Model processes, and possible contributions from physics beyond the Standard Model. Moreover, the final state, which consists of two same-flavor, opposite-charge lepton pairs, provides a clean detector signature with relatively limited backgrounds, making it an ideal candidate for probing the theory with high precision. The analysis presented here measures the four-lepton differential and integrated fiducial cross-sections within an inclusive phase-space targeting all production modes. Observed detector yields are corrected for inefficiency and resolution effects, and compared to leading Standard Model calculations, which are found to be consistent with the observations. The Z → 4l branching fraction is extracted, yielding the most precise measurement to date of (4.41 ± 0.30)×10−6. Constraints are placed on possible BSM scenarios, including a model based on a spontaneously broken baryon-minus-lepton-number (B-L) gauge symmetry, and further re-interpretations are made possible through the open publication of all relevant measurements.

Boson Collider for Stress Testing the Standard Model of Elementary Particles

February 2, 2021

Aram Apyan, Fermi National Accelerator Laboratory
Host: Gabriella Sciolla

Abstract: The standard model of elementary particles has been extremely successful in describing the building blocks of matter and their interactions. The observation of a Higgs boson at the Large Hadron Collider in 2012 presented a new paradigm of precision physics aiming at the investigation of the Higgs mechanism. Since that time, many decay modes of the Higgs boson have been observed, indicating Higgs boson properties in good agreement with the simplest theoretical expectations. In this colloquium I will discuss a novel and powerful way of stress testing the standard model and studying the Higgs boson properties in the collisions of heavy vector bosons. My presentation will be illustrated with the newest results obtained from data collected at the Large Boson Collider.

Hunting for new physics using long-lived particles with ATLAS at the LHC

January 27, 2021

Please note that this colloquium takes place at the special date and time of 10:00am on a Wednesday.

Katherine Pachal, Duke University
Host: Gabriella Sciolla

Abstract: We have a great mathematical model for how the known elementary particles interact, but there are phenomena such as dark matter that it doesn't explain. New particles light enough to be created at the LHC could resolve many of these issues. We have been looking in LHC data for dark matter and other new particles for years, and found nothing, but we have been looking in the most obvious places. There are equally plausible but harder to access corners where we are only starting to search. The most interesting involve "long-lived" particles (LLPs): particles that can travel a measurable distance in the detector before decaying to dark matter and/or other objects. Our search program looking for LLPs is still young and growing, with many places where investment in technical improvements could pay off with big leaps in discovery potential. This colloquium will discuss the methods and challenges involved in doing these searches and what makes them the most exciting targets for the LHC Run 3.

Searching for the Higgs Boson decaying to Dark Matter at the LHC

January 25, 2021

Please note that this colloquium takes place at the special date and time of 10:00am on a Monday.

Douglas Schaefer, University of Chicago
Host: Gabriella Sciolla

Abstract: To date the Standard Model (SM) of particle physics has proven incredibly accurate in its predictions, but it falls short in its ability to account for a vast portion of non-electromagnetically interacting mass in our Universe. The numerous astrophysical observations that corroborate the existence of this non-interacting dark matter (DM) point to the need for a theory Beyond the SM (BSM) to explain its abundance. The discovery of the Higgs boson provides a new particle that couples to mass through Yukawa couplings, which might include new undiscovered DM particles through BSM physics. I will present the latest search results for the Higgs boson coupling to DM from the ATLAS Collaboration using 139 fb-1 of 13 TeV center-of-mass pp collision data. I will also show comparisons to direct-detection experiment results for DM cross-sections as well as future prospects for Higgs to invisible searches.

Quantum Sensors for Direct Detection of Light Dark Matter

January 21, 2021

Please note that this colloquium takes place at the special date and time of 10:00am on a Thursday.

Noah Kurinsky, Fermi National Accelerator Laboratory
Host: Gabriella Sciolla

Abstract: The identification of dark matter (DM) is a central question for high-energy physics and one of the strongest motivations for physics beyond the standard model. In light of the lack of evidence for supersymmetry at the LHC, theoretical models for particle DM have veered towards ideas like a ‘dark sector’, where there may be many different light DM particles, including some that mediate the interaction with normal matter. All of these light DM candidates would deposit meV-scale to eV-scale energies in an earth-bound detector. The challenge for experiments is to develop detectors that can measure the sub-eV energy deposits produced by such potential DM interactions. In this talk, I will discuss the roadmap for detector R&D down to meV energy thresholds for background-free, gram-day exposures, in the context of the experiments I am currently involved in. These include a broadband axion search in the 0.1-10 THz energy range, and the expansion of phonon-mediated detectors to sub-eV thresholds beyond my current SuperCDMS HVeV program.

This work also has broad synergies with understanding qubit decoherence times for superconducting quantum computing, which is the current limitation on complexity of modern quantum computers. I will touch on recent results which relate efficient phonon detection for radiation detection to qubit responses to external radioactivity and near-term plans at Fermilab to continue these studies in NEXUS and QUIET, our low-background test facilities. The R&D for radiation-hardened qubit architectures goes hand in hand with massively multiplexed phonon sensors, and I will discuss how both dark matter and quantum science can be studies in the same experimental infrastructure.

Dark Matter Photography with DAMIC

January 19, 2021

Please note this colloquium takes place at a special time of 10:00 am.

Daniel Baxter, University of Chicago
Host: Gabriella Sciolla

Abstract: The DAMIC experiment exploits the exceptional energy and spatial resolution of charge coupled devices (CCDs) to search for dark matter particles scattering in silicon. I present the latest results from the 11 kg-day exposure collected with the DAMIC experiment at SNOLAB, which includes major improvements in the understanding of bulk and surface backgrounds down to energies as low as 50 eV. In addition to producing competitive limits on dark matter scattering, DAMIC finds a small excess near threshold over the background prediction. DAMIC-M will build on the success of DAMIC at SNOLAB to further probe this excess of events using lower threshold Skipper CCDs, lower backgrounds, and increased exposure. The first low background prototype detector for DAMIC-M will produce world-leading limits on dark matter scattering within the next year.

 

Fall 2020

Discovery through the lens of the computational microscope

December 1, 2020

Juan Perilla, University of Delaware
Host: Michael Hagan

Abstract: The essential conundrum of modern biology, namely the question of how life emerges from myriad molecules whose behavior is governed by physical law alone, is embodied within a single cell—the quantum of life. The rise of scientific supercomputing has allowed for the study of the living cell in unparalleled detail, from the scale of the atom to a whole organism and at all levels in between. In particular, the past three decades have witnessed the evolution of molecular dynamics (MD) simulations as a “computational microscope”, which has provided a unique framework for the study of the phenomena of cell biology in atomic (or near-atomic) detail. Here I present an overview of our efforts to determine the molecular details during the life-cycle of multiple infectious diseases using the computational microscope.

The mechanics of robust tissue folding

November 23, 2020

Hannah Yevick, Massachusetts Institute of Technology
Host: Jané Kondev

Abstract: Tissue folding is a ubiquitous shape change event during development whereby a cell sheet bends into a curved 3D structure. In this talk, I will show that this mechanical process is remarkably robust, and the correct final form is almost always achieved despite internal fluctuations and external perturbations inherent in living systems. While many genetic and molecular strategies that lead to robust development have been established, much less is known about how mechanical patterns and movements are ensured at the population level. I will describe how quantitative imaging, physical modeling and concepts from network science can uncover collective interactions that govern tissue patterning and shape change.

Actin and myosin are two important cytoskeletal proteins involved in the force generation and movement of cells. Both parts of this talk will be about the spontaneous organization of actomyosin networks. First, I will present how out-of-plane curvature can trigger the global alignment of actin fibers and a novel transition from collective to individual cell migration in culture. I will then describe how tissue-scale cytoskeletal patterns can guide tissue folding in the early fruit fly embryo. I will show that actin and myosin organize into a network that spans a large region of the embryo. Redundancy in this supracellular network encodes the tissue’s intrinsic robustness to mechanical and molecular perturbations during folding. Moving forward, deciphering the physical basis of robustness in living systems will inspire new ways to engineer and control biological tissues and bioinspired active materials that can spontaneously fold into predetermined complex 3D shape.​

Search for Non-Abelian Majorana particles as a route to topological quantum computation

November 17, 2020

Jay Deep Sau, University of Maryland
Host: Brian Swingle

Abstract: Majorana zero modes are fermion-like excitations that were originally proposed in particle physics by Ettore Majorana and are characterized as being their own anti-particle. In condensed matter systems Majorana zero modes occur as fractionalized excitations with topologically protected degeneracy associated with such excitations. For over a decade the only candidate systems for observing Majorana zero modes were the non-Abelian fractional quantum Hall state and chiral p-wave superconductors. In this talk, I will discuss a recent set of proposals for realizing Majorana zero modes in a large class of spin-orbit coupled, time-reversal symmetry broken superconducting systems. The simplicity of this class of systems has resulted in several experimental attempts, which have successfully observed preliminary evidence for the Majorana zero modes in the form of zero-bias conductance peaks and the fractional Josephson effect. Following this I will review recent experimental progress on spin-orbit coupled superconductors. I will end by describing what a topological quantum computer based on Majorana modes might look like in the future.

Dynamical Phase Transitions in Cold Atomic Gases

November 10, 2020

Ana Maria Rey, University of Colorado, Boulder
Host: Brian Swingle

Abstract: Non-equilibrium quantum many-body systems can display fascinating phenomena relevant for various fields in science ranging from physics, to chemistry, and ultimately, for the broadest possible scope, life itself. The challenge with these systems, however, is that the powerful formalism of statistical physics, which have allowed a classification of quantum phases of matter at equilibrium does not apply. Precisely engineered ultracold gases are emerging as a powerful tool to shed light on the organizing principles and universal behaviors of dynamical quantum matter. One emerging paradigm where ultracold atomic systems have shown to be a powerful resource is the case of dynamical phase transitions (DPTs). DPTs share many aspects of standard equilibrium phase transitions but differ in many fundamental ways with them. They also allow for the generation of metrologically useful steady states. I will discuss the observation of DPTs in different but complementary systems: trapped quantum degenerate Fermi gases, trapped bosonic gases and long lived arrays of atoms in an optical cavity. I will show how these systems can be used to simulate iconic models of quantum magnetism with tunable parameters and to probe the dependence of their associated dynamical phases on a broad parameter space. Besides advancing quantum simulation, our studies pave the ground for the generation of metrologically useful entangled states which can enable real metrological gains via quantum enhancement.

No Colloquium

November 3, 2020

Graphene Statistical Mechanics

October 27, 2020

Mark Bowick, University of California, Santa Barbara
Host: Brian Swingle

Abstract: Thermalized two-dimensional metamaterials, such as graphene at room temperature, exhibit novel mechanical properties by virtue of being bending, rather than stretching, dominated. A wide variety of responses can be obtained simply by varying purely geometrical properties such as size and shape. Furthermore, by slicing and dicing to change the topology of the sheet, one can realize long sought-after phenomena such as the thermal crumpling of elastic surfaces and multiple origami-like folding. I will discuss recent developments in this area including theory, simulations and experimental tests and challenges.

Dark Matter halos from parametric resonance and their signatures

October 20, 2020

Asimina Arvanitaki, Perimeter Institute
Host: Brian Swingle

Abstract: While there is undisputed evidence for Dark Matter, its nature and properties remain one of the biggest questions of our time. What is Dark Matter(DM)? How is it produced? Does it have interactions other than gravitational? In this talk, I will describe how a large class of bosonic particles can account for the DM of the Cosmos. These particles can be much lighter than those of the Standard Model with Compton wavelengths that are bigger than the size of our solar system or smaller than a millimeter. In the presence of attractive self-interactions, there is a parametric resonance effect in the early universe that can cause growth of structure at small scales, an effect so dramatic that can cause structures to collapse well before matter-radiation equality. The signatures of this effect span several experiments and orders of magntude in parameter space. When the DM boson is heavy, the dense DM halos can alter the optimal search strategies in direct detection experiments. When the DM boson is light, these halos may leave their imprint in searches for dark matter substructure, primordial gravitational waves and alter the star formation history of the universe.

Hands-on Astrophysics: Using Stardust Grain Analysis to Further our Understanding of Stellar Nucleosynthesis and Galactic Chemical Evolution

October 13, 2020

Reto Trappitsch, Brandeis University

Abstract: Nuclear astrophysics and the evolution of elements in the Milky Way are topics generally associated with theory, large-scale nuclear physics experiments, and astronomical observations. Primitive meteorites, which are Solar System rocks that represent the original composition of the solar nebula, however preserved micrometer-sized, bona fide stardust grains. These grains formed in the death throes of dying stars and subsequently travelled through the interstellar medium until they were incorporated into the first solids that formed in the solar nebula. Each stardust grain represents a fingerprint of its parent star. Therefore, by analyzing these samples we can directly probe the stellar environment they formed in.

In my talk I will give an overview of stellar nucleosynthesis and galactic chemical evolution and discuss how stardust grain measurements done in the lab have contributed to our understanding of these fields. Additionally I will discuss how open questions in these fields pose challenges that need to be addressed in the future with a multi-disciplinary approach combining theoretical and experimental nuclear physics, nucleosynthesis modeling, and stardust grain analyses.

 

Modern Explorations of Theoretical Physics

October 6, 2020

Henriette Elvang, University of Michigan, Ann Arbor
Host: Brian Swingle

Abstract: There is a wide range of physical phenomena -- from boiling water to pion scattering -- that can be described theoretically in a common mathematical language, namely that of quantum field theory. In recent years, there has been a resurgence of new powerful approaches to explore this theoretical framework. In the talk, without assuming any prior knowledge of field theory, I will present some of these ideas and give examples of their application in contexts that range from ferromagnets to nonlinear electrodynamics and quantum gravity.

Schrodinger’s What is Life? at 75: The Physical Aspects of the Living Cell Revisited

September 22, 2020

Rob Phillips, California Institute of Technology
Host: Jané Kondev

Abstract: 2019 marked the 75th anniversary of the publication of Erwin Schrodinger's ``What is Life?'', a short book hailed by Roger Penrose as ``among the most influential scientific writings'’ of the 20th century'’. In this talk, I will briefly review the long argument made by Schrodinger as he mused on how the laws of physics could help us understand ``the events in space and time which takes place within the spatial boundary of a living organism.'' Though often chronicled for its influence on some of the titans of the rise of molecular biology, my talk takes a different tack. I pivot to the idea of Schrodinger’s classic as a timeless manifesto. Rather than a dated historical curiosity, I argue that ``What is Life?'' is full of stylish approaches to understanding the mysterious livingworld around us and instead, might be viewed as a call to arms to tackle the unfinished business of the scientific study of living matter. To make those points, I will focus on two key case studies from my own laboratory that pursue several of the key threads laid out in Schrodinger’s book: i) how does the control of hereditary information buried within the genome look from the point of view of physics and ii) how do the spatiotemporal structures (mitotic spindle, organelles, tissues, etc.) so familiar in biology arise as the playing out of underlying laws of non equilibrium physics?

Extending classical ideas in fluid mechanics: Thin films, a novel similarity solution, nanoscale instabilities, and molecular biology

September 15, 2020

Howard Stone, Princeton University
Host: Guillaume Duclos

Abstract: Many modern research themes in science and engineering introduce new questions, some of which can be understood using fundamental concepts. We will provide examples of research discoveries from the subjects of thin film dynamics and cellular biology and in each case illuminate the phenomenon using fundamental ideas in fluid dynamics, with experiments and theory.

In the first example, we will document experimentally the time and (three-dimensional) space variations of the shape of a falling film near the edge of a vertical plate and rationalize the quantitative features using a similarity solution. This example seems particularly unusual since we are able to theoretically show that the shape is described by a nonlinear partial differential equation, involving three independent variables, yet the equation can be reduced by a similarity.