homotopy

In a mountain range, there are trails that can be contracted to a point if you walk through a flat valley. But there are also loop routes that cannot be shrunk without crossing a ridge.

In a mountain range, there are trails that can be contracted to a point if you walk through a flat valley. But there are also loop routes that cannot be shrunk without crossing a ridge.

The holographic principle is often compared to a hologram: all the information about a volume is encoded on a flat surface. In physics, this means that three-dimensional spacetime with gravity could be equivalent to a two-dimensional theory without gravity — a so-called conformal field theory (CFT).

The quantum vacuum is not empty: it is permeated by entanglement linking distant regions. But when physicists tried to compute this entanglement, it turned out to be infinite. The culprit: particles with very low energy that can smear out over arbitrarily large volumes. They are like waves with no shores to contain them.

Two cities with the same average population density. In one, a neat grid of blocks; in the other, a maze of narrow alleys and unexpected squares. On a global map they look like twins, but life inside flows quite differently.

In a crystal, electrons are not distributed randomly. Their waves obey the lattice symmetry, like ripples on water taking the shape of a hoop thrown into it. The new work shows that at high-symmetry points, this ripple must have nodes — places where electron density is strictly zero.

Cut a spin chain, and particles without an energy gap appear at the edges. They are called zero modes, and they live only at the boundary. Previously, they were counted using complex topological invariants, but now there is a simpler way.

An ordinary road map with traffic is a heap of numbers. Sending them all as is takes too long. But you can reduce the volume without losing the gist. Researchers found a clever solution: first redraw the map so that streets become points, and intersections become bridges between these points.

The book is open to the middle. Current runs along the spine, while the pages remain cool. That's roughly how a two-dimensional topological insulator works — a material where electricity flows only along the edges, while the interior stays still.

A snowflake grows, an oil slick spreads, a crowd moves — each case has its own equations. But the end result is often the same: shape, speed, and pattern follow a simple scheme that repeats across scales. This is universality — when a common law hides behind the details.

In the microscopic world, noise, random fluctuations, usually destroys fragile quantum states, much like ripples on water break the reflection in a puddle. However, in some systems, collective, coordinated noise from many particles produces the opposite effect: a stable unidirectional flow emerges.

Dim the lights. The stage belongs to a shadow theater, with just r puppeteers, but the screen obeys the laws of hyperbolic geometry. Every wave of the hand births a pattern that mathematicians call a discrete group of isometries.

A huge sheet of paper lies on the table. A cartographer tries to fit all conceivable landscapes onto it: mountain ranges, tangled gorges, endless plains. Each landscape is a finitely generated group, and the map is its Cayley graph, a graphic portrait of the group, where vertices are elements and edges are connections between them.

The frost pattern on a windowpane doesn't spread uniformly: it sprouts branches, captures new areas, sometimes sealing off voids. This growth is not a smooth expansion but a series of leaps. This image resonates surprisingly with a mathematical model devised by Charles Loewner a century ago.

A geometer runs a finger along an invisible surface, and by how the geodesics bend, lifts its fingerprint. That fingerprint can be the maximum normal curvature κ(F): the largest curvature among all geodesics passing through a point of the immersed manifold.

Imagine a philharmonic orchestra playing, but the microphone picks up only a blended roar — a tutti where double basses, flutes, and timpani all merge. Physicists have grappled with a similar task for decades, staring at fields of density, temperature, or magnetic induction.

Magnetic vortices just tens of nanometers across are like tiny tornadoes frozen in metal. Called skyrmions, they are remarkably stable: hard to destroy but easy to move. That's why engineers see them as ideal 'bits' for future memory — fast, energy-efficient, and resilient. But until now, no one knew how such a vortex is structured within the material's depth.

A sheet of rubber is squeezed from two sides. At first glance, it's just deformation. But inside, deep within the material, a hidden order awakens. This is not just wrinkles on the surface. It's the birth of special zones where vibrations become trapped, as if in a cage. Previously, it was thought that such effects require intricate lattices and periodic structures.

A thrown stone follows a smooth arc: its path is easy to predict. But if the stone skips across water, the trajectory becomes jagged and far more interesting. Quantum particles behave much the same way when their state is not a calm wave with fixed energy but a turbulent mix of several waves at once.

A quantum particle traveling around a closed ring leaves two distinct 'records' along its path. The first is like a fingerprint on a doorknob: it remains unchanged no matter how many times you turn the knob, and it only tells you about the shape and material.

Imagine a huge orchestra where every musician plays at their own tempo, yet together they create a single rhythm. Scientists studying the brain or laser networks often face a similar picture: thousands of neurons or oscillators each pulse on their own, but the whole crowd needs to be observed at once.

On a donut-shaped ring, the theory lives on the surface, while inside hides a curved space — like a saddle widening toward the edge. According to the holographic principle of Juan Maldacena, everything happening inside can be described by data on the boundary.

A hot spoon in a cup of tea — heat runs along the metal from the hot end to the cold one. Textbooks describe this process as atoms jiggling and passing on their jolts to each other. But atoms don't just jiggle: they also spin, like tiny tops. Until now, physicists had no common language to describe both motions at once. The authors of the work built such a language.

Water churns behind a rock in a river, air swirls behind an airplane wing: it seems like completely random motion. But the authors of the study found a rigid kitchen recipe in this chaos. They decomposed the fluid's rotation (angular momentum) into two independent parts.

The birth of the universe is often compared to inflating a balloon: in the first moments, it stretched rapidly. But the very "rubber" of this balloon was probably not perfectly smooth — rather, it was covered with the finest ripples.

A drop of water on a table usually sits still. But make the table slightly softer on one edge and stiffer on the other, and the drop will crawl toward the stiffer side. This phenomenon is called durotaxis: the liquid itself chooses where to flow, without pumps or electricity. This trick works because the surface under the drop is not uniform.

An ordinary black hole resembles a funnel, at whose center spacetime plunges into a point — a singularity. There, density becomes infinite and the laws of physics break down. But what if, instead of a plunge, there were a smooth bottom inside? The authors of the study constructed exactly such a model.

In a cold cloud of atoms chilled to near absolute zero, energies can behave like threads that weave into braids. These braids are not just a pretty image: their pattern is tied to topology, the branch of mathematics that studies properties preserved under continuous deformations.

When you press your finger on a piece of soft putty, it obediently dents and then slowly returns to its original shape. But if you press on a window pane, it either holds or cracks — no smooth deformation. Both materials are amorphous: their atoms are not arranged in a strict crystal lattice like in metal. Yet they behave completely differently.

Light can be more than just a stream of particles; it can be a wave with a specific shape over time. Sometimes this shape changes so rapidly that capturing it is like photographing a bullet in flight. Scientists have devised a way to measure such fleeting quantum states of light. For this, they use a device called a balanced homodyne detector.

The world of physics works like a music box: there's a speed knob. Turn it slowly, and the melody is strict — balls roll along trajectories (classical mechanics). Speed it up — the notes blend into a chord: a particle is in many places at once, all probabilities (quantum regime). The secret of switching is in the parameter κ.

A flashlight beam in fog draws a cone, but it can't carve a figure out of a cloud. A hologram, though, is like a stencil for light: it knows in advance where to gather into bright points and where to remain shadow. Now such a stencil makes liquid resin solidify throughout the entire volume at once, without layer-by-layer printing.

Snap the bottom of a soda can: it will dent inward and stay that way. Press again: it pops outward. That's bistability: a system with two stable states, like a light switch. An elastic arch is the simplest example. It can curve upward or downward, and to flip it, you need to apply force.

Open systems exchange energy with their surroundings, like a guitar string fading into the air. Scientists built such a system out of light, twisting the beams into corkscrews and forming an artificial lattice — a maze where losses can be controlled at every step. Light in this maze always moves at the maximum possible speed, but its energy can fade or grow.

A black hole in anti-de Sitter space is not a void but rather a pot of boiling liquid. This liquid has two phases: small and large black holes, like steam and water. At a certain pressure, they merge at a critical point, just as water at 374 °C and 218 atmospheres loses the distinction between liquid and gas.

A mechanical testing lab is like a kitchen where a chef tastes a dish to guess the recipe. For an anisotropic material, one that is stiffer along its fibers than across them, a single taste is not enough. You need three or more separate tests, each with its own specimen. Engineers have spent years refining specimen shapes by hand, relying on intuition and experience.