# Topological Quantum Field Theory Explained

Topological quantum field theory is a highly abstract mathematical framework that studies quantum physics by looking exclusively at the overarching shape and connectivity of a space, entirely ignoring traditional measurements like distance, angles, and time. In the real world, this theoretical math perfectly describes exotic, two-dimensional states of matter where quantum information is inherently shielded from local environmental noise. Because of this built-in stability, the theory serves as the fundamental blueprint for topological quantum computing, a revolutionary approach that seeks to process information by braiding the physical paths of quasi-particles.

## The Geometry of the Quantum World

To comprehend topological quantum field theory (TQFT), one must first untangle standard quantum field theory from the mathematical concept of topology. Modern theoretical physics is built upon standard quantum field theory, which describes the universe as a complex tapestry of interacting, omnipresent fields [cite: 1]. In this framework, what we perceive as solid particles—such as electrons or photons—are actually tiny ripples or localized excitations within their respective fields. However, standard quantum field theory is strictly bound to geometry. To calculate the probability of particles moving, colliding, or scattering, physicists rely heavily on a "metric," which is a mathematical tool used to measure distances, angles, and the passage of time across the background of spacetime [cite: 2, 3]. If the underlying spacetime background is stretched, compressed, or curved, the physical predictions generated by the theory will change entirely.

Topology, conversely, is a branch of mathematics concerned solely with the global, structural properties of a space that survive continuous deformation. In the eyes of a topologist, distance and geometry are irrelevant. A coffee mug and a doughnut are considered mathematically identical because they both possess exactly one hole; you can theoretically stretch, twist, and mold the shape of one into the other without tearing the material. 

A topological quantum field theory merges these two realms, creating a special class of quantum field theory designed to compute topological invariants [cite: 4, 5]. In a TQFT, the correlation functions—the mathematical equations that determine the probabilities of different quantum events occurring—are completely metric-independent [cite: 3, 4]. This means that if you were to continuously deform the spacetime in which the quantum fields exist, the outcomes of the quantum processes would remain completely unaltered [cite: 4]. 

Because it ignores local geometry, a TQFT describes a hypothetical universe devoid of "local degrees of freedom" [cite: 2]. In such a universe, it is impossible to measure a particle moving from point A to point B at a specific speed, because the fundamental concepts of distance and speed simply do not exist. Instead, the theory only registers global, structural changes to the space itself. Consequently, applying TQFT to the flat Minkowski spacetime used in standard particle physics yields trivial results, because a flat plane can be contracted to a single point without tearing. To find meaningful applications, mathematicians and physicists apply TQFT to highly curved, complex spacetimes, such as Riemann surfaces and multi-dimensional manifolds [cite: 4].

## Bridging Space, Time, and Quantum States

To make TQFT mathematically rigorous, physicists rely on a powerful structural analogy that forms a bridge between Einstein’s general relativity—the physics of spacetime—and quantum theory, which dictates the physics of states and probabilities [cite: 2]. This framework translates the continuous flow of time into rigid algebraic operations.

### The Concept of Cobordisms

In general relativity, the physical space of the universe at any given frozen moment in time is represented mathematically as a manifold. If a physicist wants to observe how that space evolves from an initial starting time to a final ending time, they must examine the specific chunk of spacetime that bridges the two moments. Mathematicians refer to this connecting spacetime manifold as a cobordism [cite: 2, 6]. 



While a cobordism is essentially just a multidimensional geometric shape, it can be conceptualized as the physical process of time passing. One of the most critical features of this mathematical model is that it allows for explicit topology change [cite: 2]. A cobordism might, for instance, represent a process where two entirely separate spatial dimensions—such as two isolated circles—collide, overlap, and eventually merge to form a single, unified circular space.

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### The TQFT Dictionary

In standard quantum mechanics, the condition of any physical system is described by a "state," which lives in an abstract mathematical realm known as a Hilbert space. When a quantum system naturally evolves over time or interacts with its environment, a mathematical "operator" is used to transform the initial state into the final state [cite: 2]. 

Topological quantum field theory acts as a perfect translation dictionary between the geometric cobordisms of relativity and the abstract operators of quantum mechanics [cite: 2, 7]. In this translation, physical space is perfectly analogous to a quantum state, and the spacetime cobordism connecting two spaces is perfectly analogous to the operator connecting two states. Therefore, the physical passage of time in TQFT is literally the mathematical transformation of the quantum state itself [cite: 2]. 

Because a TQFT lacks a background metric, time cannot be measured in standard units like seconds or minutes. The theory only tracks structural shifts in the topology of space. If time passes but the fundamental topology of space remains completely unchanged, physicists refer to this as an "identity cobordism." In the strict rules of a TQFT, experiencing an identity cobordism is physically identical to waiting zero seconds [cite: 2]. The state of the universe simply does not change at all unless the overarching topology of space itself undergoes a transformation.

### The Cobordism Hypothesis

To mathematically solidify this relationship, TQFTs are defined as "functors" that systematically assign specific Hilbert spaces to geometric manifolds, and linear operators to cobordisms [cite: 2, 4]. In 1995, mathematical physicists John Baez and James Dolan proposed a radical idea known as the Cobordism Hypothesis, a concept that was later formalized and proven by mathematician Jacob Lurie in 2008 [cite: 5, 8, 9]. 

The hypothesis suggests that a fully extended topological quantum field theory is uniquely and completely determined by the algebraic data it assigns to a single, zero-dimensional point [cite: 5, 8, 9]. Physically, this idea relates directly to the fundamental principle of "locality" in quantum mechanics. It proves that if researchers can fully map and understand the microscopic rules governing an infinitely small point in a topological field, they possess the mathematical tools to "glue" those points together to understand the behavior of the entire universe, regardless of how complex its overarching topology becomes [cite: 9, 10].

## Comparing the Great Frameworks of Physics

Because theoretical physics relies on dense jargon and complex mathematics, topological quantum field theory is frequently confused with other foundational frameworks, most notably standard Quantum Field Theory and String Theory. While they are related mathematical cousins that borrow heavily from one another, they are designed to describe completely different aspects of physical reality [cite: 1, 11, 12, 13]. 

Standard quantum field theory is the bedrock of the Standard Model of Particle Physics, providing the backdrop for the universe's grand theater by laying down the fields on which particles interact [cite: 1]. It excels at describing electromagnetism and the strong and weak nuclear forces, but it completely falters when attempting to incorporate gravity. When physicists try to force the metric-heavy equations of general relativity into standard QFT, the math breaks down into unsolvable infinities [cite: 1, 14].

String Theory emerged as a potential remedy to this gravitational disconnect. By abandoning the notion of zero-dimensional point particles and instead proposing a universe filled with one-dimensional vibrating strings of energy, String Theory naturally incorporates gravitons—the theoretical particles that carry the force of gravity [cite: 1, 14, 15]. However, String Theory requires the existence of multiple extra, curled-up spatial dimensions and relies heavily on a background spacetime geometry [cite: 1, 11]. While standard QFT has been exhaustively validated by decades of particle accelerator experiments, String Theory remains largely theoretical, lacking direct empirical validation at currently accessible energy scales [cite: 11, 16, 17].

Topological quantum field theory operates differently from both. Rather than trying to unite all fundamental forces or plot the exact collisions of particles, it strips away the background geometry entirely. It serves as a simplified, highly idealized model of quantum gravity by completely removing the concept of distance, allowing physicists to study how quantum states react purely to changes in the shape of space [cite: 4, 5, 14].

| Feature | Standard Quantum Field Theory (QFT) | String Theory | Topological Quantum Field Theory (TQFT) |
| :--- | :--- | :--- | :--- |
| **Core Concept** | Particles are localized excitations (ripples) in universal fields [cite: 1, 16]. | Particles are different vibrational modes of 1D strings in higher dimensions [cite: 1, 11, 17]. | Quantum states depend only on global connectivity, immune to local geometry [cite: 4, 7]. |
| **Role of Gravity** | Fails to incorporate gravity. Predicts infinities when applied to general relativity [cite: 1, 11]. | Aims to unify all forces, including gravity, into a "Theory of Everything" [cite: 1, 11, 14]. | Acts as a simplified "toy model" for quantum gravity by removing distances [cite: 4, 5, 14]. |
| **Dependence on Spacetime** | Highly dependent on metric (distances, time intervals) [cite: 3, 4]. | Dependent on background spacetime geometry and extra curled-up dimensions [cite: 1, 11]. | Metric-independent; stretching spacetime has no effect on outcomes [cite: 3, 4]. |
| **Current Applications** | The Standard Model of Particle Physics; highly validated by collider experiments [cite: 1, 11, 16, 17]. | Theoretical mathematics; lacks direct empirical validation at current energy scales [cite: 1, 11, 16]. | Modeling robust phases of matter (topological insulators, anyons) and quantum computing [cite: 4, 18, 19, 20]. |

## Anyons and the Rules of Flatland

While TQFT was initially a purely mathematical pursuit utilized by string theorists and geometric topologists, condensed matter physicists soon realized that it perfectly describes certain phenomena in the physical world. Specifically, TQFT governs the behavior of unique materials that are functionally restricted to two spatial dimensions [cite: 4, 19, 21]. 

### Bosons, Fermions, and the 3D Limit

In our standard three-dimensional universe, every fundamental particle that exists belongs to one of two strict categories: fermions or bosons [cite: 21, 22]. Fermions, which include particles like electrons and quarks, are the building blocks of solid matter. If a physicist were to swap the physical positions of two identical fermions, their underlying quantum wavefunction would change its mathematical sign, resulting in a phase shift of -1. Bosons, which include particles like photons, are responsible for transmitting fundamental forces. If two identical bosons are swapped, their wavefunction remains completely unchanged, resulting in a phase shift of +1. 

Because of the geometric constraints of three-dimensional space, swapping identical particles twice is mathematically and physically identical to never having moved them at all. In 3D space, any entangled paths or "knots" created by moving particles around each other can always be untangled without the paths crossing, meaning the statistical possibilities are firmly locked into the binary choice of fermions or bosons [cite: 21, 22, 23].

### Entering the 2D Plane

However, if physical particles are restricted to a strictly two-dimensional flat plane—such as the microscopic surface of certain superconducting materials—the geometric rules of three-dimensional physics completely break down [cite: 22, 23]. 

In a strictly 2D environment, swapping two particles twice is not the equivalent of leaving them alone. Because there is no third dimension to lift a particle "over" another to untangle them, their paths can permanently loop and knot around one another. Quasiparticles that emerge in these two-dimensional systems are called anyons, deriving their name from the fact that their wavefunctions can shift by literally *any* complex quantum phase when swapped, rather than being restricted to just +1 or -1 [cite: 21, 22, 24]. 

The behavior of these anyons is governed mathematically by the "braid group," an algebraic structure that tracks the specific history of how particles have moved around each other, contrasting sharply with the simple permutation group that governs standard 3D particles [cite: 22, 24]. By studying fractional quantum Hall effect states—where electrons in a 2D plane are subjected to extreme magnetic fields and near-absolute-zero temperatures—physicists have been able to observe the exotic quantum properties of these emergent anyon quasiparticles [cite: 19, 21, 22, 25].

### Abelian vs. Non-Abelian Anyons

Anyons are broadly divided into two distinct classes based on how their symmetries respond to being moved around each other: Abelian and non-Abelian anyons.

Abelian anyons are the simpler of the two. When you swap the positions of two Abelian anyons, their overall quantum state simply picks up a global mathematical phase shift [cite: 25, 26]. While the phase changes, the core identity of the quantum state remains fundamentally intact. Because their braiding only produces a phase shift, Abelian anyons are considered highly useful for creating incredibly stable quantum memory systems, but they are insufficient for building a universal quantum computer because they do not provide the ability to fully manipulate and rotate the quantum state [cite: 25, 26]. 

Non-Abelian anyons, however, represent the holy grail of topological physics. If you swap the positions of two non-Abelian anyons, you do not simply alter their phase; the braiding action fundamentally forces the quantum system to transition into an entirely new, distinct quantum state [cite: 21, 23, 26]. Crucially, the order in which non-Abelian anyons are swapped dictates the final outcome. Swapping particle A around particle B will yield a completely different physical reality than swapping particle B around particle A [cite: 27]. This complex, non-commutative behavior allows physicists to use the physical movement of the particles as a mechanism to execute complex computational algorithms.

## Braiding: How to Compute with Topology

The unique mathematical properties of non-Abelian anyons provide the direct physical mechanism required for building a topological quantum computer. To understand why this is revolutionary, one must look at the fatal flaw plaguing modern quantum engineering.

### The Hardware of Topological Quantum Computers

In a traditional quantum computer, digital information is encoded in the fragile properties of individual, isolated particles—such as the physical spin of an electron or the energy level of a trapped ion. Because these particles are incredibly sensitive, the slightest fluctuation in ambient temperature, a stray magnetic field, or even cosmic radiation can cause the particle to accidentally flip its spin. This instantly destroys the encoded data in a catastrophic process known as quantum decoherence [cite: 28, 29]. To combat this, traditional quantum computers require massive, resource-heavy software algorithms that utilize thousands of extra "physical qubits" just to constantly monitor and correct the errors of a single "logical qubit." 

In a topological quantum computer, the information is not stored in the physical properties of the particles themselves. Instead, the data is non-locally encoded in the global, structural *relationships* between multiple non-Abelian anyons [cite: 20, 30]. Because the information is distributed across the entire topology of the system, a local disturbance hitting a single anyon cannot corrupt the data. 

### Tying Quantum Knots

To process information and perform calculations, physicists take these non-Abelian anyons and physically swap their positions on a 2D surface over a specific period of time. If you were to trace the continuous paths of these particles through three-dimensional spacetime—utilizing two spatial dimensions and one time dimension—their chronological world-lines would literally wrap around each other, forming distinct physical braids [cite: 23, 27, 30, 31].

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Each specific braid configuration engineered by the researchers corresponds directly to a specific quantum logic gate, forming the foundational operations required to run an algorithm [cite: 30, 32]. 

### Hardware-Level Fault Tolerance

This braiding process yields an unparalleled advantage: hardware-level fault tolerance. Because the quantum state is fundamentally tied to the macroscopic knot made by the braided world-lines, small, highly localized errors cannot alter the system. If an anyon wiggles slightly off its intended path due to ambient heat or electromagnetic interference, the fundamental topology of the knot remains entirely unchanged [cite: 20, 23, 30, 33]. Just as you cannot untie a physical knot in a shoelace by simply jiggling the string, the universe cannot unravel the quantum data without a massive, systemic disruption. This inherent resilience suggests that a topological quantum computer could drastically bypass the crippling error-correction overhead that currently stalls the commercial viability of standard quantum computers [cite: 20, 30, 34].

## The Race to Build Topological Hardware

While topological quantum field theory provides a mathematically beautiful framework, building a physical machine capable of reliably isolating, controlling, and braiding non-Abelian anyons has proven to be one of the most formidable engineering challenges of the 21st century. 

### Trapped Ions and the Quantinuum Milestone

In May 2023, the quantum computing company Quantinuum, working in deep collaboration with researchers from Harvard University and Caltech, achieved a historic milestone that was subsequently published in the journal *Nature* in February 2024. Utilizing the unique capabilities of their H2 trapped-ion quantum processor, the team successfully simulated a non-Abelian topological state across a network of 27 qubits [cite: 33, 35, 36]. 

By deploying a shallow adaptive circuit to manipulate the system's wavefunction on a kagome lattice, the researchers were able to create anyons and deliberately move them along complex paths in spacetime, forming structures akin to Borromean rings [cite: 35, 36]. Through anyon interferometry, the team successfully detected the first intrinsically non-Abelian braiding process [cite: 33, 35, 37]. While this achievement was fundamentally a simulation executed *on* a highly advanced standard quantum computer, rather than a natively topological material, it served as critical proof that non-Abelian topological order can be manipulated and verified with a fidelity exceeding 98.4% [cite: 33, 35, 36]. 

### Microsoft’s Majorana 1 Architecture

Conversely, Microsoft has placed a massive corporate bet on skipping standard superconducting qubits entirely, opting instead to build native topological qubits from scratch. In February 2025, Microsoft made global headlines by announcing the "Majorana 1," an 8-qubit quantum processing unit that the company boldly claims utilizes true, hardware-protected topological qubits capable of eventually scaling to a million qubits on a single desktop-sized chip [cite: 38, 39, 40].

Microsoft's ambitious approach relies heavily on mastering an exotic phenomenon known as Majorana zero modes (MZMs) [cite: 38, 41, 42]. First theorized in 1937 by Italian physicist Ettore Majorana, a Majorana fermion is a unique particle that acts as its own antiparticle [cite: 41, 43]. By fabricating bespoke materials atom by atom, Microsoft layered an indium arsenide semiconductor with an aluminum superconductor to create an entirely new class of material they dubbed a "topoconductor" [cite: 38, 39, 44, 45]. 

When these topoconductor nanowires are cooled to near absolute zero and subjected to intense magnetic fields, an electron is effectively split into two distinct Majorana zero modes localized at the extreme opposite ends of the wire [cite: 38, 42]. The quantum information—specifically the "parity" of the electron—is hidden non-locally in the vast space between these two endpoints, heavily shielding it from the surrounding environment [cite: 38, 42]. Microsoft claims that this architecture allows them to read and control the qubits using simple digital microwave pulses connected to quantum dots, bypassing the delicate, error-prone analog controls required by competing trapped-ion or superconducting systems [cite: 38, 39, 42].

### Scientific Controversy and Skepticism

Despite the immense corporate fanfare, Microsoft's Majorana 1 claims have been met with intense scientific debate and widespread skepticism from the broader physics community [cite: 46, 47]. 

While the data supporting the Majorana 1 chip was published in the prestigious journal *Nature*, independent physicists have pointed out severe caveats regarding the findings [cite: 46, 47]. Most notably, the editorial team at *Nature* attached a highly unusual, explicit disclaimer to the top of the paper, firmly stating that the manuscript *does not* represent definitive evidence for the actual physical presence of Majorana zero modes [cite: 46, 47]. Instead, the journal clarified that the work was published merely because it introduced a novel device architecture that *might* enable future fusion experiments [cite: 47].

Furthermore, leaked peer-review reports revealed that during the first round of review, three of the four independent referees actively declined to recommend the paper for publication. The reviewers cited questionable hypotheses and argued that the physical presence of Majoranas was forced into the theoretical modeling by hand, rather than being organically proven by the data [cite: 46, 47]. Physicists have also raised alarms regarding the system's actual coherence times, noting that while certain measurements showed stability, other vital operations degraded in mere microseconds due to the unwanted overlapping of the Majorana modes, casting severe doubt on whether true "topological protection" had been achieved [cite: 46].

This current skepticism is heavily compounded by Microsoft's recent history. In 2018, Microsoft researchers published a landmark paper similarly claiming the definitive discovery of Majorana particles. That paper had to be fully retracted after independent researchers audited the raw data and found it had been improperly manipulated to display the exact quantized values expected by the theory [cite: 47, 48]. While some physicists currently view the Majorana 1 chip as a remarkable engineering feat that genuinely pushes the boundaries of materials science, many prominent critics warn that the company's marketing apparatus has vastly outpaced the empirical reality of the underlying physics [cite: 46, 47].

## Beyond Computing: The Rise of Topological Materials

While the race to build a fault-tolerant quantum computer dominates mainstream headlines, the mathematical principles of topological quantum field theory have quietly revolutionized the broader field of materials science. 

The most prominent success story in this domain is the discovery of topological insulators, a breakthrough that was awarded the Nobel Prize in Physics in 2016 [cite: 49, 50]. A topological insulator is a highly unusual material—such as bismuth mercury telluride—that acts as a perfect insulator on its interior, completely blocking the flow of electricity, but acts as a perfect conductor along its exterior surfaces or edges [cite: 49, 50]. 

Because the electrons traveling on the surface of a topological insulator are topologically protected by the overarching geometry of their quantum state, they are forced to move in one continuous direction. They cannot scatter, bounce backward off impurities, or lose energy to heat [cite: 49, 50]. This zero-resistance conduction has sparked a massive global race to develop ultra-low-energy transistors and dissipationless electronics [cite: 50, 51].

Recent discoveries continue to dramatically expand the boundaries of topological materials. In late 2025, researchers utilizing advanced scanning tunneling microscopy at Princeton University identified the world's first "topological excitonic insulator," utilizing a relatively obscure quantum material (Ta2Pd3Te5). This discovery marked a breakthrough state where correlated particles and topological protection naturally coexist to form Bose-Einstein condensates within a three-dimensional macroscopic material [cite: 52]. 

Similarly, international research teams at the Max Planck Institute have identified highly tunable "Kagome metals" that exhibit giant anomalous Hall effects, and scientists at Aalto University have developed quantum-inspired algorithms to simulate massive super-moiré quasicrystals [cite: 49, 51, 53]. By mapping the quantum metric of these materials—which guides electrons as if they were being pulled by a localized gravitational field—researchers are unlocking new ways to steer chiral fermions purely through geometry [cite: 53, 54, 55]. Ultimately, these advanced topological materials could eventually lead to entirely dissipationless electronics, providing a critical solution to drastically reduce the surging heat and energy demands of modern AI data centers and classical computing architecture [cite: 51].

## Bottom line

Topological quantum field theory represents a profound mathematical departure from traditional physics, opting to completely ignore the local geometry of the universe in order to focus on its fundamental, unchangeable connectivity. By applying these abstract, metric-independent equations to strictly two-dimensional physical systems, scientists have uncovered exotic quasi-particles called anyons that can fundamentally change their quantum state simply by being braided around one another. While the physical realization of a fully fault-tolerant topological quantum computer remains fraught with extreme engineering challenges and intense scientific controversy, the underlying TQFT framework has already irreversibly transformed our modern understanding of quantum phases and next-generation materials science.

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61. [TQC and Braiding Operations](https://arxiv.org/html/2307.01892v2)
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64. [Ising Anyons and Topological Computation](https://www.eurekalert.org/news-releases/1093185)
65. [Latest breakthroughs in Quantum Computing (2026)](https://gilkut.net/quantum-computing/quantum-news/latest-breakthroughs-in-quantum-computing/)
66. [Microsoft Quantum Breakthrough Analysis](https://thenewstack.io/microsoft-makes-quantum-computing-breakthrough-with-new-chip/)
67. [Quantum Computing Arriving Soon](https://www.securitymagazine.com/articles/101413-quantum-computing-could-arrive-soon-due-to-microsofts-new-chip)
68. [Replication Crisis in Quantum Computing Milestones](https://www.sciencedaily.com/releases/2026/03/260328043600.htm)
69. [Microsoft's Quantum Breakthrough Explained](https://www.miragenews.com/microsoft-claims-quantum-breakthrough-explained-1411943/)
70. [Majorana Zero Modes and Topological Qubits](https://gilkalai.wordpress.com/2025/02/21/majorana-zero-modes-and-topological-qubits/)
71. [Microsoft Claims Topological Quantum Computing Again](https://hackaday.com/2025/02/20/microsoft-again-claims-topological-quantum-computing-with-majorana-zero-mode-anyons/)
72. [Spacetime Torsion and MZMs](https://www.researchgate.net/publication/389418995_Majorana_Zero_Modes_in_Microsoft_Quantum_Chips_The_Fundamental_Role_of_Spacetime_Torsion)
73. [Reddit Review of Microsoft's Nature Paper](https://www.reddit.com/r/stocks/comments/1ithdwd/the_new_microsoft_work_is_moot_perspective_from_a/)
74. [Microsoft's Majorana 1 and Quantum Singularity](https://www.forbes.com/sites/luisromero/2025/02/19/microsofts-majorana-1-a-step-closer-to-quantum-computing-singularity/)
75. [Topological Quantum Processor Milestone](https://news.ucsb.edu/2025/021760/topological-quantum-processor-marks-breakthrough-computing)
76. [Quantum Visualisation Techniques at Oxford](https://www.ox.ac.uk/news/2025-05-30-new-quantum-visualisation-techniques-could-accelerate-arrival-fault-tolerant-quantum)
77. [Materials for Quantum Technologies Roadmap](https://arxiv.org/abs/2406.07720)
78. [UCSB Physicists and Majorana 1](https://www.eurekalert.org/news-releases/1074452)
79. [Quantum Shortcut for Simulating Quasicrystals](https://www.sciencedaily.com/releases/2026/05/260512202355.htm)
80. [Topological Excitonic Insulator Discovery](https://www.eurekalert.org/news-releases/1101507)
81. [Quantum Metric in Topological Insulators](https://www.youtube.com/watch?v=tojCKuxcPas)
82. [Photonic Topological Insulators at ITMO](https://news.itmo.ru/en/news/9108/)
83. [Metals, Insulators, and Topological Discovery](https://www.youtube.com/watch?v=jQ_ihxXcqpg)
84. [Magnetic Topological Insulators](https://www.youtube.com/watch?v=JlH-MQ3O7ks)
85. [Symposium on Geometry and Topology of Quantum Materials](https://www.fkf.mpg.de/8817964/geometry-and-topology-of-quantum-materials)
86. [Topological Quantum Chemistry at Max Planck](https://www.cpfs.mpg.de/tqc)
87. [Topological Materials Research at MPI](https://www.pks.mpg.de/condensed-matter/research-areas/topological-materials)
88. [Max Planck Press Releases on Kagome Metals](https://www.cpfs.mpg.de/1622021/pressreleases)
89. [PhD Applications in Germany](https://www.youtube.com/shorts/K7CmFTz6Bwg)
90. [Perimeter Institute Overview](https://perimeterinstitute.ca/)
91. [Holomorphic-Topological Field Theories Conference](https://events.perimeterinstitute.ca/e/1214)
92. [Perimeter Institute Training](https://perimeterinstitute.ca/training/online-training)
93. [Quantum Fields and Strings at Perimeter](https://perimeterinstitute.ca/news/25-years-quantum-fields-and-strings)
94. [Perimeter Institute 2025 Annual Report](https://perimeterinstitute.ca/2025-annual-report)

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36. [researchgate.net](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFdycZfZSnZbEiw91eeEj1yZ3SrIxg_XGKr_NUDXcBEOGUV4JZJdc4fcPUzThHj8LRK92J0A-FlO-G-sRVa5blgyqwNM4_ncrkVTQI7apn1LfPEfELXZ6IW3QnKytbQIE1_HNUOYmM8ug5MEi-QoSAXOIuGu7uEJEMJJhnKZ_lT15IE1JOzbl8V3cjA36pd85HYv5dc-KwOw57yhf3GQ47PWMTMalCLTxHEvaGu)
37. [spacefed.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHA66zVrUh0zdf-YQ1kRigbhsWhM6d-P0kUuC02zWlx7sy2sDr_Hnn0PPqDJOMStwikJUjo7YLMpawR8D90zEpKZykI3k_M4gf27cNROkSNsYRtLo-hrn3LnSU7PeSMsawjae-VsF4iJrgOA-jV9Fcq5NzS4nrrrlhZLzuCoUZeoqG0NXoMp-ulFJN3YLfbw4Fg)
38. [microsoft.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQH71OwgKdI3AO1Fr4D6JeNrwn1_Ct7AWHZ6ahEyNaesHegw0f8jb0HGXNW783uikFILgls9qgy-Xc9vom1C-y-ZRONDmhLm4I0Pr42tDSa2ksfQ6AWP70URjGcpXkSEqzCNEr-zn8oUPD4YOjpxbhdDrjtkbdqgYPRwUuouVmERSICE1A42UyNwgP2vXDV036LLmGB77b3GxwnJSU5x02J_-L6gmI8w18XIBGCk_9IKCHxZ6phAfXH54tsShmJIzDYFblRDdvhaRdzj3xbiiWQ=)
39. [microsoft.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFZigNSwBiZ19jtY65HGcBcomUFw8Ta5bNxdMxZ_i7yvi3faa4SggDdK41PWvGCxaGy6NhvsepGZjzOAAU4AkdTAzyG94oJ06LNBsnNviPQGvCs48QvoKyivNFOEneykGj_yk2bYlddxxBbZxF6QQSXTN5nvr-m323ZWY-r9lp7s9COb52Pze4iRbP3B5ikSaVq38fb6G69eC2KUBIK_9e6byAfzqVvmzLv3k2vzw==)
40. [techmonitor.ai](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQE6bVAoUH9VBj1br0CGDRVEetdjNaYzLnuNyMjYH2Wofphz_ye2l_XUsNswzql-Nn72xz_wQyq4lmpR-r6ysQkJh14exPAIEkawHesnsXnd7WFblOmGDgfqJ4xlwVfpQQShMHv18BPTYJR5yDrXBEL-I93QB__zT0OOjX_zGCBWozdji6INXvTUJQ5lKJ99hp3ZVlIsvISw6bLV9EJfckYZNw==)
41. [wordpress.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHKz5-KkS8t3t5_kk8jUVZVQsJyH6iTAb_O0vUvOtvzQug2XxnNvR2RFNT_QKsP_qKxK4RP5K3Fq5F2JWuiy_D0gtwT6YJB9RPS6q0G6HveJVpxr7pn75h4RPGxSh50B6cpRGGKZ1p3gcF0N_-PC1SXUvAMvMql_DcWBEV6ILX8Tt2q0A1VN4QgJstm)
42. [forbes.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHWb2ZEQbQX_HMhjgcYGJIYmKiel84ivpfe59WFkRIQtFXv9Ft0Qs4l0sFuRXJcQZ7zPk6SQ00n3ZrKjqL1p7hHhQT8VniswGPCiacFtfYS6x0nFeeRq4aNPmu_6TvatauGy0F4Sr2JqK_AjhYkX_DY28BSOlevFILnZOcCQoHYheyzIhbdlaZljq6q0VKjtW9J_yWy8f1IxcIjZIaZ4oS7DoXYrCLYlVNKTNdCUCI=)
43. [miragenews.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHDrOkbMkdLlAPWv14W3z-EKGGhhAeQjb2mE-gjwwOj_AYQP3mwuUU608LpuMi3wqVgzHxzCqaPTZS5_R9VNQEKT0hj7TcreL_A-FbY7VCmb6dlwmjI3jy7A8-QFmILKxzNXxoyUScRqbcuhvF4iUyBBC-eeZSk3JctpsKZ3YAPSlu0Ish2s_yItg==)
44. [ucsb.edu](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHc0ifIdHZmftGR7exOIeF3Z_TtBlGUZ3mZn7WxIO8Qwn0xkNaCPs6XOcCtBHL8d1Q1ZIXbn6PMEmImWH6cUd0WHdoM22if3oSAvq0vVMmXVfklgptbAMRXdJelzAxlDOFWAo1wdjI93YuAPRzExkEOyg9o9eZrdk2fyTqztH_TJ-o7AKslpbKPGqcZNv-RZt6ufA==)
45. [eurekalert.org](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHwxGW816E5GeNfIWf_ME00mSUZjfpM2ZpOxOAydUzRE7f4U_zatoI1TT5LmKklW6wFk0-OHQ9CS1UAh_4eRHuK6gvnJ3XIYAfgdLa-DLsPM5IoYVVfITSNC53Nw4OQ9T1JGzhKclU=)
46. [Link](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQEDD2gVTZRMiG2YontNajHYHKNKlO_q0EdR97hXcYBnzUr6NCqZanbO8W9Q1pDVbO2H8ih4JDxoa9_iWBMATvfnkxuL80QcxvEZbE9JyvEGeipJbTMCzSkK)
47. [reddit.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFU67-wcm-eqaS4bc7_wz2lkVETMJjkY5xjqXoxG8gN0WOkYn3CVddSKbXJJpJkQyixBRBM3UX1sOB42iB0fHrRQZ-DcY8PVHy3wc0X9Po87VFQDukiIuhbQTKOG6Dmca27hSHFr5-5vQcHaAN1q_6lWD1pbO_wbZSy2zXhAstpOvc4tzMQDqmC9mbLb247hfov9d3YcK65msE=)
48. [hackaday.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQHbjWk5nNCKv1JC3-oDWV7aPvX-KThlOx2B7X5xn2PUigC0_KfrUT8nW_6TsNn4FSnAKOTgZmINBsAT0PsVPipSyrwCqY-OTCHEEzYy3PsxB0EOraBnDaUtFUWvd9q8H3XNrqamniWoVh-LC0j4-5bDr4ziBqx2q56aqD0kGNPC4JieOaNy3Rut_EtYI4xCchUf6w0BON8DbpeFgwkPI0ld9SGvovJdLxyfMQ==)
49. [itmo.ru](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFnVO-hTcrcZ3zuBgTknichw-EK7Y0xmJ89FRdI_UwneaYIypsEDNxuKrZvP4F76j7tDCHGgTX0oxD29PHo7ORqaUbUXR20B8_G3EXDZJ-ikhOm-ZFFHB0V)
50. [youtube.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQF7Iu3nSkPOiGzoM8FCKUUZ_xFHafklGmfrwdNWN63X3mZlM8nPAcq2x2iortTNdhdTj0utD4QHHMUtJKftLLNlUkb6nHf0I5L2gia7m5KxeGpolhlvZunoxB7GOqSaadwA)
51. [sciencedaily.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQGHY7tqxmMyjZ0VlSA9dd-RdBh3xn5zncwAxQrkdl55CGba12r7H-BVnx9LH16hK--jicOPRd6SDru8S9qglgpfUeZAYsOTuIczlKkUzmu7cBsL3-V5CfUs2WE4LPBE_c6lxFV1eB-k8-tNs0q04NBnQAjR7w==)
52. [eurekalert.org](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFsn0xNi7mUJHDucoEU85PhBV4f2WNscy1gQRKR6NZ66eoYWPQSixoo85VdkGCKSQ687aicic5SA3pwlxQ7SlaWxpWdDjYXGGtc9Sj7b4UeN8En-tEOmoH9xvbY8tpG5F6143i8ewc=)
53. [mpg.de](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQEJH_5pCd4037Q02S0WTX_yH3JUhiC1xj3nszvmx5UhGHcDXqdoXl7w8MGOgEiCAaHIoCWjv9iIA6z3du7KEBtvFT2PLA1Y7f8KFZQR0GZi8mvOGcaHmrGTozyy0ClfnbJggp8=)
54. [youtube.com](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQFu0VVXJCIXiru4EQ9YP600t46dVHSkuLGJpor3bEd4JxaklmdIGJ6DIwR3XNhY1k0MdyTK7lEITm0xTtuySlwCM_jOdnJJGjfeMaY5oH7vQu47kIbGo4wB3PHyZyx4Oy_0)
55. [mpg.de](https://vertexaisearch.cloud.google.com/grounding-api-redirect/AUZIYQH1FV9_b6RGkBHA4RR7jAbAKDFHaCit5Ua5LM44Dx1KTSS5nuq_AB1R0wCtl8uNso5zdIwMS56x66rM4CQQK3rFu1-Z8hIh53YA483CVuZeozE=)