The hottest Topology Substack posts right now

And their main takeaways
Category
Top Science Topics

Surfaces and sequence graphs

A Piece of the Pi: mathematics explained • 66 implied HN points • 31 Jan 26
🔬 Science Topology
  1. You can build a graph by placing n vertices in a cycle and linking them according to the rank order of the first n terms of a real sequence, and as n grows these sequence graphs reveal striking geometric patterns.
  2. Graphs coming from the Kronecker sequence (multiples of the golden ratio mod 1) can be drawn on a torus without crossings, typically after removing the edge from n−1 to 0.
  3. Graphs from the van der Corput sequence embed into the Chamanara surface — a highly singular, infinite‑handle (“Loch Ness monster”) surface made by identifying shrinking boundary segments of a square — and finite approximations avoid the worst singularities so they can be visualized.

The surreal line is topologically compact!

Infinitely More • 25 implied HN points • 28 Nov 25
🔬 Science Topology
  1. Compactness in mathematics means that a set can be covered by a limited number of open sets, making it easier to work with. This concept is important in various areas of math like topology and analysis.
  2. The surreal numbers initially seem to lack compactness compared to real numbers, showing that many of the typical properties may not apply directly.
  3. However, by looking at the surreal numbers from a different perspective, we can discover surprising instances of compactness that we didn't expect.

Rubik’s abstract polytopes

A Piece of the Pi: mathematics explained • 18 implied HN points • 03 Mar 25
🔬 Science Topology
  1. Rubik's Cube can be made in different shapes, like a tetrahedron or dodecahedron, instead of just the classic cube. These variations have their own names, like the Megaminx for the dodecahedron.
  2. A new study explains how to think about these puzzles in more dimensions, counting how many different ways they can be turned or rearranged. This includes understanding shapes like the hypercube.
  3. The math behind a Rubik's Cube shows interesting patterns, like counting sticks and pieces, and this can help us understand all kinds of shapes and designs better.

Hexagonal knot mosaics

A Piece of the Pi: mathematics explained • 24 implied HN points • 03 Nov 24
🔬 Science Topology
  1. Hexagonal knot mosaics are a way to represent knots on a hexagonal board. You can use different types of tiles to create them.
  2. There are three categories of hexagonal mosaics: standard, semi-enhanced, and enhanced. Each type has different rules about how crossings can occur.
  3. Research has shown the maximum number of crossings you can achieve in these mosaics. Enhanced mosaics can have the most crossings, while standard ones allow the least.

Tips for new community members

Quantum Formalism • 39 implied HN points • 09 Mar 22
🚌 Education Topology
  1. Start with the 'Foundation Module' YouTube playlist for basics on finite-dimensional Hilbert spaces and quantum mechanics postulates
  2. Consider auditing crash courses on topics like Topology & Differential Geometry for Lie Groups and Group Theory for advanced knowledge
  3. Exploring topics like smooth manifolds and Group Theory can be valuable not just in quantum computation but also in applied fields like ML and Cryptography
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Linear Algebra Is Not Enough!

Quantum Formalism • 59 implied HN points • 01 Jun 20
🔬 Science Topology
  1. Linear Algebra is foundational in quantum mechanics but may not be enough if dealing with infinite-dimensional Hilbert spaces.
  2. To fully understand quantum mechanics, one needs to delve into Functional Analysis along with other mathematical branches like Topology, Measure Theory, and Group Theory.
  3. The newsletter focuses on explaining the mathematical formalism of quantum mechanics through easily understandable posts on relevant topics.

Lecture 3 Replay

Quantum Formalism • 19 implied HN points • 02 Aug 21
🚌 Education Topology
  1. The lecture was split into 3 parts for easy replay: Opening Discussion & Post Crash Course Program, Introduction to Metric Spaces (PART A), and Introduction to the Metric Topology (PART B).
  2. Viewers can watch the replay on YouTube after networking issues during the live session.
  3. The registration page for the next live session on the Topology Crash Course is available at the provided link.

Influential Mathematicians: Maurice René Fréchet

Quantum Formalism • 19 implied HN points • 23 Jul 20
🔬 Science Topology
  1. Maurice René Fréchet, a disciple of Jacques Hadamard, made significant contributions to mathematics through his work on metric spaces and abstract spaces, laying the groundwork for modern mathematical formalism, including quantum mechanics.
  2. Fréchet's research on functional analysis has influenced the development of the quantum formalism, allowing for the creation of abstract concepts crucial in understanding quantum mechanics.
  3. The Riesz–Fréchet representation theorem plays a key role in making mathematical sense of Dirac's bra-ket notation used in quantum mechanics, showcasing the impact of Fréchet's work in this field.

Lecture 3 Registration

Quantum Formalism • 0 implied HN points • 26 Jul 21
🚌 Education Topology
  1. Registration for Lecture 3 on topology is open via a provided link.
  2. Reviewing basic concepts from real analysis like open intervals and closed intervals is recommended for better understanding.
  3. Joining the Discord community server is encouraged, and a webinar on the application of Maths and Computing in various fields is scheduled.

Lecture 2 Replay

Quantum Formalism • 0 implied HN points • 16 Jul 21
🚌 Education Topology
  1. The lecture replay covers a topology crash course.
  2. The session was split into 3 parts: Intro & Industry Fellowship Announcement, Main Lecture, and Extra About Measure Theory.
  3. The speaker promised a future crash course on Measure Theory & Functional Analysis to refine the definition of sigma-algebra in constructing one from finite topologies.

Tessellating the Globe

Bram’s Thoughts • 0 implied HN points • 18 Jan 24
🔬 Science Topology
  1. The world jigsaw puzzle design has imperfections when trying to tessellate it without visible seams.
  2. Two geometries proposed to fix the issue involve dividing the globe into hemispheres and deforming them into squares or into triangles corresponding to a tetrahedron's faces.
  3. Alignment of these geometries with the equator and land bodies is an interesting challenge.

Replay + Invitation to Lecture 2

Quantum Formalism • 0 implied HN points • 05 Jul 21
🚌 Education Topology
  1. Lecture one is now available on YouTube, split into Welcome & Agenda, PART A (Course Introduction & Comments), and PART B (the actual Topology lecture).
  2. The next lecture on Topology is scheduled for Friday, July 16, with registration open. Joining the Discord community is recommended to connect with others and set up study groups.
  3. There was a gap before the next lecture to allow time for understanding topological concepts and solving challenges. This was decided based on a poll to give participants more time for preparation.

Discord Server Invitation + Background Poll

Quantum Formalism • 0 implied HN points • 30 Jun 21
🚌 Education Topology
  1. Join the Quantum Formalism Discord server for a topology crash course.
  2. The crash course will cover core concepts like topological spaces, closed sets, subspace topology, neighborhoods, and Hausdorff spaces.
  3. The course will include elements of Differential Geometry in the manifold section and participants are asked to fill out a short survey to help adjust the level of abstraction.

Lecture #1 Reschedule

Quantum Formalism • 0 implied HN points • 15 Jun 21
🚌 Education Topology
  1. The first lecture on Topology Crash Course has been rescheduled due to a funeral service on June 25
  2. Alternative dates offered are June 24, June 29, and July 2, all at 5pm BST
  3. Participants are reminded to review basic set-theoretic and real analysis concepts mentioned during the last Q&A session

Today's Session Reminder

Quantum Formalism • 0 implied HN points • 28 May 21
🚌 Education Topology
  1. There will be a live session at 5 pm BST today for a crash course on topology.
  2. Practical tips will be shared to help overcome potential abstraction barriers.
  3. The session recording will be available on YouTube for those who cannot attend.

Lie Theory Prerequisite

Quantum Formalism • 0 implied HN points • 28 Apr 21
🔬 Science Topology
  1. The post provides a crash course motivation and schedule for Lie Theory, encouraging viewers to watch preparatory material on YouTube and offering usage cases outside of mathematics for motivation.
  2. Highlighted articles and studies demonstrate real-world applications of Lie Theory in areas like quantum computation, deep learning, and unitary operators in quantum mechanics.
  3. The presenter provides access to slides and recommended study materials on GitHub, emphasizing the importance of preparation before the upcoming course session on Lie Theory.

Latest Community updates

Quantum Formalism • 0 implied HN points • 12 Apr 21
🚌 Education Topology
  1. The Lie Theory prerequisite mini-series will focus on point-set topology, metric spaces, and basics of differentiable manifolds.
  2. Reviewing basics of set theory, including intersections, unions, Cartesian products, and maps between sets, is recommended for the upcoming lectures.
  3. While the Lie Theory module may not be sufficient to understand Eric Weinstein's 'Geometric Unity' paper, it provides a foundational knowledge base that can ease the understanding of complex topics in differential geometry and topology.

Certificate of Attendance versus Certificate of Completion + Quantum Webinar

Quantum Formalism • 0 implied HN points • 04 Nov 21
🚌 Education Topology
  1. Two certificates are being issued for the crash course: Certificate of Attendance for those who attended the entire course, and Certificate of Completion for those who comprehend the basics discussed.
  2. The crash course covers topics like topological and smooth manifolds.
  3. A Quantum Webinar by Dr. Ryuji Takagi from Nanyang Technological University in Singapore is scheduled, presenting their recent ArXiv paper 'Fundamental limits of quantum error mitigation'.

Lecture 6 (Manifolds) Replay

Quantum Formalism • 0 implied HN points • 02 Oct 21
🚌 Education Topology
  1. The lecture 6 replay on manifolds is available on the YouTube playlist for the crash course.
  2. Registration for lecture 7 is now open via a link provided.
  3. Joining the Discord community server is encouraged by the author for further engagement.

Lecture 6 Registration

Quantum Formalism • 0 implied HN points • 29 Sep 21
🚌 Education Topology
  1. Lecture 6 registration for the topology crash course is now open at the provided crowdcast link.
  2. The lecture will feature a cartoon joke at the end, involving a flat earth theorist and a mathematician (topologist).
  3. Joining the Discord community server is highly encouraged for more engagement and interaction.

Lecture 5 Replay

Quantum Formalism • 0 implied HN points • 12 Sep 21
🚌 Education Topology
  1. The replay of lecture 5 is available on YouTube, split into 3 parts for convenient viewing and discussing path-connectedness, open covers, and compactness.
  2. Registration is open for lecture 6 on topology, focusing on the manifold section.
  3. You are encouraged to join the Discord community server as part of the Quantum Formalists group for further discussions and engagement.

Lecture 5 Invitation

Quantum Formalism • 0 implied HN points • 06 Sep 21
🚌 Education Topology
  1. The upcoming lecture will focus on a topology crash course before moving on to topological manifolds.
  2. Registration is available for the lecture happening this Friday at 5 pm BST.
  3. You're encouraged to join the Discord community server for further engagement.

Lecture 4 Replay

Quantum Formalism • 0 implied HN points • 16 Aug 21
🚌 Education Topology
  1. The replay of lecture 4 is available on YouTube split into 3 parts for easy viewing.
  2. Registration for lecture 5 is open, leading into the much-anticipated manifold section.
  3. Lecture 4 covers topics such as Topological Bases, Product Topology, Topological Invariants, and Connectedness.

Lecture 4 Registration

Quantum Formalism • 0 implied HN points • 10 Aug 21
🚌 Education Topology
  1. Lecture 4 registration for the topology crash course is now open
  2. Participants will be guided towards understanding topological manifolds in upcoming lectures
  3. There's an invitation to join the Discord community server