The first two that you should study are:
List of handbooks open as of 2022 at: www.maths.ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/handbooks-synopses Kudos, e.g. unlike the physics course of the University of Oxford which paywalled them. 2022 one: www.maths.ox.ac.uk/system/files/attachments/UG%20Handbook%202022.pdf
The Oxford mathematics Moodle has detailed course listings, and most PDFs are not paywalled.
E.g. the 2024 course:
- Year 1: everything seems mandatory:
- Michaelmas Term
- Introduction to University Mathematics
- Introduction to Complex Numbers
- Linear Algebra I
- Analysis I
- Introductory Calculus
- Probability
- Geometry
- Hilary Term
- Trinity Term
- Groups and Group Actions
- Analysis III
- Statistics and Data Analysis
- Constructive Mathematics
- Michaelmas Term
- Year 2:
- Mandatory big courses:
- long options:
- Rings and Modules
- Integration
- Topology
- Differential Equations 2
- Numerical Analysis
- Probability
- Statistics
- Fluids and Waves
- Quantum Theory
- short options
- Number Theory
- Group Theory
- Projective Geometry
- Integral Transforms
- Calculus of Variations
- Graph Theory
- Mathematical Modelling in Biology
- Year 3: pick any 8 courses. Does not say which courses exist in PDF but we can get them from courses.maths.ox.ac.uk/course/index.php?categoryid=814 of the Oxford mathematics Moodle:
- Michaelmas
- B1.1 Logic (2024-25)
- B2.1 Introduction to Representation Theory (2024-25)
- B3.2 Geometry of Surfaces (2024-25)
- B3.5 Topology and Groups (2024-25)
- B4.1 Functional Analysis I (2024-25)
- B5.2 Applied Partial Differential Equations (2024-25)
- B5.3 Viscous Flow (2024-25)
- B5.5 Further Mathematical Biology (2024-25)
- B6.1 Numerical Solution of Partial Differential Equations (2024-25)
- B6.3 Integer Programming (2024-25)
- B7.1 Classical Mechanics (2024-25)
- B8.1 Probability, Measure and Martingales (2024-25)
- B8.4 Information Theory (2024-25)
- B8.5 Graph Theory (2024-25)
- BO1.1 History of Mathematics (2024-25)
- BOE Other Mathematical Extended Essay (2024-25)
- BSP Structured Projects (2024-25)
- Hilary
- B1.2 Set Theory (2024-25)
- B2.2 Commutative Algebra (2024-25)
- B2.3 Lie Algebras (2024-25)
- B3.1 Galois Theory (2024-25)
- B3.3 Algebraic Curves (2024-25)
- B3.4 Algebraic Number Theory (2024-25)
- B4.3 Distribution Theory (2024-25)
- B4.2 Functional Analysis II (2024-25)
- B5.1 Stochastic Modelling of Biological Processes (2024-25)
- B5.4 Waves and Compressible Flow (2024-25)
- B5.6 Nonlinear Dynamics, Bifurcations and Chaos (2024-25)
- B6.2 Optimisation for Data Science (2024-25)
- B7.2 Electromagnetism (2024-25)
- B7.3 Further Quantum Theory (2024-25)
- B8.2 Continuous Martingales and Stochastic Calculus (2024-25)
- B8.3 Mathematical Models of Financial Derivatives (2024-25)
- B8.6 High Dimensional Probability (2024-25)
- SB3.1 Applied Probability (2024-25)
- BO1.1 History of Mathematics (2024-25)
- BOE Other Mathematical Extended Essay (2024-25)
- BSP Structured Projects (2024-25)
- Michaelmas
- Year 4: pick any 8 courses (up to 10 if you're crazy). Does not say which courses exist in PDF but we can get them from courses.maths.ox.ac.uk/course/index.php?categoryid=814 of the Oxford mathematics Moodle:
- Michaelmas
- C1.1 Model Theory (2024-25)
- C1.4 Axiomatic Set Theory (2024-25)
- C2.2 Homological Algebra (2024-25)
- C2.4 Infinite Groups (2024-25)
- C2.7 Category Theory (2024-25)
- C3.1 Algebraic Topology (2024-25)
- C3.3 Differentiable Manifolds (2024-25)
- C3.4 Algebraic Geometry (2024-25)
- C3.7 Elliptic Curves (2024-25)
- C3.8 Analytic Number Theory (2024-25)
- C4.1 Further Functional Analysis (2024-25)
- C4.3 Functional Analytic Methods for PDEs (2024-25)
- C5.2 Elasticity and Plasticity (2024-25)
- C5.5 Perturbation Methods (2024-25)
- C5.7 Topics in Fluid Mechanics (2024-25)
- C5.11 Mathematical Geoscience (2024-25)
- C5.12 Mathematical Physiology (2024-25)
- C6.1 Numerical Linear Algebra (2024-25)
- C6.5 Theories of Deep Learning (2024-25)
- C7.1 Theoretical Physics (C6) (2024-25)
- C7.5 General Relativity I (2024-25)
- C8.1 Stochastic Differential Equations (2024-25)
- C8.3 Combinatorics (2024-25)
- CCD Dissertations on a Mathematical Topic (2024-25)
- COD Dissertations on the History of Mathematics (2024-25)
- Hilary
- C1.2 Gödel's Incompleteness Theorems (2024-25)
- C1.3 Analytic Topology (2024-25)
- C2.3 Representation Theory of Semisimple Lie Algebras (2024-25)
- C2.5 Non-Commutative Rings (2024-25)
- C2.6 Introduction to Schemes (2024-25)
- C3.2 Geometric Group Theory (2024-25)
- C3.5 Lie Groups (2024-25)
- C3.6 Modular Forms (2024-25)
- C3.9 Computational Algebraic Topology (2024-25)
- C3.10 Additive Combinatorics (2024-25)
- C3.11 Riemannian Geometry (2024-25)
- C3.12 Low-Dimensional Topology and Knot Theory (2024-25)
- C4.6 Fixed Point Methods for Nonlinear PDEs (2024-25)
- C4.9 Optimal Transport & Partial Differential Equations (2024-25)
- C5.1 Solid Mechanics (2024-25)
- C5.4 Networks (2024-25)
- C5.6 Applied Complex Variables (2024-25)
- C6.2 Continuous Optimisation (2024-25)
- C6.4 Finite Element Method for PDEs (2024-25)
- C7.1 Theoretical Physics (C6) (2024-25)
- C7.4 Introduction to Quantum Information (2024-25)
- C7.6 General Relativity II (2024-25)
- C7.7 Random Matrix Theory (2024-25)
- C8.2 Stochastic Analysis and PDEs (2024-25)
- C8.4 Probabilistic Combinatorics (2024-25)
- C8.7 Optimal Control (2024-25)
- CCD Dissertations on a Mathematical Topic (2024-25)
- COD Dissertations on the History of Mathematics (2024-25)
- Michaelmas
Aptitude test scene from the Snowden 2016 film
. Source. FISA Court Order The Guardian discussion scene from the Snowden 2016 film
. Source. How is this Possible? scene from the Snowden 2016 film
. Source. Fresh Brains for You scene from the Snowden 2016 film
. Source. The last value we will likely every know for the busy beaver function! BB(6) is likely completely out of reach forever.
By 2023, it had basically been decided by the The Busy Beaver Challenge as mentioned at: discuss.bbchallenge.org/t/the-30-to-34-ctl-holdouts-from-bb-5/141, pending only further verification. It is going to be one of those highly computational proofs that will be needed to be formally verified for people to finally settle.
As that project beautifully puts it, as of 2023 prior to full resolution, this can be considered the:on the Busy beaver scale.
simplest open problem in mathematics
Quantum Computation and Quantum Information by Nielsen and Chuang by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Lists of the most promising implementations:
As of 2020, the hottest by far are:
Super quick overview of the main types of quantum computer physical implementations, so doesn't any much to a quick Google.
He says he's going to make a series about it, so then something useful might actually come out. The first one was: Video "How to Turn Superconductors Into A Quantum Computer by Lukas's Lab (2023)", but it is still too basic.
The author's full name is Lukas Baker, www.linkedin.com/in/lukasbaker1331/, found with Google reverse image search, even though the LinkedIn image is very slightly different from the YouTube one.
To get an intuition for it, see the sample computation at: en.wikipedia.org/w/index.php?title=Ackermann_function&oldid=1170238965#TRS,_based_on_2-ary_function where in this context. From this, we immediately get the intuition that these functions are recursive somehow.
Publicly released documents from the Los Alamos National Laboratory are marked with this identifier. This is for example the case of each video on ther YouTube channel: www.youtube.com/@LosAlamosNationalLab. E.g. Video "Historic, unique Manhattan Project footage from Los Alamos by Los Alamos National Lab" is marked with "LA-UR 11-4449".
www.osti.gov/biblio/1372821 contains "How to Get an LA-UR: Using RASSTI to Release Your Work" which is of interest: permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-UR-17-26023. That document documents the acronym's expansion, plus it leaks some internal-only URLs such as lasearch.lanl.gov/oppie/service.
TODO is there somewhere you can search for the document for a given identifier? Some PDFs are listed at: sgp.fas.org/othergov/doe/lanl/index2b.html
Sample implementations:
www.nature.com/articles/d41586-023-02600-xThey overreached it seems.
Almost since it began, however, the HBP has drawn criticism. The project did not achieve its goal of simulating the whole human brain — an aim that many scientists regarded as far-fetched in the first place. It changed direction several times, and its scientific output became “fragmented and mosaic-like”, says HBP member Yves Frégnac
Apparently posted to
ba.jobs.offered Usenet newsgroup?The prototypical example is the Busy beaver function, which is the easiest example to reach from the halting problem.
CIA 2010 covert communication websites USA spying on its own allies by Ciro Santilli 40 Updated 2025-08-08
Ciro Santilli 40 Updated 2025-08-08Being Brazilian, Ciro Santilli was particularly curious about the existence of a Brazil-focused mentioned in the Reuters article, as well as in other democracies.
WTF the CIA was doing in Brazil in the early 2010s! Wasn't helping to install the Military dictatorship in Brazil enough!
Here are the websites likely targeting democracies based on their language and content found so far, defining a democracy as a country with score 7.0 or more in the Democracy index 2010:In English, so more deniable:"Almost democracies":Ciro couldn't help but feel as if looking through the Eyes of Sauron himself!
- France (6: affairesdumonde.com, guide-daventure.com, lesummumdelafinance.com, football-de-luxe.com, romulusactualites.com, suparakuvi.com)
- Germany(2: dedrickonline.com, neighbour-news.com)
- Italy (2: attivitaestremi.com, garanziadellasicurezza.com, podisticamondiale.com)
- Spain (3: armashoy.com, montanismoaventura.com, ordenpolicial.com)
- Brazil (2: noticiasmusica.net, vejaaeuropa.com)
- South Korea (1: economicnewsbuzz.com)
- Poland (1: boxingstop.net)
- Japan (1: snapnewsfront.net)
- Canada (2: kanata-news.com, mynewscheck.com)
- Philippines (1: half-court.net)
- India (1: amishkanews.com)
- Croatia (2: european-footballer.com, stara-turistick.com)
- Thailand (1: thefairwaysaregreen.com)
- Peru (1: todosperuahora.com)
It is worth noting that democracies represent just a small minority of the websites found. The Middle East, and Spanish language sites (presumably for Venezuela + war on drugs countries?) were the huge majority. But Americans have to understand that democracies have to work together and build mutual trust, and not spy on one another. Even some of the enlightened people from Hacker News seem to not grasp this point. The USA cannot single handedly maintain world order as it once could. Collaboration based on trust is the only way.
Snowden's 2013 revelations particularly shocked USA "allies" with the fact that they were being spied upon, and as of the 2020's, everybody knows this and has "stopped caring", and or moved to end-to-end encryption by default. This is beautifully illustrated in the 2016 film "Snowden" when Snowden talks about his time in Japan working for Dell as an undercover NSA operative:
NSA wanted to impress the Japanese. Show them our reach. They loved the live video from drones. This is Pakistan right now [video shows American agents demonstrating drone footage to Japanese officials]. They were not as excited about that we wanted their help to spy on the Japanese population. They said it was against their laws.And we did not stop there. Once we owned their communications systems, we started going after the physical infrastructure.We'd slip these little sleeper programs into power grids, dams, hospitals. The idea was that if the day came when Japan was no longer an ally, it would be "lights out".But Austria?!
But Austria?! scene from Snowden (2016)
. Source. Another noteworthy scene from that movie is Video 2. "Aptitude test on communication networks scene from the 2016 Snowden film", where a bunch of new CIA recruits are told that:thus somewhat mirroring what actually happened with these real world websites.
Each of you is going to build a covert communications network in your home city [i.e. their fictitious foreign target location written on each person's desk such as Berlin, Istanbul and Bangkok, not necessarily where they were actually born], you're going to deploy it, backup your site, destroy it, and restore it again.
There are unlisted articles, also show them or only show them.