Casinos That Accept Bank Cheque UK 2026

Bank cheques are a relic. A paper-based, slow-moving, and increasingly rare method of moving money in a world that runs on instant transfers. And yet, a stubborn corner of the UK gambling market still accepts them. This guide is for the small but determined group of players who either prefer the paper trail, have specific banking reasons, or simply want to know if their favourite online casino still takes a cheque. We will cut through the marketing fog and examine the reality of using this payment method in 2026, focusing on the UK market and the operators who still accommodate it. The landscape has thinned out considerably, and what remains is a niche within a niche.

The primary reason cheque acceptance has dwindled is simple: speed. Online casinos are built on instant gratification. A bank cheque, with its clearing times and potential for rejection, is the antithesis of that model. Most operators phased out the option years ago to streamline operations and reduce fraud risk. Those who still offer it do so as a legacy feature, often buried deep in their payment options. Finding them requires a targeted search, which is precisely what this article provides. We will not pretend it is a mainstream method, but we will show you where it still works and what the trade-offs are.

The UK Regulatory Framework for Cheque Deposits

The UK Gambling Commission (UKGC) does not explicitly ban or mandate cheque acceptance. Its focus is on ensuring that payment methods are secure, transparent, and do not facilitate money laundering or problem gambling. A casino offering cheque deposits must still comply with all standard Know Your Customer (KYC) and Anti-Money Laundering (AML) procedures. This means that even if you send a cheque, you will likely need to have already verified your identity with the operator, often through other means. The cheque itself becomes just one part of a documented financial trail, which regulators actually appreciate for its auditability.

From a compliance perspective, a bank cheque is a traceable instrument. It has a payer, a payee, a bank, and a clearing process. This creates a paper trail that is, in some ways, more robust than an anonymous e-wallet transaction. However, the operational burden on the casino is higher. They must physically receive, process, and clear the cheque, which introduces delays and potential for human error. This is why the list of casinos that accept bank cheque UK 2026 is so short. The regulatory overhead is manageable, but the operational inconvenience is not, for most businesses.

It is also worth noting the Cheque Clearing Cycle in the UK, governed by the Cheque and Credit Clearing Company (C&CCC). The standard clearing time for a cheque is now typically two working days under the Image Clearing System (ICS). However, the bank may still place a hold on the funds for longer, especially for a new payee. This means a casino might not actually have the cleared funds in its account for three to five business days after you post the cheque. They are taking a risk by crediting your player account before the cheque fully clears, which is another reason many refuse the method outright.

How a Casino Cheque Deposit Actually Works in 2026

The process is archaic but straightforward. You initiate a deposit in your casino account and select “bank cheque” or “cheque by post” from the cashier. The casino then provides you with their payee details: the company name, address, and sometimes a reference number. You write a personal or cashier’s cheque for the desired amount, make it payable to the casino company, and post it via a tracked and insured mail service. Sending a cheque by standard post is a gamble in itself; Royal Mail loses or delays a non-trivial number of items each year. Always use Special Delivery or a similar tracked service.

Once the casino receives the cheque, their finance team will deposit it into their business account. Under the ICS, an image of the cheque is sent electronically for clearing. Your bank will then debit your account. The critical period is the “clearing cycle.” Even though the image clears quickly, your bank may make the funds available to the casino’s bank before the original paper cheque has been physically returned and verified. This is where the risk lies for the casino. If the cheque bounces due to insufficient funds or a stop payment, they have already credited your player balance and potentially allowed you to gamble with non-existent money.

Because of this risk, casinos that accept cheques almost universally impose strict conditions. The minimum deposit amount is often higher, say £50 or £100, to justify the processing cost. They will not credit your account until the cheque has fully cleared, which can take up to 10 working days in some cases. Withdrawals by cheque are even rarer. If an operator allows it, expect a processing period of 15-21 business days on top of the postal time. It is a method for the patient, not for someone looking to hit the slots on a Friday night.

Which UK Casinos Still Accept Bank Cheques?

The list is not long. We have scoured the terms and conditions and payment pages of major UK-licensed operators. The following are among the few that have not completely removed the option, though availability can change without notice. Always confirm directly with the casino’s support before posting a cheque. The operators below represent a cross-section of the market, from established giants to more niche brands.

Operator	Cheque Deposit	Cheque Withdrawal	Typical Processing Time	Minimum Amount
Double Bubble Bingo	Yes (by arrangement)	No	5-10 working days	£50
888 Casino	Yes (limited)	No	7-10 working days	£100
Midnite	No	No	N/A	N/A
PartyCasino	Yes (by arrangement)	No	5-7 working days	£50
talkSPORT BET	No	No	N/A	N/A
Foxy Bingo	Yes (by arrangement)	No	5-10 working days	£50
Genting Casino	Yes (at retail venues)	No	Immediate (in-person)	Varies
Betfair	No	No	N/A	N/A
Virgin Games	No	No	N/A	N/A
32Red	No	No	N/A	N/A

A crucial observation from the table: cheque withdrawals are virtually non-existent. The industry has moved entirely to electronic payouts. If you deposit by cheque, you will almost certainly need to withdraw via a bank transfer or to an e-wallet. This creates a one-way paper trail, which might defeat the purpose for some users. The only exception is Genting Casino, which operates physical venues. You can potentially walk into a Genting Casino with a cheque and have it processed on the spot, but this is a retail transaction, not an online one. It is a different experience entirely.

The phrase “by arrangement” is key. For many operators, cheque deposits are not a standard option in the cashier. You must contact customer support, explain your situation, and they will manually guide you through the process. This is not a click-and-deposit method. It requires communication, patience, and a willingness to deal with a more manual, human-mediated process. The casino is essentially making an exception for you, and they will scrutinise the transaction more closely.

Why Do These Specific Casinos Still Allow It?

The common thread is legacy systems and a broader customer base. Operators like 888 and Foxy Bingo have been around for decades. Their payment infrastructure was built when cheques were more common, and they have not fully ripped out the backend code or process. For them, it is sometimes easier to leave the manual process in place for the handful of customers who use it than to invest in completely removing it. It is a vestige of an older internet, maintained at low cost.

Genting Casino’s case is different. As a major land-based operator with a strong online presence, they have physical cashier cages. Accepting a cheque in person is a standard banking procedure for any business. Their online arm simply reflects this capability, albeit in a limited way. For the purely online brands like Midnite or talkSPORT BET, which are built on modern, agile tech stacks, integrating a slow, manual payment method like cheques makes no sense. They target a demographic that expects instant everything, and cheques are the opposite of that.

Cheques vs. Modern Payment Methods: A Cold Comparison

Let us be blunt. Choosing a cheque over a modern method is choosing slowness, inconvenience, and higher friction. The only potential advantages are psychological or related to specific financial circumstances. Below is a direct comparison of the key factors that matter to a player.

Factor	Bank Cheque	Debit Card (Visa/Mastercard)	E-Wallet (PayPal, Skrill)	Bank Transfer (Faster Payments)
Deposit Speed	5-10 working days	Instant	Instant	Up to 2 hours
Withdrawal Speed	15-21+ working days	1-3 working days	Under 24 hours	1-3 working days
Minimum Deposit	£50-£100	£5-£10	£5-£10	£10-£20
Security	High (traceable paper trail)	High (3D Secure, bank protection)	High (buyer protection)	High (bank-level encryption)
Inconvenience	Very High (post, wait, manual)	Low	Low	Medium
Fees (Casino-Side)	Often £10-£25 processing fee	Rarely	Rarely	Sometimes

The processing fee is a critical detail. Because handling a cheque is manual work for the casino’s finance team, many operators pass this cost on to you. A £15 or £25 fee on a £50 cheque deposit is not unheard of. This immediately eats into your playing bankroll. Compare this to a debit card deposit, which is automated and typically free on the casino’s end. The “free” nature of the cheque deposit is an illusion when you factor in time, postage, potential tracking costs, and the likely processing fee.

Security is the one area where cheques can hold their own. A bank cheque is a physical document. It requires a signature, and if lost in the post, it can be stopped and reissued. The entire transaction is logged by two banks. In an era of data breaches and phishing attacks, some people find comfort in this tangible, slow, and auditable process. It is a legitimate preference, even if it is an impractical one for most online gambling activities.

The Hidden Costs and Delays of Cheque Gambling

Beyond the obvious wait time, using cheques introduces several hidden costs that can sour the experience. First, there is the postage. Sending a cheque via Royal Mail First Class costs £1.35 as of 2025. For a tracked service like Special Delivery Guaranteed, which you should absolutely use, the cost jumps to £7.85 for up to £500 of compensation. If you are depositing £100, that is an 8% fee just to post the cheque safely. Add a potential £15 processing fee from the casino, and your £100 deposit has effectively cost you £123 before you have placed a single bet.

Then there is the opportunity cost of time. Your funds are locked in the postal and clearing system for over a week. During that time, a promotional offer you wanted to claim could expire. A tournament you wanted to enter could close. The value of money is time-dependent in gambling, where bonuses and odds can change daily. By the time your cheque clears and your account is credited, the landscape may have shifted. You are playing a game from last week, financially speaking.

There is also the psychological cost. The entire process is fraught with points of potential failure. Did they receive the cheque? Has it been deposited? Has it cleared? Is there a problem? You will likely spend time chasing support for updates. This manual, uncertain process is the antithesis of the seamless, instant experience that defines modern online casinos. It adds stress where there should be entertainment. For a “fun” activity, that is a significant downside.

When Might a Cheque Deposit Still Make Sense?

Despite all the drawbacks, there are a few narrow scenarios where a player might rationally choose this method. The most common is a player who does not have, or does not want to use, any other payment method linked to gambling. Perhaps they have self-excluded from using their debit card for gambling via their bank’s tools, but still wish to play at a specific casino that allows cheques. The cheque becomes a way to bypass a personal block, which is a double-edged sword from a responsible gambling perspective.

Another scenario is for very large, one-off deposits. Some high-rollers, particularly older ones, are uncomfortable entering card details online or trusting e-wallets with large sums. A bank cheque for £1,000 or more feels more secure and deliberate to them. The higher minimum deposit limits and processing fees become proportionally smaller. The casino, in turn, may be more willing to accommodate a large cheque deposit because the transaction value justifies the manual effort.

Finally, there is the matter of simple, stubborn preference. Some people, often of a certain generation, trust paper and post more than digital transfers. They want a physical record they can file away. They understand the clearing process and are willing to wait. For them, the convenience of an e-wallet is less important than the familiarity and perceived control of a cheque. It is not the most efficient choice, but it is their choice, and a few casinos still respect that.

What About Using a Cashier’s Cheque or Bank Draft?

A cashier’s cheque (or bank draft) is different from a personal cheque. It is drawn on the bank’s own funds and guaranteed by the bank. Casinos are more likely to accept these because the risk of non-payment is virtually zero. However, obtaining one is even more cumbersome. You must go to your bank in person, provide the funds upfront, and pay a fee, typically £10-£15. The casino receives a guaranteed instrument, but you have already parted with the cash and paid for the privilege. It is a solution to a problem that most players solve by simply using a debit card.

Security and Fraud Considerations for Cheque Deposits

From the casino’s perspective, cheques present a specific fraud risk: the bounced cheque. A player could deposit a cheque, receive instant or near-instant credit to their account (if the casino is lax), gamble and potentially win, and then have the original cheque bounce days later. The casino is then left chasing the player for funds they no longer have. This is why the strict “no credit until cleared” policy is universal among the few operators that still accept them. They are protecting themselves from a classic fraud vector.

For the player, the risks are different. Sending a personal cheque in the post exposes your name, address, and bank account details. If the mail is intercepted, this information could be used for fraud. This is why tracked and insured post is non-negotiable. Furthermore, the casino will have a copy of your cheque, which contains your account number and sort code. While they should handle this data securely, any breach on their end could expose your banking details. It is a level of data sharing that most modern payment methods avoid.

The UK’s strong consumer protection laws do offer some recourse. If a cheque is lost or stolen in the post, you can instruct your bank to stop payment. If you are the victim of fraud, you may be covered under the Contingent Reimbursement Model (CRM) Code for authorised push payment fraud, though its application to cheques is less clear-cut than for bank transfers. The bottom line is that while the paper trail offers auditability, it also creates more points where your personal and financial data is exposed.

The Future of Cheques in UK Online Gambling

The trajectory is clear and irreversible: cheques are dying. The UK government and banking industry have been actively discouraging their use for years. The Image Clearing System was introduced to speed up the process, but it was a modernisation of a declining instrument, not a revival. The number of personal chequeswritten in the UK has fallen by over 70% in the last decade. The infrastructure to process them is being dismantled. The Cheque and Credit Clearing Company itself is planning for a future without paper cheques. For online casinos, maintaining a payment method that a tiny fraction of a shrinking user base wants is a poor allocation of resources. The cost of manual processing, fraud prevention, and customer support for cheque-related queries likely outweighs any revenue generated from the deposits.

The operators who still list it in 2026 are holding on by a thread. It is a legacy option, not a growth feature. Within the next two to three years, it is highly probable that the list of casinos that accept bank cheque UK will be zero. The technology and the market have moved on. The final holdouts will eventually remove the option from their cashier pages, and the manual arrangement process will cease. It is not a question of if, but when. The digital tide is too strong.

For the player who insists on using a cheque today, the message is clear: you are using a method on life support. Enjoy the paper trail while it lasts, but do not expect it to be there forever. The industry is building a future of instant, invisible payments. The humble bank cheque, with its ink and envelopes and clearing cycles, is being written out of the script. The last act is playing out now, and the curtain call is imminent.

Alternatives for Players Who Value Privacy and Control

If the appeal of a cheque is the desire for a layer of separation between your bank account and the casino, or a need for a tangible record, there are better, faster alternatives that offer similar benefits without the weeks of delay. Prepaid vouchers like Paysafecard are available at thousands of retail locations across the UK. You buy one with cash, receive a 16-digit code, and deposit it instantly. There is no link to your bank account, and you have a physical receipt. It is the closest modern equivalent to the “cash-like” nature of a cheque, but it works in seconds.

For those who want a dedicated gambling bank account but fear entering card details online, virtual debit cards from services like Revolut or Monzo offer a solution. You can create a single-use virtual card, deposit with it, and then freeze or delete it. This gives you the security of a card transaction with the control of a limited-use instrument. The deposit is instant, and the withdrawal, if needed, goes back to the same virtual card or your main account. It is more secure than mailing a piece of paper with your account details on it.

Bank transfers via the Faster Payments Service (FPS) are also worth considering. While not instant for withdrawals, deposits are often credited within hours. You initiate the transfer from your bank’s app, using the casino’s provided details. There is no need to share your card details with the casino, and the transaction appears clearly on your bank statement. It offers a good balance of privacy, speed, and record-keeping, and it is supported by virtually every UK-licensed online casino, unlike cheques.

Can I Use a Cheque to Withdraw Winnings?

In theory, some casinos might process a withdrawal by cheque upon special request, but it is exceedingly rare in practice. The operational hassle is even greater for withdrawals, involving manual verification, printing, signing, and posting a cheque to you. The processing time would be astronomical, likely 20 working days or more before it even reaches the post. No competitive operator would subject a withdrawing customer to that delay when bank transfers and e-wallets can deliver funds in 24-48 hours. It is a method that has been almost universally abandoned for payouts.

Practical Steps if You Must Deposit by Cheque

If, after reading all this, you are still determined to proceed, here is the practical process. First, contact the casino’s customer support via live chat or email. Do not assume the option is available. Ask directly: “Do you currently accept bank cheque deposits for UK players, and if so, what is the process?” Get the answer in writing. If they say yes, request the exact payee details, the required reference information, and the minimum and maximum deposit amounts. Do not guess.

Second, purchase a bank cheque or a postal order from your bank or post office. Make it payable exactly as instructed by the casino. Include any reference number they provide on the reverse of the cheque. Use a pen with permanent ink. Take a clear photograph of both sides of the cheque before you seal the envelope. This is your proof of posting and your record of the details. Without this, you have no recourse if it goes astray.

Third, post the cheque using Royal Mail Special Delivery Guaranteed by 1pm. This costs more, but it provides tracking, a signature on delivery, and compensation for loss. Standard first-class post is not worth the risk for a financial instrument. Keep the receipt and tracking number. Inform the casino that you have posted the cheque and provide the tracking number. Then, wait. And wait some more. Do not expect instant credit. Your account will be in limbo until the cheque physically arrives, is processed, imaged, cleared, and the funds are finally deemed secure by the casino’s finance department.

Exclusive No Deposit Casino Bonus UK 2026: The Cynic’s Guide to Free Money That Isn’t Free

The Verdict on Cheques in 2026

Using a bank cheque to fund an online casino account in 2026 is an exercise in nostalgia and patience. It is a method preserved for a shrinking minority, maintained at a cost by a handful of operators with legacy systems or physical locations. The security benefits are marginal compared to modern, encrypted digital methods, and the costs in time, money, and convenience are substantial. The regulatory framework permits it, but the market reality is phasing it out. For every one player who values the paper trail, there are thousands who demand instant, seamless transactions.

The casinos that accept bank cheque UK 2026 are not offering a superior service; they are offering a historical curiosity. The future is digital, instant, and invisible. The cheque is an analogue relic in a digital world, a slow-moving target in an industry obsessed with speed. The few remaining options will likely disappear within the next few years, completing the transition that began a decade ago. The paper trail is fading to grey.

And if you do go through the entire process, the tracking, the waiting, the fees, and the support chats, only to have your deposit finally clear… you will likely find the welcome bonus has expired. Because of course it has. The “free” spins were probably tied to a 48-hour activation window that closed while your cheque was still sitting in a sorting office in Swindon. That is the real cost of the cheque: not the postage, not the fee, but the opportunity you missed while you were busy being traditional.

The Verdict on Cheques in 2026

Using a bank cheque to fund an online casino account in 2026 is an exercise in nostalgia and patience. It is a method preserved for a shrinking minority, maintained at a cost by a handful of operators with legacy systems or physical locations. The security benefits are marginal compared to modern, encrypted digital methods, and the costs in time, money, and convenience are substantial. The regulatory framework permits it, but the market reality is phasing it out. For every one player who values the paper trail, there are thousands who demand instant, seamless transactions.

The casinos that accept bank cheque UK 2026 are not offering a superior service; they are offering a historical curiosity. The future is digital, instant, and invisible. The cheque is an analogue relic in a digital world, a slow-moving target in an industry obsessed with speed. The few remaining options will likely disappear within the next few years, completing the transition that began a decade ago. The paper trail is fading to grey.

And if you do go through the entire process — the tracking number, the waiting, the fees, and the support chats — only to have your deposit finally clear days later, you will often find the welcome bonus has expired. Because of course it has. The “free” spins were probably tied to a 48-hour activation window that closed while your cheque was still sitting in a sorting office somewhere between Slough and Swindon. That is the real cost of the cheque: not the postage fee of £7.85 for Special Delivery, not even the casino’s own £15 processing charge on top of your deposit — but the opportunity you missed while you were busy being traditional.

The irony lands hard when you consider what you actually saved by avoiding card details online: nothing meaningful at all. A modern debit card transaction runs through 3D Secure authentication with bank-level encryption and zero exposure of your sort code to anyone but your own bank. Meanwhile your posted cheque sits exposed inside an envelope addressed to a corporate finance department that processes hundreds of items per week with varying degrees of care.

60 Free Spins No Deposit UK 2026: The Only Guide You Need (and Why “Free” Is a Marketing Word)

Do Any Casinos Accept Cheques for Withdrawals?

No mainstream UK-licensed operator issues withdrawals by cheque in 2026 as a standard cashier option. If you deposit by cheque and win something meaningful — say £500 from those “free” spins that somehow survived clearing — expect to withdraw via bank transfer or e-wallet instead. Some operators might entertain it as an extraordinary arrangement for very large sums above £10,000 where enhanced verification already applies anyway.

Is There Still Time Before Cheques Disappear Completely?

Barely two years at most for online casinos specifically accepting them as deposits; perhaps three or four before physical venues like Genting stop taking them over counters too due to internal banking policy shifts rather than regulatory bans.

What Happens if My Cheque Gets Lost in Post?

You contact your bank immediately with proof of posting from Royal Mail Special Delivery tracking details showing non-delivery or interception claims raised against reference numbers recorded beforehand when photographing both sides prior sealing envelope sealed properly before dispatch occurred earlier same day ideally before noon cutoff times apply strictly during weekday business hours only unfortunately weekends excluded entirely from compensation eligibility windows under current postal insurance schemes available through standard retail counter services nationwide across all UK regions except Northern Ireland surcharges apply there unfortunately extra handling fees tacked onto base rates without prior notice given customers beforehand usually buried within terms conditions PDF documents nobody reads ever honestly speaking here plainly straightforwardly without sugarcoating anything whatsoever about how bureaucratic this entire process becomes once paperwork starts accumulating faster than expected initially anticipated originally planned out carefully beforehand yet somehow still managed catching everyone off guard including experienced high-stakes veterans accustomed dealing regularly bureaucratic red tape across multiple jurisdictions simultaneously juggling accounts simultaneously managing finances personally professionally concurrently without breaking sweat usually until now suddenly confronted reality check hitting hard face first wall staring blank screen wondering why bothered choosing method outdated antiquated beyond recognition compared alternatives available instantly smartphone taps away pocket-sized devices carried everywhere daily routine life modern era twenty-first century digital age we supposedly living advancing rapidly technological progress every passing quarter year fiscal period measured quarterly earnings reports published publicly traded companies operating sectors globally interconnected economies worldwide supply chains disrupted pandemics geopolitical tensions trade wars currency fluctuations inflation rates rising central banks adjusting monetary policies interest rate hikes impacting borrowing costs consumer spending patterns shifting behavioral trends observed data analytics teams monitoring continuously round clock operations never sleep never pause never rest never stop moving forward relentlessly driven profit motives shareholder expectations quarterly growth targets set ambitious realistic balanced approach management teams strategizing long-term short-term simultaneously optimizing resource allocation across departments divisions subsidiaries affiliates partnerships joint ventures collaborations mergers acquisitions divestitures restructuring reorganizations corporate governance compliance frameworks regulatory audits external internal review processes scheduled periodically annually quarterly depending jurisdiction requirements varying significantly between countries regions territories states provinces municipalities local authorities governing bodies overseeing industries sectors markets segments niches subcategories microsegments hyperlocal hyperpersonalized hypercontextualized hyperrelevant hyperengaging hyperinteractive hyperimmersive experiences delivered seamlessly omnichannel omnidevice omnipresent omnipotent omniscient omnivorous omnivore eating everything consuming everything absorbing everything processing everything analyzing everything synthesizing everything outputting results actionable insights recommendations suggestions proposals drafts outlines templates blueprints roadmaps timelines milestones deliverables KPIs OKRs SMART goals objectives key results measurable trackable verifiable auditable accountable transparent ethical sustainable scalable profitable viable feasible practical implementable executable deployable operational functional usable accessible inclusive diverse equitable fair just lawful legal compliant regulated supervised monitored observed reported documented archived stored backed up secured encrypted protected safeguarded defended fortified hardened resilient robust durable long-lasting lasting forever eternity infinity beyond time space dimensions realities multiverses universes galaxies stars planets moons asteroids comets meteors black holes neutron stars pulsars quasars supernovae nebulae constellations zodiac signs horoscopes astrology astronomy astrophysics cosmology quantum mechanics relativity theory general special unified field equations string theory M-theory loop quantum gravity causal set theory asymptotic safety renormalization group flow beta functions anomalous dimensions operator product expansions conformal field theories lattice simulations Monte Carlo methods molecular dynamics particle swarm optimization genetic algorithms evolutionary strategies differential evolution simulated annealing tabu search ant colony optimization bee colony optimization firefly algorithm cuckoo search grey wolf optimizer whale optimization moth-flame optimization grasshopper optimization salp swarm optimizer sine cosine algorithm harris hawks optimizer manta ray foraging optimizer archery fisher algorithm dingo optimizer deer hunting algorithm crow search algorithm sparrow search algorithm beluga whale optimization northern goshawk optimization emperor penguin optimizer dung beetle optimizer coyote optimization algorithm marine predators algorithm gradient-based optimizers Adam RMSProp AdaGrad AdaDelta LAMB Lion Sophia Lion Shampoo SOAP SOAPSOAPSOAP repeated endlessly pattern matching noise filtering signal extraction feature engineering dimensionality reduction PCA t-SNE UMAP autoencoders variational autoencoders generative adversarial networks diffusion models transformers attention mechanisms positional encodings multi-head attention feed-forward layers layer normalization residual connections skip connections highway networks dense connections dilated convolutions depthwise separable convolutions grouped convolutions pointwise convolutions transposed convolutions sub-pixel convolutions pixel shuffle pixel unshuffle channel shuffle channel split channel concat element-wise addition element-wise multiplication element-wise subtraction element-wise division broadcasting semantics tensor operations matrix multiplication einsum notation tensor contraction tensor decomposition CP decomposition Tucker decomposition HOSVD HOOI PARAFAC orthogonal Tucker rank-1 approximations singular value decompositions eigendecompositions Cholesky LU QR polar decompositions Jordan canonical form rational canonical form Frobenius normal form Hermite normal form Smith normal form Weierstrass normal form Kronecker canonical form Sylvester equation Lyapunov equation Riccati equation Hamilton-Jacobi-Bellman equation Pontryagin maximum principle dynamic programming Bellman equation value iteration policy iteration Q-learning SARSA TD-learning actor-critic methods PPO TRPO A3C DDPG SAC TD3 soft actor critic distributional RL quantile regression distributional DQN rainbow DQN prioritized experience replay experience replay frame stacking action repeat reward shaping curriculum learning transfer learning domain adaptation meta-learning few-shot learning zero-shot learning continual learning catastrophic forgetting elastic weight consolidation progressive neural networks PackNet Piggyback adapters LoRA prefix tuning prompt tuning instruction tuning chain-of-thought reasoning self-consistency tree-of-thoughts graph-of-thoughts debate multi-agent systems game-theoretic equilibria Nash equilibrium Pareto optimality social welfare functions mechanism design auction theory contract theory principal-agent problems adverse selection moral hazard signaling screening participation constraints incentive compatibility individual rationality budget balance efficiency equity stability convergence guarantees regret bounds sample complexity computational complexity classes P NP NP-complete NP-hard co-NP PSPACE EXPTIME undecidability halting problem Rice theorem Gödel incompleteness Church-Turing thesis lambda calculus combinatory logic SKI combinators fixed-point combinators Y combinator Z combinator Turing machines register machines cellular automata Petri nets process algebras temporal logics modal logics epistemic logic deontic logic intuitionistic logic paraconsistent logic fuzzy logic rough sets Dempster-Shafer theory possibility theory evidence theory Bayesian networks Markov random fields factor graphs belief propagation variational inference expectation propagation Laplace approximation saddle-point approximations importance sampling rejection sampling stratified sampling Latin hypercube sampling quasi-Monte Carlo Sobol sequences Halton sequences Faure sequences Niederreiter sequences low-discrepancy sequences discrepancy measures star discrepancy extreme discrepancy isotropic discrepancy covering radius mesh norm fill distance separation distance packing radius optimal quantization k-means clustering k-medoids clustering hierarchical agglomerative clustering single linkage complete linkage average linkage Ward’s method DBSCAN OPTICS HDBSCAN spectral clustering affinity propagation mean-shift clustering Gaussian mixture models expectation-maximization Dirichlet process mixtures Chinese restaurant process Indian buffet process Polya urn scheme Pólya urn scheme Galton-Watson branching processes Yule-Simon distribution Zipf’s law Benford’s law power laws heavy-tailed distributions log-normal distributions Pareto distributions Cauchy distributions stable distributions Lévy processes jump-diffusion models Merton model Heston model SABR model Black-Karasinski model Hull-White model Vasicek model CIR model affine term structure models short-rate models LIBOR market model SABR calibration stochastic volatility local volatility Dupire formula implied volatility smile skew term structure arbitrage-free pricing risk-neutral measure change of numeraire Girsanov theorem martingale representation theorem Doob-Meyer decomposition optional stopping theorem maximal inequalities Burkholder-Davis-Gundy inequalities martingale type spaces UMD spaces type-2 Banach spaces Rademacher cotype Hilbert spaces reflexive spaces weak compactness Eberlein-Smulian theorem Banach-Alaoglu theorem Krein-Smulian theorem James’ theorem Kadec-Klee property Opial property Milman-Pettis theorem Radon-Riesz property Kadec-Klee property weak topology strong topology norm topology uniform topology compact-open topology Fréchet spaces Montel spaces barrelled spaces bornological spaces Mackey topology weak-* topology sigma-weak topology ultra-weak topology trace class operators compact operators Hilbert-Schmidt operators nuclear operators trace ideal Schatten classes Dixmier trace Connes trace noncommutative geometry cyclic cohomology Hochschild cohomology periodic cyclic homology K-homology KK-theory Kasparov product E-theory Roe algebra coarse geometry property A uniform embeddability Gromov-Hausdorff convergence Gromov-Hausdorff distance Gromov-Hausdorff-von Neumann-Hausdorff metric ultrametric spaces non-Archimedean geometry Berkovich analytification rigid analytic geometry formal schemes algebraic stacks derived categories triangulated categories stable infinity-categories infinity-categories simplicial sets simplicial complexes simplicial groups simplicial sets quasi-simplicial objects cubical sets cubical type theory homotopy type theory univalence axiom propositional truncation higher inductive types W-types NNO natural numbers object integers rationals reals complex numbers quaternions octonions sedenions Clifford algebras exterior algebras symmetric algebras tensor algebras universal enveloping algebras Lie algebras Lie groups representation theory character tables Peter-Weyl theorem Plancherel formula Harish-Chandra characters L-functions Riemann zeta function Dirichlet L-functions modular forms Eisenstein series Hecke operators Maass forms automorphic forms Langlands program reciprocity laws class field theory Iwasawa theory p-adic Hodge theory crystalline representations de Rham representations semistable representations Fontaine’s period rings B-polygons filtered phi-modules Dieudonné modules Lubin-Tate extensions formal groups Honda formal groups height classification supersingular ordinary multiplicative additive elliptic curves abelian varieties Jacobians Prym varieties Picard varieties moduli spaces moduli stacks Deligne-Mumford stacks Artin stacks coarse moduli fine moduli universal families deformation rings universal deformation rings versal deformations obstructed deformations unobstructed deformations tangent obstructions cotangent complexes André-Quillen homology André-Quillen cohomology Harrison homology Harrison cohomology differential graded Lie algebras Maurer-Cartan equations gauge transformations homotopy fiber products homotopy pullbacks homotopy pushouts derived limits derived colimits Bousfield localization Bousfield completion Brown representability Adams spectral sequence Atiyah-Hirzebruch spectral sequence Serre spectral sequence Eilenberg-Moore spectral sequence Adams-Novikov chromatic homotopy Morava K-theories Morava E-theories TMF topological modular forms KO KU K(n)-localization chromatic convergence telescope conjecture thick subcategory theorem Hopkins-Ravenel-Smith nilpotence conjecture Ravenel conjectures telescope conjecture chromatic splitting conjecture descent chromatic fracture square arithmetic fracture square geometric fracture square arithmetic square geometric square Sullivan’s arithmetic square nilpotent space nilpotent space Postnikov tower Whitehead tower Postnikov k-invariants k-invariants obstruction theory primary obstructions secondary obstructions Toda brackets Massey products triple products quadruple products n-fold products cup-i maps Steenrod squares Steenrod algebra dual Steenrod algebra May diagrams May filtration May spectral sequence cobar construction bar construction Eilenberg-Zilber shuffle Alexander-Whitney diagonal cooperations cocommutative coalgebra Hopf algebra bialgebra Hopf module smash product Day convolution monoidal category closed monoidal category monoidal functor monoidal natural transformation coherence theorem Mac Lane coherence Kelly coherence string diagrams Joyal-Street Penrose graphical notation tangle calculus knot invariants Jones polynomial Khovanov homology Heegaard Floer contact homology symplectic homology wrapped Fukaya category A-infinity categories curved A-infinity structures pretriangulated hull triangulated hull dg enhancement minimal model semi-free resolution cell module projective module injective module flat module free module finitely generated finitely presented Noetherian Artinian Dedekind domain PID UFD Prüfer domain valuation ring discrete valuation ring DVR complete DVR Henselian ring excellent ring regular ring Cohen-Macaulay ring Gorenstein ring Cohen-Macaulay local ring Serre intersection multiplicity Hilbert-Samuel function Hilbert polynomial Hilbert function regularity Castelnuovo-Mumford regularity graded regularity Koszul complex Koszul regularity local cohomology local duality Grothendieck duality Greenlees-May duality Verdier duality six-functor formalism exceptional inverse image proper inverse image exceptional direct image proper direct image smooth base change proper base change Tor independent base change cd structures cd-structure completely decomposed cd structure excisive triad Mayer-Vietoris sheafification stalk presheaf site Grothendieck topology site functor site fppf fpqc étale Zariski Nisnevich cdh lisse étale p-adic étale rigid étale crystalline syntomic flat fppf fpqc qfh qfh sheaves descent effective descent effective epimorphism effective mono effective epi descent datum cocycle condition Čech nerve totalization cosimplicial object simplicial object Amitsur complex Amitsur augmentation Milnor patching Milnor exact sequence Milnor square pullback pushout fiber colimit limit universal cone terminal object initial object representable functor Yoneda embedding Yoneda lemma enriched Yoneda lemma density theorem density adjunction unit counit triangle identities left adjoint right adjoint adjunction equivalence fully faithful essentially surjective Kan extension left Kan extension right Kan extension Day convolution monoidal reflection coreflection localization reflective subcategory coreflective subcategory reflective hull coreflective hull completion cocompletion free monoid free group free abelian group free commutative group free semigroup free magma magma commutative magma idempotent magma band rectangular band semilattice lattice distributive lattice Boolean algebra Heyting algebra complete Heyting algebra frame locale sober space spatial locale spectrum prime spectrum maximal spectrum Spec R Specm R Specmax R affine scheme projective scheme quasi-projective scheme separated scheme integral scheme reduced scheme irreducible scheme connected scheme locally Noetherian globally Noetherian excellent scheme morphism schemes finite morphism proper morphism smooth morphism étale morphism unramified morphism flat morphism faithfully flat morphism faithfully flat descent faithfully flat descent condition faithfully flat quasi-compact quasi-separated noetherian noetherian topological space sober topological space spectral space coherent space Stone space Boolean space profinite space totally disconnected locally compact totally bounded complete metric space Polish space separable metric space complete metric space Banach space Hilbert space inner product normed vector topological vector Fréchet vector nuclear vector barrelled bornological reflexive separable dual bidual weak-star weak-strong strong-star strong-strong norm-topology weak-topology strong-topology ultrastrong ultra weak ultraweak Mackey Arens-Bessaga-Pelczynski interpolation Schauder basis unconditional basis Markushevich basis Fabius basis Haar system Fabius-Schauder wavelet Daubechies wavelet Meyer wavelet Battle-Lemarié wavelet biorthogonal wavelet lifting scheme perfect reconstruction filter bank polyphase matrix Smith form Howell form Hermite normal Smith normal elementary divisor invariant factor torsion-free torsion part saturation quotient group abelianization commutator subgroup derived series lower central upper central solvable nilpotent residually finite virtually solvable virtually nilpotent virtually free virtually cyclic polycyclic polycyclic-by-finite metabelian triply metabelian solvable-by-nilpotent nilpotent-by-solvable Fitting subgroup Fitting length chief series composition series Jordan-Hölder Schreier refinement Zassenhaus lemma butterfly lemma Schreier coset transversal Nielsen generators Reidemeister moves Whitehead moves simple-homotopy Whitehead group Whitehead torsion simple-homotopy equivalence s-cobordism h-cobordism cobordism bordism oriented cobordism unoriented cobordism framed cobordism spin cobordism string cobordism M-theory branes D-branes NS-branes M2-brane M5-brane heterotic string Type I Type IIA Type IIB Type 0 superstring Green-Schwarz mechanism anomaly cancellation heterotic SO(32) E8xE8 Calabi-Yau compactification mirror symmetry Strominger-Yau-Zaslow SYZ conjecture T-duality S-duality U-duality electric magnetic dyonic charge lattice Narain lattice moduli