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Self-hosting is awesome, because you own the hardware, and you have control over access and uptime. However, unless you've also written the software that runs your self-hosted services, there's still much of your homelab that depends on a third party to function.

TL;DR: Live stream Bayern Munich vs. Arsenal in the Champions League for free on RTÉ Player. Access this free streaming platform from anywhere in the world with ExpressVPN.

The Champions League isn't short of huge matchups, but this week feels like an extra special treat. Fresh from PSG vs. Barcelona, we're now blessed with Bayern Munich vs. Arsenal. Both sides are unbeaten after two rounds, and this could easily have been the final last season.

We're expecting a fascinating clash between two teams laden with stars. We're hoping for big performances from the likes of Harry Kane and Declan Rice. If it's anything like last night's battle at the Parc des Princes, we're in for a treat.

If you want to watch Bayern Munich vs. Arsenal in the Champions League from anywhere in the world, we have all the information you need.

When is Bayern Munich vs. Arsenal?

Bayern Munich vs. Arsenal in the Champions League kicks off at 3 p.m. ET on Oct. 21. This fixture takes place at the Allianz Arena.

How to watch Bayern Munich vs. Arsenal for free

Bayern Munich vs. Arsenal is available to live stream for free on RTÉ Player.

RTÉ Player is geo-restricted to Ireland, but anyone can access this free streaming platform with a VPN. These tools can hide your real IP address (digital location) and connect you to a secure server in Ireland, meaning you can unblock RTÉ Player to stream the Champions League for free from anywhere in the world.

Live stream Bayern Munich vs. Arsenal for free by following these simple steps:

1. Subscribe to a streaming-friendly VPN (we recommend ExpressVPN)

2. Download the app to your device of choice (the best VPNs have apps for Windows, Mac, iOS, Android, Linux, and more)

3. Open up the app and connect to a server in Ireland

4. Visit RTÉ Player

5. Watch Bayern Munich vs. Arsenal for free from anywhere in the world

Opens in a new window Credit: ExpressVPN ExpressVPN (1-Month Plan) $12.99 only at ExpressVPN (with money-back guarantee) Get Deal

The best VPNs for streaming are not free, but most do offer free trials or money-back guarantees. By leveraging these offers, you can access free live streams of the Champions League without actually spending anything. This obviously isn't a long-term solution, but it does give you enough time to stream Bayern Munich vs. Arsenal (plus more Champions League fixtures) before recovering your investment.

What is the best VPN for RTÉ Player?

ExpressVPN is the best choice for bypassing geo-restrictions to stream live sport on RTÉ Player, for a number of reasons:

• Servers in 105 countries, including Ireland

• Easy-to-use app available on all major devices, including iPhone, Android, Windows, Mac, and more

• Strict no-logging policy so your data is secure

• Fast connection speeds free from throttling

• Up to 10 simultaneous connections

• 30-day money-back guarantee

A two-year subscription to ExpressVPN is on sale for $68.40 and includes an extra four months for free — 81% off for a limited time. This plan includes a year of free unlimited cloud backup and usually offers a generous 30-day money-back guarantee. Alternatively, you can get a one-month plan for just $12.99.

Live stream Bayern Munich vs. Arsenal in the Champions League for free with ExpressVPN.

Your Pixel phone came with a nice USB-C cable, but Google didn't include a power adapter in the box. So, you resorted to using one that happened to be sitting around your house, and your expensive Pixel phone is charging half as fast as it should.

If your favorite website is down for maintenance today (October 21), you now know why. The developers of the PHP scripting language have released emergency patches for two "critical" vulnerabilities that hackers can use to compromise web servers.

Early Black Friday Samsung deals at a glance Best TV deal Samsung 65-Inch Class S85H Series OLED 4K TV $1,297.99 (save $300) Get Deal Best gaming monitor deal Samsung 27-inch Odyssey OLED G5 $329.99 (save $170) Get Deal Best Samsung unlocked phone deal Samsung Galaxy S26 Ultra (256GB) $979.99 (save $420) Get Deal Best Samsung storage deal Samsung SSD 990 PRO 2TB $409.99 (save $220) Get Deal

Black Friday has come earlier than ever this year. We have great options if you're shopping for outdoor gear, kitchen upgrades, and the best toys of the year. Black Friday creeping earlier and earlier isn't such a bad thing. That simply gives us shoppers more time to find the perfect gift at the best price.

Samsung's lineup of TVs, gaming monitors, phones, and internal storage are all seeing nice discounts. Shopping these early deals means you'll be in great shape for streaming new fall shows, playing upcoming game releases, and keeping family entertained during their holiday visits.

SEE ALSO: Early Black Friday 2026 deals: See holiday gift guides from Amazon, The Home Depot, Nordstrom, Ulta, and more

We've collected the best Samsung deals below if you're ready to get holiday shopping off your to-do list.

Best Samsung TV deal Opens in a new window Credit: Samsung Samsung 65-Inch Class S85H Series OLED 4K TV $1,297.99 at Amazon
$1,597.99 Save $300   Get Deal at Amazon Why we like it

Since winter is hibernation season, it's important to go in with plenty of entertainment options. You have your streaming services all set up and gaming options on hand, but if your TV isn't up to the task, the entertainment can suffer. Before the holidays begin, upgrade to the 65-inch Samsung Class S85H Series OLED 4K TV. The OLED display is perfect for clarity and crisp images, whether that's Christmas movies or playing the new GTA 6.

Best Samsung gaming monitor deal Opens in a new window Credit: Samsung Samsung 27-inch Odyssey OLED G5 $329.99 at Amazon
$499.99 Save $170   Get Deal at Amazon Why we like it

Without going wild on price or size, the 27-inch Samsung Odyssey OLED G5 gaming monitor is the perfect winter upgrade. Refresh rate reaches 180Hz and response time measures 0.03ms for a smooth gaming experience. Plus, it'll be a great additional monitor for working from home during the holdiays.

TL;DR: Live stream Real Madrid vs. RB Leipzig in the Champions League for free on Virgin Media Player. Access this free streaming platform from anywhere in the world with ExpressVPN.

The Champions League group stage is starting to take shape, and it's no surprise to see Real Madrid in a strong position. José Mourinho's team are sitting on six points after two games, three points ahead of their next opponents, RB Leipzig. The visitors are coming off an embarrassing 6-1 defeat at the hands of Bayern Munich in the Bundesliga, so they'll be desperate to improve.

We expect Real Madrid to take the victory in this matchup. It's not going to be straightforward, but Real Madrid's star names should be able to overcome the German visitors.

If you want to watch Real Madrid vs. RB Leipzig in the Champions League from anywhere in the world, we have all the information you need.

When is Real Madrid vs. RB Leipzig?

Real Madrid vs. RB Leipzig in the Champions League kicks off at 3 p.m. ET on Oct. 21. This fixture takes place at the Santiago Bernabéu.

How to watch Real Madrid vs. RB Leipzig for free

Real Madrid vs. RB Leipzig is available to live stream for free on Virgin Media Player.

Virgin Media Player is geo-restricted to Ireland, but anyone can access it for free with a VPN. These tools can hide your real IP address (digital location) and connect you to a secure server in Ireland, meaning you can unblock Virgin Media Player to stream the Champions League for free from anywhere in the world.

Live stream Real Madrid vs. RB Leipzig for free by following these simple steps:

1. Subscribe to a streaming-friendly VPN (we recommend ExpressVPN)

2. Download the app to your device of choice (the best VPNs have apps for Windows, Mac, iOS, Android, Linux, and more)

3. Open up the app and connect to a server in Ireland

4. Visit Virgin Media Player

5. Watch Real Madrid vs. RB Leipzig for free from anywhere in the world

Opens in a new window Credit: ExpressVPN ExpressVPN (1-Month Plan) $12.99 only at ExpressVPN (with money-back guarantee) Get Deal

The best VPNs for streaming are not free, but most do offer free trials or money-back guarantees. By leveraging these offers, you can access free live streams of the Champions League without actually spending anything. This obviously isn't a long-term solution, but it does give you enough time to stream Real Madrid vs. RB Leipzig (plus more Champions League fixtures) before recovering your investment.

What is the best VPN for Virgin Media Player?

ExpressVPN is the best choice for bypassing geo-restrictions to stream live sport on Virgin Media Player, for a number of reasons:

• Servers in 105 countries, including Ireland

• Easy-to-use app available on all major devices, including iPhone, Android, Windows, Mac, and more

• Strict no-logging policy so your data is secure

• Fast connection speeds free from throttling

• Up to 10 simultaneous connections

• 30-day money-back guarantee

A two-year subscription to ExpressVPN is on sale for $68.40 and includes an extra four months for free — 81% off for a limited time. This plan includes a year of free unlimited cloud backup and usually offers a generous 30-day money-back guarantee. Alternatively, you can get a one-month plan for just $12.99.

Live stream Real Madrid vs. RB Leipzig in the Champions League for free with ExpressVPN.

As concern grows regarding the dangers of artificial intelligence, AI regulation has become a hot topic in both Silicon Valley and Washington, D.C.

AI giant Anthropic, the maker of Claude, has been pitching its own set of AI safety rules to lawmakers, and Politico recently obtained a confidential draft of the company's proposals.

So, what's in Anthropic's proposed AI safety bill? Let's just say at least some of these proposals is eye-opening, to say the least.

According to the document obtained by Politico, Anthropic's AI safety bill would only allow the government to block the release of an AI model if it posed a "foreseeable, imminent risk" of causing more than $1 billion in damages or the death or serious injury of more than 50 people.

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Furthermore, the risk from the AI model must be related specifically to a cyberattack, a developer losing control over a model, or the creation or release of a biological weapon.

How would such risks be determined?

According to the proposal seen by Politico, independent evaluators would periodically analyze an AI lab and its models. If the evaluators find that there is an "imminent enumerated risk" from a model, they would report it to the U.S. Commerce Department within 72 hours. The Commerce Department would then decide if the threshold for a risk requires a request for a judge to block the release of the model. 

Under Anthropic's bill, only a federal judge in Washington, D.C., could suspend a model. In fact, the bill specifically prohibits any other "department, agency, or officer" from having the authority to suspend an AI model.

Interestingly, what's not prohibited is who funds the independent evaluators. In fact, Anthropic's proposal explicitly states that evaluators can be paid by AI labs, as long as the payment does not influence their findings, according to Politico.

This raises an obvious question: How would one prove that an evaluator isn't influenced by where their funding comes from? In addition, it seems that, based on this reported bill, $999,999,999 in damages or the death of 49 people would not trigger the process necessary to pause a model. 

As numerous outlets have pointed out, Anthropic's AI bill closely resembles bills that recently passed in California, like SB 53. With that said, though, AI companies such as Anthropic all provided input that led to the creation of the final bill.

With the number of "rogue" AI agent incidents seemingly increasing, it does seem like AI safety laws and regulation are a necessity at this point. But it's unclear just how much of a role AI companies themselves should play in crafting those laws, especially if we're seeing proposals that include supposedly neutral regulators being paid by the very companies they are supposed to regulate.

Want to learn more about getting the best out of your tech? Sign up for Mashable's Top Stories and Deals newsletters or get Mashable push alerts today. 

In early September 2026, Microsoft announced Project Zenith, an effort to provide developers with a ready-to-code, distraction-free Windows configuration out of the box, equipped with a suite of useful developer tools.

SAVE $94.50: As of Oct. 21, get the Samsonite Octiv Pro 3-Piece Hardside Luggage for $535.49 at Amazon, down from its usual price of $629.99. That's a discount of 15%.

Opens in a new window Credit: Amazon Samsonite Octiv Pro 3-Piece Hardside Luggage $535.49 at Amazon
$629.99 Save $94.50   Get Deal at Amazon

Thinking of getting some traveling in over the holidays? You'll want to be prepared with a good set of luggage to help get you from point A to point B. Amazon is already offering several deals you can shop ahead of the rush to make sure you're in good shape to see family and the world throughout the rest of the year, and we've found one deal you won't want to miss.

As of Oct. 21, get the Samsonite Octiv Pro 3-Piece Hardside Luggage for $535.49 at Amazon, down from its usual price of $629.99. That's $94.50 off and a discount of 15%.

SEE ALSO: It's Black Friday again — here's how to use a Visa gift card to shop

This 3-piece lightweight hardsided luggage set has everything you need to pack for your trip, from start to finish. There's a perfectly-sized carry-on bag as well as both medium and large checked luggage bags. All are expandable and use spinner wheels as well, so you can carry them around the airport or to your destination with ease.

The telescopic handle on each piece of luggage can adjust to 8 different lengths so you can comfortably adapt, no matter what your height. Plus, the spinner wheels absorb shock to make for steadier movement across the airport or on bumpy terrain if you've got to hoof it to your car. There's also a TSA-approved lock on each bag for another layer of added protection, just in case you need additional security.

Pack up all your things with peace of mind and save some cash while you're at it with this deal and you'll be more than ready to travel now that the year is coming to an end.

Halloween is about a week and a half away, and I'm definitely in the spooky szn spirit (now that it's socially acceptable to be; please see my thoughts on premature pumpkin spicing). Meanwhile, brands across every industry are doing the most to get your attention and your cash.

One brand that's yet to let me down is Hello Nancy, and they've got the cutest Halloween treat for their customers (literally no tricks in this one): a brand-new three-toy Halloween collection launching with a special-edition blind box drop. More on the toys below.

SEE ALSO: I've tested 100+ sex toys. Here are the 15 most mind-blowing toys I've ever owned.

This is Hello Nancy's first-ever mystery box, and each one comes with two toys: a brand-new, exclusive Halloween toy valued at $149.99 and a Hello Nancy fave worth at least $99, according to the brand. That's over $248 worth of goodies for $79!

If you want first dibs, you can sign up for priority access — but you'd better hurry. Only 5,000 mystery boxes are available, and as of this writing, just 970 were left. The priority-access link expires today, Oct. 21, and the public sale opens tomorrow, Oct. 22, at 8 a.m. ET. According to the website, there won't be a restock, so this isn't one to leave sitting in your cart. My review sample is already in the mail, so I'll report back.

Bonus: 10 "Golden Lem" tickets are hidden at random among the boxes. If you get one, you can redeem it for an exclusive gold-plated collectible inspired by Hello Nancy's Lem toy (read: a display piece, not a functioning toy). It's like Charlie and the Chocolate Factory, but like, so much better.

Meet the monsters

So, who's potentially coming home with you? The Halloween collection includes three new toys: Jack, Lemmy, and Medusa. You won't get to pick which one lands in your mystery box, but here's a peek at the lineup.

Jack The cutest little sucker. Credit: Hello Nancy

Jack is a pumpkin-shaped suction vibrator with no face (thank God). I haven't tried it yet, but if it's anything like the Lem, I'd be pretty happy to find this one in my box. It has 11 levels of pulsating air suction, plus seven vibration speeds and three patterns, and you can use the suction and vibration separately or together. It's also 100 percent waterproof and charges via USB-C.

Lemmy Quite the mouth on this one. Credit: Hello Nancy

Lemmy is a little freakier-looking, but the design totally makes sense. The soft lips form a seal around your clitoris, while a little "tongue" inside does the licking. There are three suction levels and a long-suction mode for those of you who like a longer pull, plus three licking speeds and six patterns. You can use the suction and the licking separately or together.

Medusa Looks scary. I'm into it. Credit: Hello Nancy Medusa

Medusa is the scariest-looking of them all (rightfully so), but it's also the one I'm most hoping to get. This dual-stimulation toy has an insertable pink tentacle that swirls inside you and a vibrating yellow tip that stimulates your clitoris. It has three speeds and seven patterns, and you can use either function on its own or together.

If you miss out on a mystery box, the Halloween toys will also be available individually — so yes, you can get all three.

The team behind PHP released emergency security updates on Wednesday to fix two critical vulnerabilities, a day after details about the flaws leaked online.

The UK’s National Cyber Security Centre (NCSC) is urging organizations to install the fix immediately.

What is PHP?

PHP is a programming language that runs on web servers. When you load a webpage, PHP often runs behind the scenes, retrieving data from databases and building the page before it reaches your browser. BleepingComputer notes that it powers many websites, including content management systems, online stores, and custom web services. According to the NCSC, PHP is also built into a range of other products.

The NCSC says the two flaws, tracked as CVE-2026-109618 and CVE-2026-109619, could allow an attacker to run their own commands on a PHP-based server without logging in. This is called remote code execution, and it’s one of the most serious kinds of security bugs.

According to the NCSC, an attacker could overload PHP’s memory to corrupt it, then, under certain conditions, use that corruption to take control of the server. One of the flaws involves how PHP processes form submissions and file uploads.

PHP rated each flaw 9.2 out of 10 on CVSS, the industry’s standard severity scale. Security firm watchTowr described them as a perfect 10, BleepingComputer reported, and so did an anonymous Hacker News post that has since been deleted. Researchers have started calling the bugs "PHP2SHELL."

The fixes were originally planned for a coordinated release later this week, so companies would have time to prepare. Then, information about the flaws leaked early. Speaking to BleepingComputer, watchTowr CEO Benjamin Harris said the vulnerabilities are "indeed real" and affect PHP versions 5 through 8. The firm said it had reproduced one of them.

PHP moved its release up to 10 a.m. ET Wednesday. As of Tuesday, watchTowr said it hadn’t seen anyone exploiting the flaws. By Wednesday, Cloudflare CTO Dane Knecht said on X that the company was seeing early attacks, according to BleepingComputer.

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The NCSC says organizations running PHP 5 or later are affected and should update to the latest fixed version. Anyone still using an older, unsupported version should move to a supported one.

Wikipedia impacted by PHP vulnerability

Wikipedia went down for a time on Wednesday, and the Wikimedia Foundation, the nonprofit that runs the site, confirmed to Mashable that the outage was tied to the PHP vulnerabilities.

In a statement to Mashable through email, the foundation said it is working to address the flaws in PHP, which it described as the programming language behind most wikis, including its own projects. "This issue affects all known versions of PHP, and is not specific to the Wikimedia Foundation," the statement said.

The foundation added that all of its PHP-based systems are affected, along with volunteer-run systems that use PHP. It has also taken several other systems offline as a precaution until they can be safely patched. Wikipedia was back up as of this writing.

Who else is affected, and what should you do?

If you only browse the web, there’s nothing for you to install. The fix is up to the people who run websites.

If you run a website, blog, or online store, check whether it uses PHP and make sure it’s up to date. If a hosting company manages your site, contact them to confirm they’ve applied the patch.

For organizations, the NCSC recommends:

• Finding every system that runs PHP, starting with the ones reachable from the internet.

• Watching for updates from vendors whose products include PHP.

• Taking systems offline if they can’t be fixed quickly.

• Checking any system that couldn’t be updated right away for signs of a break-in.

Cloudflare has added emergency firewall rules designed to block suspicious requests linked to the flaws. The rules are available to all its customers, including those on free plans.

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There's a new Android beta out there for your perusal.

Google has released the second beta for the third quarterly platform release of Android 17, available for installation on Pixel devices now. You can install it now by enrolling your device in Google's beta program. Sometimes, these betas come packed with lots of exciting new toys to play with, on top of various bug fixes. This is not necessarily one of those times.

SEE ALSO: iPhone vs. Android on privacy-invasive bloatware: Which lets users remove pre-installed apps?

According to Google's release notes, as well as some reporting by 9to5Google, there are a couple of small new features and a bunch of fixes in this update. One new feature is the ability to require a phone unlock in order to change quick settings like Bluetooth, WiFi, and airplane mode. Normally, you can do that from the lock screen. There are also new themes, custom icons, and weather effects for your phone's wallpaper to play around with.

As for the rest of the update, it sounds like it's mostly unexciting (but probably necessary) bug fixes. Google fixed an issue that caused phones to crash and reboot during normal use, as well as another one that caused devices to shut down unexpectedly during WhatsApp video calls. Apparently, paired smartwatches were also seeing an issue where they would continue ringing even after a phone call was answered. That has been fixed, among other smaller issues.

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A24's You Can See Everything has swiftly become the must-see movie in theaters across the U.S.

Opening on Oct. 16, the critically acclaimed documentary from Nathan Fielder and Lance Oppenheim about Elizabeth Holmes made a remarkable $15 million over the weekend, coming in #2 behind Street Fighter. More incredibly, You Can See Everything managed to earn the biggest opening of any documentary of the decade, while running nearly three hours and debuting in limited release.

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Yes, the internet is obsessed with You Can See Everything, championing the doc as a movie one must see in theaters with a crowd. But opening in fewer than 1,000 theaters, the film has become a hard-to-get ticket. The New York Times reported the film sold out over 2,000 showtimes so far.

"You Can See Everything showtimes" is a trending search term on Google, and aspiring moviegoers are expressing frustrations online about sold-out screenings.

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And yet those are the lucky ones, as others on X lamented that You Can See Everything is not playing at a theater near them.

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But there's good news for those desperate to get tickets.

When is You Can See Everything expanding to more theaters?

A24 took to X to announce You Can See Everything will expand to more than 2,500 screens in its second weekend.

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And if you don't know why A24 used an image of hands holding a yellow bowl to accompany the announcement, just you wait.

Mashable has reached out to A24 for more details on You Can See Everything's theatrical expansion, and will update accordingly.

You Can See Everything is now playing in theaters. Find available showtimes at Fandango. 

TL;DR: Dunkin' is giving away 1 million free coffees on Oct. 21. Enter the code SIZEDOESNTMATTER in your app's "Offers & Rewards" section. The free coffee certificate is valid for five days after the promo code is entered, with a limit of 1 redemption per customer.

Here we go again.

Dunkin’ is giving away 1 million free coffees on Oct. 21. Is that déjà vu? Dunkin' does this sort of thing fairly regularly, and the process of claiming your freebie remains the same.

How to get a free coffee from Dunkin'

The promo code SIZEDOESNTMATTER was dropped by Dunkin on Oct. 21. The first 1 million customers to enter the promo code in the “Offers & Rewards” section of the Dunkin’ Donuts app can redeem a free coffee. The free coffee certificate is valid for five days after it is entered.

Here are the steps you need to take to claim:

• Make sure you're a Dunkin' Rewards member

• Open the Dunkin' app and go to the "Offers & Rewards" section

• Enter the code SIZEDOESNTMATTER to claim your free coffee

I absolutely love my Bambu Lab 3D printers—but they could be a lot better. Not in hardware. I think the hardware is fantastic. But the software that runs the printers could use a lot of work. Thanks to the open source community, my Bambu Lab 3D printers got a free upgrade that made them a lot smarter.

Microsoft researcher and collaborator combine new mathematical ideas with human-directed AI to resolve a central problem in high-dimensional geometry and probability At a glance

• Fundamental scientific breakthrough: Researchers resolved the Kannan-Lovász-Simonovits (KLS) conjecture, a 31-year-old problem widely regarded as one of the central open questions in high-dimensional geometry, with implications for statistics, optimization, algorithms, and computer science.
• A universal geometric principle: The result establishes a universal geometric principle for log-concave distributions, showing that simple flat cuts can capture their narrowest bottlenecks within a fixed factor, regardless of dimension. 
• Human-AI collaboration: The breakthrough emerged from a human-directed agentic research process in which AI systems explored mathematical possibilities while researchers evaluated, pruned, and redirected the search.
• Independent confirmation: Multiple independent researchers have validated this research, identifying its central criterion as the key breakthrough, followed by a machine-checked formal verification. 

For decades, a fundamental question about the difficulty of computing in high-dimensional spaces has resisted proof. New research resolves this question, removing a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing.

The Kannan–Lovász–Simonovits (KLS) conjecture was proposed in 1995. It asks whether, for a broad class of high-dimensional probability distributions, the best flat or hyperplane cut is always nearly as effective as the best cut of any shape. More precisely, KLS predicts that the two differ by no more than a universal constant, regardless of the number of dimensions. This would establish a dimension-free geometric principle with important consequences for high-dimensional geometry, probability, and theoretical computer science. Because the geometry of these cuts governs how quickly random walks can explore a distribution, KLS also plays a central role in the theory of high-dimensional sampling.

Figure 1: An illustration of the KLS Conjecture (simplified) on regular tetrahedron.

Since 1995, mathematicians have accumulated evidence that the conjecture was true, but a complete proof remained elusive. Now, two researchers, including Xinzhi Zhang from the Machine Learning and Optimization group at Microsoft Research, have proved the KLS conjecture (opens in new tab). They have shown that the relationship holds with a fixed bound, regardless of how many dimensions a shape has. Before this work, the best known result (opens in new tab), established in July 2026, allowed that bound to grow slowly as the number of dimensions increased. This work establishes a fundamental geometric principle: high-dimensional spaces are better behaved than previously known, and the obstacles to navigating them do not grow without bound as dimensions increase. 

The importance of the KLS conjecture was recognized across the field long before it was resolved. In its 2017-18 Annual Report, the Mathematical Sciences Research Institute (MSRI) (opens in new tab) called it “a central open problem in high-dimensional convex geometry.” The Simons Institute (opens in new tab) described it as “one of the most important open problems in convex geometry.” In the proceedings of the 2022 International Congress of Mathematicians (ICM) (opens in new tab), Ronen Eldan noted its broad implications for high-dimensional convex and computational geometry. Quanta Magazine (opens in new tab) wrote that KLS “lies at the heart of many questions in statistics and computer science.”

Technical details

The authors began by improving the July 2026 result in order to make the mathematical bound more precise. Working with AI agents in a structured, human-directed process, they explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors. This produced a new mathematical criterion that became the foundation of the final proof.

The authors then discovered that the new criterion could be applied recursively. Each improved bound could be fed back into the same argument to produce a still tighter one. In the October 1, 2026 version of the paper, this reduced the dependence on dimension to the iterated logarithm. This function grows extraordinarily slowly, but it still grows with the dimension and therefore did not yet fully resolve the conjecture.

The final step was to repeat the entire recursive refinement, rather than applying it only once. This idea was inspired by the authors’ earlier work on fast algorithms for linear programming. The authors proved that the small errors introduced at each stage shrink fast enough that their total remains bounded by a universal constant, independent of the dimension. This eliminated the dependence on the number of dimensions and completed the proof in the October 4, 2026 version of the paper.

The recursive method, its repeated application, and the proof that the accumulated errors remain bounded are original contributions of the researchers. AI tools helped explore candidate approaches, carry out detailed derivations, prepare the manuscript, and perform internal proof checks. The researchers determined the mathematical direction, selected the promising ideas, assembled them into the final argument, and verified the proof.

Independent confirmation

On the same day that the authors posted their complete proof, a separate group of leading experts posted a full proof (opens in new tab) that they developed in parallel. Their work built upon the findings in the first version (opens in new tab) of the Microsoft paper, calling it “the crucial step in the proof.” A third proof (opens in new tab) which appeared separately on the same day relies on a closely related idea. In addition, a Lean 4 formalization (opens in new tab) was subsequently released, combining the key criterion from the first Microsoft paper with additional research. 

Independent proofs are among mathematics’ strongest forms of validation because they demonstrate that a result does not rest on a single chain of reasoning. The subsequent appearance of alternative approaches and formal verification further strengthens confidence that the field has uncovered a genuine underlying principle rather than a fragile technical argument. The authors have also shortened their proof and verified the streamlined argument in Lean; they plan to release both after final review. Together, these developments provide complementary forms of confirmation.

Human-AI guidance

In the process of proving KLS, this research also demonstrates the power of having researchers direct AI rather than simply asking AI to solve a complex problem through a single end-to-end run of a frontier model or agent. The authors used an agentic search process with iterative, human-assisted “pruning”: As multiple agents explored variations of an idea in parallel, the researcher examined their intermediate result, eliminated unpromising directions, and passed the most promising result to the next round. Roughly one hundred research trajectories were explored before the key breakthrough emerged. 

In a separate track of the project, the authors made repeated unguided attempts to solve the same problem with several leading AI models, using roughly the same total token budget as the main process. That effort fell short of the goal. The eventual success depended on both the capabilities of the AI systems and the mathematical judgment of the researchers. 

Conclusion

Taken together, resolving KLS settles a central problem in high-dimensional geometry. A wide range of results that previously relied on the conjecture as an assumption can now be stated unconditionally. The proof therefore provides a stronger foundation for research in convex geometry, probability, and theoretical computer science.

The work also demonstrates how carefully guided human-AI collaboration can contribute to genuine scientific discovery while reinforcing, rather than replacing human insight. It highlights the continuing importance of scientific judgment in identifying promising directions, recognizing conceptual connections, and seeing when a partial insight points toward a deeper result. The authors are now simplifying the proof and extending its formal verification so that the mathematics and theoretical computer science communities can more easily examine the result and build on it.

Acknowledgements 

Thank you to our co-author Zhao Song for his contributions to this work, and Victor Reis (opens in new tab) and Ishai Menache for helpful discussions throughout the project. 

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The post Solving a 31-year-old grand challenge in mathematics – the KLS conjecture appeared first on Microsoft Research.

When it comes to Halloween decorating, Arby's just raised the steaks.

The fast-food chain is launching a limited-edition "Meat Sweats" tracksuit ahead of the spooky holiday, but not just any tracksuit: one "made specifically for 12-foot skeletons," according to an email sent to the press.

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That could only refer to "Skelly," the viral giant skeleton animatronic that the Home Depot has sold since 2020; this year, it was updated with app controls. Tyler Pelfrey, manager of brand communications at the Home Depot, tells Mashable that Arby's designed the tracksuit before approaching the home improvement retailer about a collab.

"We love to see how the original Halloween giant, Skelly, has taken on a life of his own — inspiring creative designs in homes across the country, viral social content and now, brand collaborations," Pelfrey said. "We’re excited for the potential to continue to find new ways to celebrate Skelly’s status as a Halloween cultural icon for years to come."

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The Meat Sweats suit features an all-over print of folded roast beef and an Arby's logo on its chest. Priced at $46.66, it will be available exclusively in Arby's online merch store starting this Wednesday, Oct. 21 at noon ET.

Skelly has built up quite the wardrobe over the years. The Home Depot released Santa, scarecrow, and witch costumes for the oversized animatronic in 2021, and many Skelly owners have DIYed their own outfits for it.

Opens in a new window Credit: Arby's Arby's Skeleton Sweat Set $46.66 at ArbysShop.com
Launches Wednesday, Oct. 21 at 12 p.m. ET. Learn More at ArbysShop.com

UPDATE: Oct. 21, 2026, 9:30 a.m. EDT This story was updated with comment from the Home Depot.

Managing the constant influx of emails manually can feel like a full-time job. That’s why it makes sense to automate your inbox. With features like quick steps and email filters, you can streamline routine tasks and let Outlook do the heavy lifting. Here's how I automate my Outlook inbox.

TL;DR: The Pokémon TCG 30th Celebration Elite Trainer Box is available for $59.99 at Amazon. Due to high demand, shoppers need to request an invite to buy.

Opens in a new window Credit: The Pokémon Company Pokémon TCG 30th Celebration Elite Trainer Box $59.99 at Amazon
  Shop Now at Amazon

Pokémon TCG 30th Celebration Elite Trainer Box preorders first dropped for $89.99 at Amazon. We've then been stuck at $159.99 for some time, but now Amazon has dropped a stunning deal.

The Pokémon TCG 30th Celebration Elite Trainer Box is currently listed for just $59.99 on Amazon, with shipping and sales handled directly by Amazon. Demand for the product remains high, so Amazon is still routing orders through its "request invite" system.

At $59.99, Amazon’s price is way under TCGplayer’s $120 market price. We expect the deal to sell like wild, so we suggest that anyone interested should request an invite to buy immediately. Seriously, do not delay. This deal could disappear at any time.

The 30th Celebration Elite Trainer Box offers nine 30th Celebration booster packs, an exclusive full-art foil Nidorina promo card, 16 foil Basic Energy cards, 65 custom card sleeves, damage-counter dice, a condition marker, card dividers, a collector’s player guide, and a code card for Pokémon TCG Live.

Shop the Pokémon TCG 30th Celebration ETB for just $59.99 at Amazon.

The best early Black Friday Home Depot deals Best Early Black Friday Home Depot Refrigerator Deal LG 3-Door French Door Refrigerator $1610 (Save $1489) Get Deal Best Early Black Friday Home Depot Range Deal GE 4-Burner Free-Standing Electric Range $649 (Save $289) Get Deal Best Early Black Friday Home Depot Washer Deal LG Stackable SMART Front Load Washer $902 (Save $350) Get Deal Best Early Black Friday Home Depot Tool Deal DeWalt 20V Max Cordless Drill and Impact Driver Combo Kit $179 (Save $60) Get Deal

Looking to pick up new home appliances, tools, or something to make your home life easier? Look no further, because The Home Depot’s early Black Friday sale is already live, more than a month ahead of the Nov. 27 holiday.

The retailer has already slashed prices on a number of tools, appliances, smart home gear, and much more, so you don't have to wait until you're stuffed to the gills with turkey to score some fantastic deals.

We’ve been sorting through some of the early deals to find the ones that are most worth your money, and there’s a lot to work with. If you’ve got a renovation planned, a kitchen to upgrade, or a house to decorate before the in-laws show up, this is a good time to start filling a cart.

Keep reading to check out some of the best early Black Friday Home Depot deals if you want to beat the crowds both online and offline.

Best early Black Friday Home Depot refrigerator deal Opens in a new window Credit: The Home Depot LG 3-Door French Door Refrigerator $1,610 at The Home Depot
$3,099 Save $1,489   Get Deal at The Home Depot Why we like it

This 28-cubic-foot French door refrigerator fits a 36-inch opening, so it's compact enough for smaller kitchens while still offering plenty of storage space. Its stainless finish wipes clean without showing fingerprints, and a vent at the top of each door pushes cold air toward the bins where condiments and drinks usually warm up first. At the bottom, the full-width freezer drawer has a sliding upper tray that keeps smaller items from getting lost in the pile. Plus, it makes slow-melting round Craft Ice alongside regular cubes, so you’ve got the right ice for a cocktail or water bottle without a separate machine. It's a great addition to your kitchen and won't cost an arm and a leg.

Best early Black Friday Home Depot range deal Opens in a new window Credit: The Home Depot GE 4-Burner Free-Standing Electric Range $649 at The Home Depot
$929 Save $280   Get Deal at The Home Depot Why we like it

Looking to get more cooking done? Be more efficient with this 30-inch electric range with four elements under a smooth glass cooktop, including one that switches between a 6-inch and 9-inch ring. It has a a steam clean cycle that loosens light spills with water and a short heating run instead of the hours and heat of a traditional self-clean. Plus, its stainless steel knobs are a nice touch that you can match to other aspects of your kitchen.

Best early Black Friday Home Depot washer deal Opens in a new window Credit: The Home Depot LG Stackable SMART Front Load Washer $902 at The Home Depot
$1,299 Save $397   Get Deal at The Home Depot Why we like it

Give yourself a laundry upgrade with this stackable front loader. It has a TurboWash 360 cycle that can get a full load of clothes clean in under 30 minutes by spraying from five jets at once. The machine even connects to LG’s ThinQ app, which lets you start a cycle from your phone or get a notification when the load finishes. Get through that massive basket of clothes or towels in record time with the extra space and additional features and spend less at the same time.

Best early Black Friday Home Depot cordless tool deal Opens in a new window Credit: The Home Depot DeWalt 20V Max Cordless Drill and Impact Driver Combo Kit $179 at The Home Depot
$239 Save $60   Get Deal at The Home Depot Why we like it

The drill/driver and impact driver in this DeWalt kit are both compact enough to fit between studs and under cabinets, so it can handle a variety of jobs. It has an LED light that illuminates whatever you’re working on as well, so you don't have to struggle with visibility when handling each project. The drill uses a 1/2-inch ratcheting chuck that holds bits tight without a key, while the impact driver’s hex chuck lets you load and swap bits with one hand. You also get two 20V Max batteries, a charger, and a carrying bag to make a complete tool kit that's perfect for DIY jobs or handing over to repair enthusiasts.

More early Black Friday Home Depot deals

• Milwaukee M18 18V Lith-Ion Cordless Compact Blower w/ 2.0 Ah Battery — $129 $239 (save $110)

• DeWalt 20V MAX Cordless Brushless Screw Gun (Tool Only) — $149 $199 (save $50)

• DeWalt 20V MAX XR Cordless 1/2 in. Impact Wrench (Tool Only) — $199 $279 (save $80)

• DeWalt 20V XR Battery Starter Pack (8.0Ah + 4.0Ah with Charger) — $199 $329 (save $130)

• Milwaukee M12 12V Cordless Green 250 ft. 3-Plane Laser Level Kit — $599 $649 (save $50)

• Milwaukee M18 18V Cordless Combo Kit (8-Tool) with (2) Batteries and Charger — $649 $1,298 (save $649)