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• There’s big drama in math land.
• Two mathematicians used several AI models, including OpenAI’s, to help advance their research into fluid equations.
• OpenAI later announced a solution to famous math question, and said it couldn’t rule out the pair’s work “helped improve” their AI.

The big drama in AI land today is decidedly nerdy stuff.

But the hubbub highlights a concern everyone should care about: Can what you type into AI help train a machine that one day beats you at your own game?

On Monday, Tristan Buckmaster, an NYU mathematician, and Levent Alpöge, a mathematician who works at Anthropic, released papers claiming new results related to a famously thorny scientific question: Can the math used to predict the movement of water and air suddenly hit a breaking point?

Solving that problem could lead to a $1 million prize from the Clay Mathematics Institute. While the pair didn’t claim to have achieved that particular breakthrough, Terence Tao, a well-known mathematician, called the work “remarkable” in a Mastodon post.

But it was the four-page statement that Buckmaster published alongside the findings, which included some explosive allegations involving OpenAI, that captured people’s attention. He alleged that OpenAI learned of the pair’s findings — and told him an internal AI model had made progress on solving the $1 million question.

On Tuesday, OpenAI formally claimed it had achieved a solution to the Navier-Stokes Millennium Prize Problem — “one of the deepest problems at the frontier of mathematics.”

Buckmaster — who said that he and Alpöge used AI models from OpenAI and Anthropic to assist in their research — said he does not know whether OpenAI used the pair’s data or chats. OpenAI said “no specific user data was accessed in order to solve this problem,” but said it couldn’t rule out that “de-identified data derived from their usage of our products helped improve our models.”

The episode has set off a dispute over scientific credit, AI training data, and the paper trail behind an AI-assisted discovery. Scientific American said the drama has set up the “biggest day in math in at least two decades.”

What is Buckmaster alleging about OpenAI?

Buckmaster and Alpöge concluded that they found examples where today’s math doesn’t accurately describe smooth air and water flows. They used several AI models, including Claude and Codex, throughout their work.

Buckmaster said that after he told an OpenAI mathematician that he and Alpöge planned to post their work, OpenAI staff, including computer scientist Sebastian Bubeck, told him that their internal model had produced a proof of the Navier-Stokes question (the one that carried the $1 million prize).

Basically, Buckmaster and Alpöge were publishing work that could pave the way to achieving that bigger breakthrough. OpenAI, according to Buckmaster, signaled on the call that it had achieved that bigger breakthrough.

He also said he was offered possible publication arrangements and that OpenAI’s Bubeck wanted Alpöge removed as an author because he works at the company’s rival, Anthropic.

“I have not seen OpenAI’s proof,” Buckmaster wrote. “I do not know whether our data was used. I am not accusing anyone of anything. I am stating what I was told, when, and what was proposed to me. I am stating it because the alternative is to let a sequence of announcements say something I know to be false.”

On Tuesday, OpenAI’s Bubeck pushed back on some of the spiciest elements of Buckmaster’s statement in a post on X, denying he had “ever asked Levent to be removed from authorship of his own work.” He said he tried to “coordinate our releases” of the mathematical announcements, and that OpenAI was motivated to try to solve the problem after “viral Twitter rumors that Anthropic had resolved 2 Millenium problems.”

That followed an earlier post in which he said that “false and inflammatory allegations” against him were circulating.

The dispute has roiled the online math and AI communities. Sholto Douglas, a member of Anthropic’s technical staff, said it was “extremely sad.” Others highlighted the most explosive allegations in the press releases or posted a rundown of the drama.

Questions of scientific credit when AI is used

There are a lot of complexities within this math saga, especially around who gets recognition for what.

Buckmaster credits mathematicians Diego Cordoba and Luis Martinez-Zoroa for helping establish the pair’s underlying approach with their 2024 study. From there, Buckmaster said his work with Alpöge used LLMs to advance their findings.

That account raises new questions about how, or if, researchers should share credit with AI labs — especially if their models are trained on other breakthroughs from leading mathematicians.

It also poses another tricky dilemma: What should AI companies do with the data entered by their customers — and are they using it to compete directly against mathematicians and scientists?

OpenAI, Anthropic, Buckmaster, and Alpöge didn’t respond to requests for comment from Business Insider.

What do OpenAI’s terms say?

Buckmaster said OpenAI wasn’t clear with him on whether its AI models had been trained on the pair’s work. But OpenAI’s policies make it clear that is allowed.

OpenAI’s terms of service state that the company trains on data submitted by its users unless they opt out.

“When you share your content with us, it helps our models become more accurate and better at solving your specific problems and it also helps improve their general capabilities and safety,” the company wrote in a March post.

Users can opt out via the company’s privacy portal by selecting “do not train on my content.”

Buckmaster’s statement said he paid for the tools himself, but it does not identify the particular product, account type, or whether the pair opted out.

OpenAI claims big math breakthrough

OpenAI formally addressed the situation on Tuesday afternoon while announcing that it had solved the Navier-Stokes problem.

The company separately congratulated Buckmaster and Alpöge on their related work, but said that neither its in-house researchers nor its AI agents saw the pair’s work “through any means” until its public release. “No specific user data was accessed in order to solve this problem,” the company said.

But, addressing Buckmaster’s question about AI training data, the company said it was possible the AI models had learned from the pair’s inputs.

“While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models,” OpenAI said. “However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs. unforced).”

The latest from OpenAI suggests the drama is far from over.

The saga is a good reminder that AI companies are constantly working to improve their models — and, unless you explicitly opt out or have an enterprise account that stipulates as much, that includes what you type into them.

Ever wonder how successful leaders across industries get it all done? Business Insider’s “Power Hours” series explores that question by diving into the daily routines of top executives, founders, creatives, athletes, and celebrities — from aerospace and transportation to venture capital and the creator economy.

We’ve profiled leaders like Peter Beck, Rocket Lab’s CEO, who starts his day before dawn and avoids unnecessary meetings, and David Risher, the CEO of Lyft, who winds down with matcha in the afternoons and takes undercover driving shifts to inform his work.

Each story offers an inside look at how these high performers meticulously structure their waking hours to achieve their personal and professional goals. Besides giving us a window into their workdays, the featured leaders discuss exercise, rest, and tips for staying sharp, conserving energy, boosting productivity, and more.

Read stories from the series here and bookmark this page for the latest additions.

If you’re a leader in your industry and you have a fascinating routine to share, reach out to editor Lauryn Haas at lhaas@businessinsider.com.

• Instinct, an invitation-only AI assistant going viral in Silicon Valley, can handle real-world errands through text messages.
• A version of this story appeared in the BI Tech Memo newsletter.
• Sign up for the weekly BI Tech Memo newsletter here.

Here’s everything that Instinct — an invitation-only AI assistant going viral among Silicon Valley tech types — has done for me over the past few days:

• Booked a cabin for my birthday and made a dinner reservation.

• Ordered whey protein from Costco.

• Faxed (faxed!) an obscure document to my bank so I could get a late-fee refund.

• Checked whether a new dentist was in-network and made an appointment.

• Dealt with a human customer service representative at my insurance company.

• Canceled several recurring subscriptions I’d forgotten about.

• Responded to my emails.

Honestly, this service is nuts. And sure, AI tools like OpenClaw, Codex, and Claude Code can already do some of these things. But what makes Instinct different is that it lives in your text messages. There’s nothing to set up, and no technical fiddling. You send it a text, and it gets to work.

VCs see the magic, too. The WSJ reported that Instinct is raising $250 million at a $2.5 billion valuation. Its founder, Noah Shinn, is a 23-year-old Northeastern University dropout and an early employee at AI company Sierra. Shinn did not respond to multiple requests for comment.

Instinct is part of Silicon Valley’s next big bet: that AI should stop telling us how to do things and just start doing them for us, something that the industry calls “agentic AI.” My colleague Hugh Langley reported that Meta is testing something similar called Hatch.

There is, naturally, a catch. An assistant cannot run your life without gaining access to a lot of it. The more hooks that you let Instinct sink into your digital life by connecting it to your email, calendar, and anything else that requires you to log in, the more useful it is. Its terms of service also allow it to train its models on user data. Shinn did not respond to multiple requests for comment.

So here’s the part where I should tell you to tread with caution. Instead, I’m just going to leave you with an invitation to sign up. Write to me and tell me the craziest thing you tried.

Have a tip? Contact Pranav Dixit via email at pranavdixit@protonmail.com or Signal at 1-408-905-9124. Use a personal email address and a nonwork device; here’s our guide to sharing information securely.

• Sarah Friar, OpenAI’s CFO, said ChatGPT’s ads could be like ‘if Google and Meta had a baby.’
• OpenAI’s ads push is an attempt to monetize its base of 1 billion weekly users.
• Friar made the remarks at a San Francisco conference on Tuesday morning.

Congratulations to Google and Meta — they’re expecting.

OpenAI’s chief financial officer, Sarah Friar, was describing her company’s ads ambitions on Tuesday when she said that ChatGPT is like “if Google and Meta had a baby.”

Friar made the remarks before a full room of investors and analysts at Goldman Sachs’ Communacopia + Technology Conference in San Francisco.

The comment succinctly outlines OpenAI’s ambition to turn ChatGPT into a money-printing ads platform, following in the footsteps of its tech giant predecessors. In just seven months of placing ads on the chatbot, OpenAI’s ads business is now raking in cash at a rate that Friar said would equal $1 billion in revenue if mapped across a full year.

She attributed the “baby” idea to Fidji Simo, the former OpenAI executive who stepped down this year from her role running the company’s commercial operations.

Friar went on to explain that she thinks ChatGPT can use the data it learns from its users to bring the “intent” of a Google search together with the “context” that Meta has on its social media users.

“It’s high intent search, you’re telling a lot, but with the context — it’s not quite people like me, it’s me, because it has all the memory of me,” Friar said. “So when you put those two things together, the ability to have a really potent ad platform is clearly there.”

Right now, OpenAI serves basic ads underneath ChatGPT’s responses if a user is on the free or Go subscription tiers. Friar expects a more complex, more tailored format in the future. She said her team has shown her “early indications” of what a more AI-native ad format could look like as they develop the platform’s basics and scale its availability.

“This is all before we’ve really launched a format that feels truly endemic to AI,” Friar said. “That’s what I get super excited about.”

OpenAI’s focus on advertising marks a full 180 since 2024, when CEO Sam Altman said he found the idea of combining ads and AI “uniquely unsettling.”

The ambitions are a break from rival Anthropic, which has skewered the idea of ads in chatbot responses. Each of the companies is preparing to go public in highly anticipated IPOs that will test Wall Street’s appetite for AI and serve as referendums on the strength of the companies’ finances.

Both companies filed their confidential S-1 prospectuses in June, and Anthropic’s public version is expected in the coming weeks. OpenAI has not disclosed a specific timeline, but Friar has previously said that the listing will happen by the end of 2027.

OpenAI was last valued at a $852 billion in a March fundraise that injected $122 billion into its coffers.

The company is trying to parlay that cash, its 1 billion-user ChatGPT app, and its status on the frontier of AI research into a sustainable business. Friar has to balance OpenAI’s growing consumer and enterprise businesses with the AI lab’s vast expenses on AI inference, data center construction, and research.

Friar steered Square through its 2015 IPO as the company’s CFO and was Nextdoor’s CEO during its transition to the public markets via a 2021 SPAC.

Have a tip? Contact this reporter via email at scouncil@insider.com, or over text, Signal, Telegram, or WhatsApp at 415-757-8198. Use a personal email address, a nonwork WiFi network, and a nonwork device; here’s our guide to sharing information securely.

• Pam Manfredo Curtis is an industrial and organizational psychologist and executive coach.
• Parents can help their kids succeed by teaching emotional intelligence, she says.
• Pointing out kids’ strengths and asking them to identify their own can also help.

This as-told-to essay is based on a conversation with Pam Manfredo Curtis, author of “Origin of a Leader.” It has been edited for length and clarity.

It’s no surprise that we often make choices based on how we were raised. I can see it in my own life: my mom was a school psychologist. I went into psychology, too, and married a man who is also an industrial and organizational psychologist, like me.

In our field, there’s a lot of focus on empathy, tuning in to others, and helping them understand why they show up the way they do. Oftentimes, the answer lies in childhood.

I know that might sound like psychological gobbley gook, or Freudian hogwash. I’ve had clients tell me that. But after working with me, they realize that emotions are data. Leaders and companies can use that data to excel — and that emotional information is becoming more important than ever in the age of AI.

Parents worry about the wrong things

Parents often worry about the little mistakes they make. The moment when you raise your voice, or say the wrong thing, can haunt you. Luckily, those aren’t usually the things that stick with our kids over time.

What matters more is our parenting patterns. Children pay attention to how we build relationships, express love, and deal with conflict. Those teachings will stay with them long into their adulthood and career.

Teaching kids to name emotions is key

I often ask the executives whom I coach how they are feeling. Their response is usually the same: “Fine.”

I sometimes want to yell, “Fine is not an emotion!” Once, I was working with an executive of a well-known company, asking him again and again how he was feeling, and he answered, “Fine. Fine. Fine.” One day, he said “sad,” and I thought “woohoo. There’s something I can work with.”

The greatest executives I work with are able to tune into and name the emotions they’re feeling. They understand why they’re feeling the way they are, and can tell others. This is important data for them and their company — and something that can’t be replicated by AI. Teaching children to recognize their emotions will equip them for the C-suite — or any other career.

Check in with head, heart, and hands

In all my meetings, I typically start by asking people to check in on their heads, hearts, and hands. Your head tells you what you’re thinking. Your heart is about what emotion you’re feeling and why: for example, I feel nervous because I’m not sure what to expect. Your hands represent the physical — time constraints, hunger, or distracting muscle tension.

Sharing what’s in our heads, hands, and hearts builds trust. Once you’ve become vulnerable, there are no hidden agendas. It’s a great practice to do with kids, to help them connect with their emotional state.

Point out strengths — even in challenging situations

The environment in which children are raised shapes them as adults. A person raised in a stable home with high expectations and lots of love will likely become good at nurturing others. Even psychologically complex or traumatic environments can give rise to strength. An abused child who had someone advocate for them often becomes a fierce advocate for others, for example.

Each situation, and how a person responds to it, is unique. I always tried to let my children know what I thought was beautiful about them. I highlighted their strengths and validated their worries while pointing out what I really admired about them.

Giving children agency and helping them feel heard will strengthen their ability to move forward, even in the face of difficult emotions or circumstances.

• Leah Young moved from Texas to Spain in 2015 and now lives in Lyon, France.
• Young says she’s happy with her life in France, which includes access to affordable, high-quality healthcare.
• Her one-bedroom apartment costs her $1,104 each month, and she spends $70 a week on food.

This as-told-to essay is based on a conversation with Leah Young, a 34-year-old from Dallas who moved to Spain in 2015. Young is the founder of Fearless Abroad, a company that helps Americans working remotely relocate to Europe. The conversation has been edited for length and clarity.

A lot of people love the idea of traveling to new places and experiencing something exciting. What excited me about living abroad was the adventure. When I was younger, I’d taken a few short trips, but there was still so much more of the world I wanted to see.

For years, I tried to make it happen. In high school and college, I looked into exchange programs and studying abroad, but my family was always like, “No way.”

Eventually, a friend mentioned a government-run teaching program in Spain that had an open call for applications. I jumped at it, even though I was working in a completely different industry. I figured it was finally a way to get to Europe.

I thought I’d spend just one year abroad, then return to Texas and live my life as I always had. But that year turned into many, and I stayed, though in different ways over time.

I was terrified to move to Europe

Preparing to leave for Europe was really scary. I had never lived abroad, and I didn’t know anyone else who had. I remember thinking, “How practical is this, really?”

The visa process was stressful. My teaching program gave me permission to come, but I still had to handle the visa process myself. I got through it by diving deep into Facebook groups and random blogs, piecing everything together and learning as much as I could.

I was really worried about whether I could afford the cost of living in Spain. Yes, I was moving with a job contract, but I would earn only €700 ($813) a month. Even in 2015, that wasn’t much. I didn’t know whether it would cover my rent, let alone leave money for anything else.

There were so many other things I had to figure out, like whether to cancel my US health insurance — my government contract came with Spanish coverage — and what to do with my phone plan, car, dishes, furniture, and everything else in my apartment.

In the end, I left the US with nothing but two suitcases, filled with all the wrong things, and a visa. The truth is, I was totally freaked out. But I went for it because I saw that other people had made it work — and, ultimately, I did, too.

Lyon was the perfect place for me

I feel like I’ve become an expert at starting over. After living in Spain for five years, I spent two years in Germany, a year in Argentina, a year in Brazil, and now I’ve been in France for a year.

I live in Lyon, about two hours from Geneva, near the Swiss border. It’s very close to the mountains, so within an hour or two, you can be skiing in the Alps, or you could be swimming in the Mediterranean Sea.

Lyon is quintessentially what you’d imagine traditional France to be. It’s a really nice second option to living in Paris. While it’s a large city, it has a small-town feel.

I’ve traveled and lived around Europe, and in some places, I found people a little closed off or hard to build relationships with. That was a real struggle for me because, as a Texan, I’m outgoing and want to make friends and build a community. But since moving here, I’ve found it much easier to make friends.

My life feels pretty affordable

Finding a place to live in Lyon can be competitive, but I got really lucky. I found my apartment within two days through word of mouth. I was chatting with someone who knew a person with an available apartment, and that’s how I got it.

I pay €950 ($1,104) a month in rent for a beautiful 678-square-foot one-bedroom apartment in a historic building dating back to the 1500s. It has fantastic stone walls, tall ceilings with wooden beams, and massive French windows overlooking the old town. I can even hear the clock tower ring every half hour.

My utility bill varies a lot. It’s pretty affordable in the summer, since I don’t use air conditioning, and I might pay only €50 ($58) a month. But winters can be expensive because it gets chilly, and I live in an old building. During the coldest months, my heating bill can be around €350 ($407).

Overall, I’d say living in Lyon is pretty affordable. I cook a lot and spend about €60 ($70) a week on groceries. I usually eat out only twice a week, and getting a glass of wine or a beer might cost €5 ($6). I also don’t have a car, nor do I need one. I spend less than $10 a week getting around on the train. Taking trips can be really cheap, too. It might cost me around €50 ($58) round trip to fly to Germany.

I don’t think I’ll ever move back to the US

I’ve had a lot of different jobs during my time abroad. When I first arrived, I was teaching. Then I switched to a student visa and went back to school. For a while, I worked remotely for a US-based company. Now, I own a business that helps other remote workers relocate to Europe.

Even so, I’ve visited my family in the US many times over the years and considered moving back, but it just hasn’t worked for me.

It’s so expensive, especially in Texas, where things are booming, and rents have gotten crazy. It always blows me away to see people paying $2,000 a month for a one-bedroom apartment.

Healthcare is another reason Europe suits me better. The quality is great, but it doesn’t feel like it costs a million dollars, as it does in the US. That feels especially valuable as I get older and think about having kids.

Of course, there are things I miss about the US, like my family and friends. I miss the ease of conversation, too — being able to crack a joke without worrying that it’ll be misunderstood. And honestly, I miss the food, especially steak and Tex-Mex. Texas has the best food in the world, if you ask me.

I do think France is my permanent home, but it’s hard to say for sure. Can anyone really tell you where they’ll be in 10 years? Of all the moves I’ve made, though, I’m happiest here. It feels like home, and I’m at a point in my life where I’m ready to settle down.