Hierarchical clustering is a data analysis method that has been used for decades. Despite its widespread use, the method has an underdeveloped analytical foundation. Having a well understood foundation would both support the currently used methods and help guide future improvements. The goal of this paper is to give an analytic framework to better understand observations seen in practice. This paper considers the dual of a problem framework for hierarchical clustering introduced by Dasgupta.

Epstein's shadow: Why Bill Gates pulled out of Modi's AI summit Microsoft founder Bill Gates has cancelled his keynote speech at India's flagship AI summit just hours before he was due to take the stage on Thursday. Gates, who has faced renewed scrutiny over his past ties to the late sex offender Jeffrey Epstein, withdrew to "ensure the focus remains on the AI Summit's key priorities", the Gates Foundation said in a statement. India's Prime Minister Narendra Modi had billed the summit as an opportunity for India to shape the future of AI, drawing high-profile attendees, including French President Emmanuel Macron and Brazilian President Luiz Inacio Lula da Silva. Instead, it has been dogged by controversy, from Gates's abrupt exit to an incident in which an Indian university tried to pass off a Chinese-made robotic dog as its own innovation. So, what exactly went wrong at India's flagship AI gathering and why has it drawn such intense scrutiny?

From threat modeling to encrypted collaboration apps, we've collected experts' tips and tools for safely and effectively building a group--even while being targeted and tracked by the powerful. Rarely in modern US history have so many Americans opposed the actions of the federal government with so little hope for a top-down political solution. That's left millions of people seeking a bottom-up approach to resistance: grassroots organizing. Yet as Americans assemble their own movements to protect and support immigrants, push back against the Department of Homeland Security's dangerous incursions into cities, and protest for civil rights and policy changes, they face a federal government that possesses vast surveillance powers and sweeping cooperation from the Silicon Valley companies that hold Americans' data. That means political, social, and economic organizing presents a risky dilemma. How do you bring people of all ages, backgrounds, and technical abilities into a mass movement without exposing them to monitoring and targeting by a government--and in particular Immigration and Customs Enforcement and Customs and Border Protection, agencies with paramilitary ambitions, a tendency to break the law, and more funding than some countries' militaries. Organizing safely in an age of surveillance increasingly requires not only technical security know-how, but also a tricky balance between secrecy and openness, says Eva Galperin, the director of cybersecurity at the Electronic Frontier Foundation, a nonprofit focused on digital civil liberties.

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Autonomous agents will soon run thousands of enterprise workflows, and only organizations with unified, trusted, context-rich data will prevent chaos and unlock reliable value at scale. AI agents are moving beyond coding assistants and customer service chatbots into the operational core of the enterprise. The ROI is promising, but autonomy without alignment is a recipe for chaos. Business leaders need to lay the essential foundations now. Agents are independently handling end-to-end processes across lead generation, supply chain optimization, customer support, and financial reconciliation. A mid-sized organization could easily run 4,000 agents, each making decisions that affect revenue, compliance, and customer experience.

In the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics. Standing in the middle of a field, we can easily forget that we live on a round planet. We're so small in comparison to the Earth that from our point of view, it looks flat. The world is full of such shapes--ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds.

In 2025, the edges of mathematics came a little more sharply into view when members of the online Busy Beaver Challenge community closed in on a huge number that threatens to defy the logical underpinnings of the subject. This number is the next in the "Busy Beaver" sequence, a series of ever-larger numbers that emerges from a seemingly simple question - how do we know if a computer program will run forever? To find out, researchers turn to the work of mathematician Alan Turing, who showed that any computer algorithm can be mimicked by imagining a simplified device called a Turing machine. More complex algorithms correspond to Turing machines with larger sets of instructions or, in mathematical parlance, more states. For example BB(1) is 1 and BB(2) is 6, so making the algorithm twice as complex increases its runtime sixfold.