Mathematicians stunned by AI's biggest breakthrough in mathematics yet An 80-year-old maths conjecture that has eluded the world's greatest mathematicians has been cracked by an artificial intelligence model built by OpenAI. The result has stunned experts and is being hailed as a seismic moment for AI's mathematical ability. "This is a problem that I didn't expect to see solved in my lifetime," says Misha Rudnev at the University of Bristol, UK. "It's absolutely a bomb." Tim Gowers at the University of Cambridge wrote that the solution is "a milestone in AI mathematics" in a blog post accompanying the work . "If a human had written the paper and submitted it to the and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that."

Staff at WiseTech have been waiting months to be told if they are among the employees the company is to cut due to advances in AI. Staff at WiseTech have been waiting months to be told if they are among the employees the company is to cut due to advances in AI. WiseTech begins redundancies - but omits'AI' from emails to Chinese employees, workers say WiseTech has begun informing staff that they will lose their jobs as part of redundancies the company has said is due to artificial intelligence advancements - although an email to staff in China omitted the word "AI" after a court case against another company in the country. Staff at WiseTech have been waiting almost three months to be told if they are among the 2,000 people the logistics software company is to cut due to advances in AI. The Australian Stock Exchange-listed company announced in late February it would lay off almost 30% of its 7,000-strong workforce across 40 countries.

From a WWII submarine sewage disaster to a deadly medieval pit toilet collapse, doing your business can come with risks. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Toilets can be surprisingly dangerous. Breakthroughs, discoveries, and DIY tips sent six days a week. In 1076, a Dutch nobleman named Duke Godfrey "the Hunchback" of Lower Lorraine was murdered in a most unusual way .

Air France and Airbus have been found guilty of manslaughter over a 2009 plane crash which killed 228 people. The Paris Appeals Court found the airline and aircraft manufacturer guilty of corporate manslaughter over the incident, in which flight AF447 between Rio de Janeiro and Paris crashed into the Atlantic Ocean. The passenger jet stalled during a storm and plunged into the water, killing all on board. A court had previously cleared the companies in April 2023 but they were found guilty after this appeal. The Airbus A330 vanished from radars during a storm, with its wreckage found after a long search of 10,000 sq km (3,860 sq miles) of sea floor.

France is already moving on from Zoom and Microsoft Teams in favor of homegrown alternatives. Other countries are quickly following suit. As tensions between President Donald Trump and Europe continue to simmer, the continent is accelerating its moves to reduce its addiction to US technology . Cities and governments are ditching Microsoft Office for open-source alternatives, shifting to European cloud hosting for local AI, and moving defense data to systems without American involvement . Nowhere has this been more clear than in France.

This week, with air raid warnings wailing in the distance, Kyiv held a funeral for two sisters. They had already lost their father who had been fighting on the front line. Their grieving mother is now the family's sole survivor. This is the human cost of the largest sustained Russian aerial assault so far - with 1,500 drones and 56 missiles fired at Ukraine within 48 hours. But the loss of life could have been even higher.

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

Inferring latent dynamics from multivariate time-series defined over topological cell complexes is crucial for capturing the complex, higher-order interactions inherent in real-world systems such as in water, sensor, and transportation networks. However, reconstructing these latent states is challenging because the signals are coupled across higher-order topologies, while high dimensionality, nonlinear observations, and unknown structures increase the difficulty. To address this, we propose a topology-aware state space framework derived from stochastic partial differential equations on cell complexes. State evolution follows heat-like topological diffusion, with perturbations propagating along boundary operators. Under partial observability, we model observations using a cell complex convolution of latent states coupled with a nonlinear mapping. We perform recursive state estimation via an Extended Kalman Filter, simultaneously learning model parameters and uncertainties through an online Expectation-Maximization algorithm. Finally, for scenarios where only lower-order topological structure is known, e.g., nodes and edges, as in critical infrastructure networks, we introduce a heuristic cell identification algorithm to explicitly infer the second-order cell structures. Validations on synthetic and real datasets from water, sensor and transportation networks demonstrate that our approach yields reliable estimates under partial observability and successfully recovers the underlying topological structures.

While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each recommended item is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens nodes evaluations.