AI-solved Mathematics. Human Understanding.

The Unique Games Conjecture: a deterministic polynomial-time reduction from 3SAT with completeness arbitrarily close to one and soundness arbitrarily close to zero over a fixed finite alphabet.

An upper bound of 9/4 for the matrix multiplication exponent over the complex numbers.

We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric. In dimensions three and four, two metric derivatives and one tensor derivative suffice, and both positive enclosing area and future timelikeness follow from the hypotheses.

We prove a power saving for the planar unit-distance problem: for some absolute \beta\lt 4/3, every set of n points in the Euclidean plane determines O(n^\beta) unordered pairs at unit distance.

We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in (n+m)^{1+o(1)} time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.

We prove \mathsf L=\mathsf{RL}=\mathsf{BPL}, resolving the derandomization problem for polynomial-time randomized logarithmic space.