A Cryptography Breakthrough Rooted in Gödel’s Incompleteness Theorem
IAS researcher Rahul Ilango has published a major conceptual advance in cryptography that relaxes protocol requirements for zero-knowledge proofs. His offering, an “effective” zero-knowledge proof, matches the security guarantees of zero-knowledge proofs in practice while doing away with the need for interaction. To remove this necessity of back-and-forth verification, Ilango turned to a surprising source of insight: Gödel’s incompleteness theorem, one of the greatest – and most astonishing – revelations in modern mathematical logic.
At the Cutting Edge of Stochastic Partial Differential Equations
Stochastic partial differential equations are used to describe and model complex systems — such as heat flow and financial markets — within which change is influenced not just by deterministic variables, but also by random perturbations or “stochasticity.”
The NSF Mathematics Institutes: At the Nexus with AI
In October 2025, a scrum of mathematicians spent a week teaching partial differential equations to a computer, using a proof assistant and programming language called Lean.
Research Highlight: Mathematical Analysis of Many-Body Quantum Simulation with Coulomb Potentials
Efficient simulation of many-body quantum systems, originally proposed as a primary motivation for building quantum computers, remains one of the most fundamental tasks in quantum computing.
The Importance of a Problem List
The new book K3: A New Problem List in Low-Dimensional Topology grew out of a 2023 AIM workshop designed to create the next version of the famous "Kirby problem lists." Organizers hope that this list, like the previous two, will inspire graduate students and experienced researchers alike, and that it will set the research agenda for the field for the next decade or more.
Digital Twins and Personalized Medicine
Digital twins are virtual, data-driven models that replicate physical systems and processes, enabling continuous simulation, prediction, and optimization. In health care, a digital twin of a patient leverages real-time clinical data and computational models to reflect physiology and disease dynamics—allowing clinicians to tailor treatments, manage disease, and enhance outcomes with unprecedented precision.
Homogeneous Dynamics Proof Opens New Doors in Number Theory
A team of mathematicians at the Institute for Advanced Study has released a breakthrough proof that sheds new light on one of number theory’s oldest and most established inquiries. Their achievement highlights the power of a surprising partnership: number theory and dynamical systems.
Effective Computation With Hourglass Plabic Graphs
Many students believe they should always expand their algebraic expressions. Yet computations are often easier if we don’t expand too early and instead remember the meaning of fragments of complex expressions. Consider the calculation of the determinant. The classic expansion formula has n! terms and is impractical beyond toy examples. But if we, for instance, spot linear dependence, we can immediately conclude the n! terms will all cancel without actually doing any laborious, error-prone arithmetic.
Rigidity and Flexibility in Geometric Constraint Systems
Geometric constraint systems (GCS) are used to model a wide range of geometric objects. These structures come with natural constraints, such as distances, angles, coplanarity, volume, or tangency, that are invariant under Euclidean isometries, namely translations, rotations, and reflections. GCS arise in numerous applications, including structural engineering, crystallography, soft-matter physics, and biochemistry. Two particularly important properties associated with GCS are rigidity and flexibility.
A Massive hypercube hidden in the Bruhat Order, Found by AI
Imagine all possible ways of shuffling numbered cards. Mathematicians organize this enormous set of permutations using a structure called the Bruhat order, which arranges them by declaring τ <σ when a single swap turns τ into σ and increases the deck’s “disorder” in a precise way. This deceptively simple rule has deep consequences in representation theory, geometry, and combinatorics. Within this highly intricate structure, the authors of Bruhat intervals that are large hypercubes [Ell+25] uncover something exceptionally rigid: a gigantic, perfectly organized hypercube hidden inside the seemingly “chaotic” Bruhat order.
