Commutative Algebra
In mathematics, studying objects through their representations is a well-tested path; the area representation theory is so named because it represents the study of a group, a ring, or any other algebraic structure via linear transformations, which are sometimes easier to work with. In algebraic geometry and commutative algebra, where the objects in question are varieties and their coordinate rings, the representations take the form of modules and sheaves.
Electrification and Green Energy Markets
As economies make the transition to green, renewable sources of electricity such as wind and solar, engineers, economists, energy producers and consumers, and regulators are struggling with how to integrate these new energy technologies into the grid. A major challenge is that wind and solar are only intermittently available and their availability may not coincide with periods or locations of high demand. To address the topic of electrification via renewables, the Institute for Mathematical and Statistical Innovation (IMSI) hosted a Long Program on “The Architecture of Green Energy Systems,” during the summer of 2024.
IAS Researcher Debuts Counterexample to Viterbo’s Conjecture
Pazit Haim-Kislev has published a counterexample to Viterbo’s conjecture, demonstrating that some convex domains can achieve larger symplectic capacities than previously thought possible. With coauthor Yaron Ostrover, Haim-Kislev has discovered surprising features to the relationship between convex geometry and symplectic capacity. Their findings signal rich potential in further exploration of the unexpected ways these mathematical invariants behave...
New Momentum and Optimism at the Convergence of Geometry and Analysis
When contemplating degeneration and mess—such as wrinkles at points or cusps, or even wrinkling behavior that propagates to infinity on otherwise smooth surfaces known as manifolds—the Swiss-French mathematician Tristan Rivière, a professor at ETH Zurich, is upbeat and hopeful. “Understanding these degenerations is inspiring,” Dr. Rivière said.
Ricci Flow, Redux
The much-vaunted Ricci flow equation was introduced by the mathematician Richard Hamilton in 1982. The equation is a tool: When applied to a manifold, a curved space in higher dimensions, the equation evolves the geometry of the space, making it smoother, more like a sphere. The Ricci flow is often compared to the heat equation, which describes how heat flows and distributes through space more evenly over time.
Algebraic Tools for Phylogenetic Networks
Phylogenetics is the field of mathematical biology concerned with recovering and describing evolutionary relationships between collections of taxa. The branching trajectory of evolution is often represented via a phylogenetic tree: vertices on the tree represent different taxa and taxa that are close to each other in the tree are evolutionarily close to each other. A major theme at the ICERM Fall 2024 program concerned the effort to move beyond trees.
Machine Learning and Carbon Capture
Chemical engineers are teaming up with data scientists to use machine learning as a transformative technology in the discovery of new materials that can be used to capture the carbon dioxide produced by power plants. Their target are metal-organic frameworks – crystalline materials that act like sponges that can selectively absorb greenhouse gases in the flues of coal-burning and natural gas-burning power plants.
With New Theory and Algorithm, IAS Researchers are the First to Compute K-Groups
Since its founding, K-theory has presented seemingly intractable obstacles to computation. K-theory is a branch of pure mathematics that studies rings, algebraic structures that index systems of numbers using two binary operations. K-theory assigns rings a series of invariants, called K-groups, that provide insight into their character. Invariants are vital to understanding core truths and relationships in mathematics. They describe properties that remain constant through alteration, shedding light on – or sparking curiosity about – a wide range of objects in algebra, geometry, and topology...
Practical Inference Algorithms for Species Networks
Phylogenetics is a field of evolutionary biology dedicated to understanding the evolutionary relationships among species. Inferring these relationships is essential for diverse applications, including conservation efforts, tracking infectious disease dynamics, and improving agricultural practices.
Understanding Moneyless Markets
Economists are using mathematics to come to a deeper understanding of how to allocate indivisible goods in markets that do not use currency.
