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.
A Second-Order Correction Method For A Parabolic-Parabolic Interface Problem
Fluid-structure interaction (FSI) problems involve the study of how fluids and solid structures behave when they come into contact with each other. These problems are important in many engineering and scientific applications, such as aeronautics, aeroelasticity, aerodynamics, biomechanics, civil engineering, and mechanical engineering. Sijing Liu introduces a second-order correction method for a parabolic-parabolic interface problem, a simplified version of FSI.
Grothendieck Shenanigans: Permutons From Pipe Dreams via Integrable Probability
If Alice and Bob each take a walk, a random one, what is the chance they will meet? If n people walk randomly in Manhattan from the Upper East Side going south or west, how often will two of them meet until they reach the Hudson? If they do not want to see each other more than once, in what relative order will they most likely arrive at the Hudson shore? If we make a rhombus like an Aztec diamond from 2 by 1 dominoes, what would it most likely look like? If water molecules arrange themselves on a square grid, what angles between the hydrogen atoms will be formed near the boundaries? And if ribosomes are transcribing the mRNA, how do they hop between the sites (codons)?
