Optimal Transport and a Unified Perspective on Set Dualities
Optimal transport is a powerful framework, used by mathematicians and economists, that studies the allocation of resources. The origin of the subject goes back to the engineer Gaspard Monge, who was interested in the problem of the optimal way of redistributing mass: given a pile of soil, how can it be transported and reshaped to form an embankment with minimal effort? Originally, the cost function of transportation was simply the distance the particles were moved, but as the world evolved, scientists became interested in other, often abstract, cost functions.
White Paper: Computational Microscopy
For more than three centuries, lens-based microscopy, such as optical, phase-contrast, fluorescence, confocal, and electron microscopy, has played an important role in the evolution of modern science and technology.
White Paper: Advancing Quantum Mechanics with Mathematics and Statistics
This white paper is an outcome of IPAM’s spring 2022 long program, Advancing Quantum Mechanics with Mathematics and Statistics. The white paper summarizes the activities and outcomes of the Long Program “Advancing Quantum Mechanics with Mathematics and Statistics” which was held at the Institute of Pure and Applied Mathematics (IPAM) from March 7 to June 10, 2022. It also briefly explores some of the current open questions and future directions in the field of electronic structure theory and computational chemistry as well as related fields that were discussed during the program.
Lean for the Curious Mathematician 2022
Interactive theorem proving software can check, manipulate, and generate proofs of mathematical statements, just as computer algebra software can manipulate numbers, polynomials, and matrices. Over the last few years, these systems have become highly sophisticated, and have learnt a large amount of mathematics.
Mathematics and Statistics Contribute to Climate Science and Policy
Scientists from across the disciplines have spent decades making the case that our planet is rapidly warming due to anthropogenic emissions of carbon dioxide (CO2) and other greenhouse gasses. The catastrophic consequences of global warming are becoming undeniably obvious. Scientists, economists, and social scientists are now collaborating to improve our understanding and predictions of how Earth’s changing climate will impact humanity and the ecological and social systems upon which life relies.
Finding vulnerabilities in machine learning—and searching for ways to protect against them
Theoretical computer scientists prove that undetectable, malicious backdoors are possible.
Gibbs measures and nonlinear wave equations
Gibbs measures are tools from probability that play a fundamental role in mathematics (partial differential equations (PDE), stochastic partial differential equations, etc.) and in physics (statistical mechanics, Euclidean field theories, etc).
The quest for uniform bounds
Laura DeMarco, Holly Krieger, and Hexi Ye employ arithmetic tools and ideas from dynamical systems to provide a uniform Manin-Mumford bound on the number of torsion points for a family of genus 2 curves.
Mathematics and Physics at the Moiré Scale
The unique electronic, optical, and mechanical properties of 2D materials have recently sparked an extraordinary level of experimental, theoretical, and computational activity in the materials science and physics communities. Interest in the mathematics community has recently emerged to develop rigorous foundations, improved models, and computational methods. IPAM sponsored a workshop on “Theory and Computation for 2D Materials” during January 13-17, 2020 that facilitated exchanges between the mathematics community and the physics communities working on 2D materials.
SQuaRE group mixes theory and computation
AIM SQuaRE group produces a new tool for research in algebraic geometry.
