Better Understanding of Betti Numbers
A simplicial complex is a set formed from points, line segments, triangles, and their higher-dimensional analogs. Betti numbers classify topological spaces according to the connectivity of simplicial complexes. The kth Betti number is the rank of the kth homology group and is the maximum number of cuts that can be made before separating a surface into two pieces. For some value n, Betti numbers...
Mathematicians solve one of the mysteries of two-dimensional shapes
Pure mathematicians like Daniel Cristofaro-Gardiner thrive on the creative challenge of imagining connections between things that seem unconnected and then using these connections to puzzle out old problems. Take the simplicity conjecture, which Cristofaro-Gardiner solved earlier this year with two of his colleagues. The conjecture emerged from the work of mathematicians who, during the...
New Machine Learning Approach for Mean-Field Games
Mean field games are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents with numerous applications ranging from economics to crowd motion. Their numerical solutions typically suffer from the curse of dimensionality. Numerical methods for mean field games were usually grid based. This effectively limited them to two and three dimensions....
Quasisymmetries for Quasiparticles
For an ordinary chemical such as water, the phase diagram shows which state the chemical likes to be in at different temperatures and pressures. Water, for example, has one liquid phase and seventeen solid phases (one of which is the cubical form that you think of as “ice”). Now, mathematicians have identified a theoretically feasible substance with an even more complicated phase diagram than...
The Math of Irreversibility
A relatively new branch of mathematics, called higher category theory, seeks to address a common but perplexing phenomenon in math as well as other sciences. “We’re trying to build tools that allow us to capture irreversibility,” says Ezra Getzler of Northwestern University, one of 47 mathematicians who participated in the 2020 spring program on “Higher Categories and Categorification” at the...
Algebraic Computer Vision Advances the 3D Reconstruction of Curves and Surfaces from Multiple Views
Contributed by: Ricardo Fabbri Given a series of ordinary photographs of coffee mugs over a table, can you guess where the photographer was located at each shot? In other words, given m central projections of piecewise smooth curves and surfaces, what are the translations and rotations of the underlying camera? The diagram below illustrates a case with m = 3 where the goal is to recover two...
Math that feels good
Mathematics and science Braille textbooks are expensive and require an enormous effort to produce — until now. A team of researchers has developed a method for easily creating textbooks in Braille, with an initial focus on math textbooks. The new process is made possible by a new authoring system which serves as a “universal translator” for textbook formats, combined with enhancements to the...
The Classification of Fusion Categories
Understanding and organizing the basic building blocks of mathematics and physics is one of the most natural of scientific endeavors. We do this with the elements in our periodic table, with descriptions of elementary particles, and with many mathematical structures. Perhaps the simplest example is the description of integers is products of primes. Paul Bruillard, Siu-Hung Ng, Eric C. Rowell,...
Triangle Tiling Billiards and a Cat in the Rain
Contributed by Olga Paris-Romaskevich Tiling billiards is a mathematical model for the movement of light in heterogeneous media made from homogeneous pieces. Objects in water, seen through a flat surface, do appear magnified when the eye is close to the surface. Anyone who has used a diving mask is aware of it. The reason for this effect is the Snell’s law of refraction: light refracts when...
Establishing a Theoretical Understanding of Machine Learning
Program at IAS aims to explain the why and how of algorithms with enormous power “It’s kind of like physics in its formative stages—Newton asking what makes the apple fall down,” says Sanjeev Arora, Visiting Professor in the Institute for Advanced Study’s School of Mathematics, trying to explain the current scientific excitement about machine learning. “Thousands of years went by before science...