Khovanov Homology of Positive Knots
While there exists a theoretical algorithm that decides in finite time if a given knot is positive or not, this algorithm has a factorial run time and is thus a purely theoretical result that is not working in practice even for very simple knots. In their project, Marc Kegel and Marithania Silvero have developed a different approach to obstruct positivity of knots.
Modularity of Generating Series
Jan Bruinier, Benjamin Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang have a breakthrough result on the modularity of generating series of divisors on unitary Shimura varieties, published in two parts as a monograph in 2020 in Astérisque.
Novel use of ergodic theory leads to breakthrough solution of conjectures of Erdős about sumsets
Mathematicians build on Furstenberg’s correspondence principle to transfer the combinatorial problem to a dynamical one
Applying computational and data science to social justice issues requires a lot of “non-mathematical work”
In order to develop new mathematical techniques to explore and address issues related to social justice, ironically, mathematicians need to do a lot of work considered by many to be “non-mathematical.”
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.