#### Institute for Advanced Study researcher mines polynomial patterns in dense sets

Sarah Peluse has spent the past few years looking for patterns in numbers. Specifically, she’s been looking for polynomial progressions, which involve sequences of numbers with a fixed common ratio plus a fixed value (x, x + y, x + y2), such as 3, 3 + 3, 3 + 32. Prior work had proved that once a set of numbers gets big enough, it reliably contains these polynomial progressions. But nobody knew just how big exactly.

#### Reduced basis approaches to speed up kinetic transport simulations

Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The physical and velocity/angular variables that the solution depends on are 6-dimensional, and the problem is inherently multiscale in nature. These features hamper the efficient and accurate numerical simulation of such problems. As part of ICERM’s Spring 2020 program on Model and dimension reduction in uncertain and dynamic systems, Yanlai Chen, Yingda Cheng, Fengyan Li, and Zhichao Peng joined together to work on a reduced order model approach that would allow efficient and robust simulations of linear kinetic transport equations.

#### Decision Making in Health and Medical Care

New models are challenging old norms for how regulatory bodies oversee the drug discovery process and approve much needed new therapies. Importantly, I-SPY 2 and now GBM AGILE, are prototypes for drug platform trials for COVID-19, Alzheimer’s Disease, Amyotrophic Lateral Sclerosis, and other diseases. Modern statistical methods, and new computational learning and decision-making tools are contributing to a hopeful future for those stricken with deadly and debilitating diseases.

#### Math & Racial Justice

In response to the groundswell of support for racial justice initiatives following the killing of George Floyd, a Black man, by a White police officer in Minnesota, this workshop addressed algorithmic bias; fair division, allocation, and representation; public health disparities; and racial inequities in mathematics education.

#### Squeezing data with the power of algorithms

Two computer scientists determine the smallest amount of data needed to estimate the mean When two fields of study collide, advances that were once all but unimaginable can be realized. At least that’s what two computer scientists showed when they applied the algorithmic tools of their field to solve the statistics problem of how to accurately guess the average value of a phenomenon from as few...

#### Eccentric binary black hole surrogate models for the gravitational waveform and remnant properties: comparable mass, non-spinning case

Black holes are fascinating objects! They become more so when two of them merge. Merging black holes radiate energy in the form of gravitational waves (GWs). GWs then propagate intergalactic distances to reach Earth. Ground-based detectors operated by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo are capable of detecting such signals. Detection of GWs can help us to...

#### Gravitational hair for extreme Kerr black holes using a GPU-accelerated mixed-precision WENO method

A breakthrough discovery was made that a special kind of black hole violates black hole uniqueness, the so-called “no hair” theorem. Specifically, a computational study of extremal black holes — holes that are “saturated” with the maximum charge or spin they can possibly carry — found that there is a quantity that can be constructed from the spacetime curvature at the black hole horizon that is...

#### Orbital dynamics of binary black hole systems can be learned from gravitational wave measurements

The collision of massive compact objects, such as black holes (BHs) or neutron stars, emit an immense amount of energy in the form of gravitational waves. With the advent of advanced gravitational-wave detectors, such as the NSF-funded Laser Interferometer Gravitational-Wave Observatory (LIGO), we are now able to directly observe gravitational waves. Encoded in these waves are information about...

#### Summer School on Data, Dynamics, and COVID-19

How did the Covid pandemic affect carbon emissions in 2020? Does having a certain blood type make one more susceptible to Covid? What is the role of the uninsured population in driving the pandemic? These and many other questions were the focus of over forty graduate students and advanced undergraduates who participated in the 2020 AIM Online Summer School on Dynamics, Data and the COVID-19...

#### A Deep-learning Solution for the Schrödinger Equation

Solving the electronic Schrödinger equation is at the heart of computing and understanding in detail the physical and chemical properties of molecules and materials. They are of paramount importance for developing new drug molecules, biofuels or nanomaterials. Unfortunately the Schrödinger equation can only be solved exactly for the hydrogen atom, and the cost of highly accurate approximations...