#### 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.

#### Mathematicians prove forced Navier-Stokes solutions are not unique in the Leray-Hopf class

For almost 200 years, scientists and engineers have used the so-called Navier-Stokes equations to describe how fluids behave. These partial differential equations arose from Newton’s second law—that an object’s acceleration depends on its mass and the net force acting upon it—and incorporate internal forces, such as pressure, and external forces, such as gravity, on the movement of fluids. Among their applications, the Navier-Stokes equations, and their variations, are the main models used today for determining currents in the oceans, airflow around airplanes and cars, smoke dispersion from chimneys, and water flow in pipes.

#### Dispersive shock waves

Shock waves can be observed both experimentally and theoretically, and manifest as a train of localized, highly oscillating waves. They are observed in water waves, plasma, optics, or Bose-Einstein condensates. Dispersive shock waves can appear when one of the relevant quantities of the model changes abruptly compared to the others. Mathematically, it is possible to grasp the formation of a dispersive shock wave by letting a particular parameter of the modeling equation tend to zero: this is called the zero-dispersion limit.

#### Universality and Integrability in Random Matrix Theory and Interacting Particle Systems

This semester, participants in the Universality and Integrability in Random Matrix Theory and Interacting Particle Systems semester program studied the broad connections between random matrix theory and interacting particle systems, which can be used to model atoms or molecules, but also to find applications in areas as dissimilar as traffic flow, epidemiology, and financial markets.

#### Connecting Configurations

Consider a straight line in a plane with a point on it. Imagine moving the point on the line and moving the line in the plane. A parameter space can be constructed so that each point in the space represents a possible point-line configuration. Moreover, if one configuration can be turned into another by moving the point and the line just a little bit, then the corresponding points of the two configurations in the space are close. This space has a very nice and symmetric geometric shape.

#### African Diaspora Joint Mathematics Workshop

The African Diaspora Joint Mathematics Workshop (ADJOINT) is a yearlong program that provides opportunities for U.S. mathematicians, especially those from the African diaspora, to form collaborations with distinguished African American research leaders on topics at the forefront of mathematical and statistical research.

#### Fluid Dynamics

Fluid dynamics is a broad area of research that has been the subject of mathematical investigation for hundreds of years, inspired by but not limited to practical applications in biology, chemistry, physics, and engineering.

#### 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.