#### Interpreting the Shape of Neural Activity

An animal’s brain encodes representations of important aspects of the world and its own internal state through its constituent neurons. This information is distributed across many different anatomical regions in the brain, each of which performs computations and passes the results to others, with the end result producing behavior. The foundational observation in topological data analysis, or the study of datasets’ geometric features, is that the coarse, mesoscale structure of data can be described in terms of circles, spheres, tori, and a galaxy of more esoteric objects.

#### Connections in Gauge Theory

Inspired by the rise of mirror symmetry in theoretical physics, Jason Lotay is investigating connections between mathematical objects called calibrated submanifolds and instantons.

#### Building Bridges with Floer Homotopy Theory

It started with a walk. Lisa Traynor and Hiro Lee Tanaka liked to hike through the fire trails near the Simons Laufer Mathematical Sciences Institute (SLMath) during their lunch hours. Both were participants in the Floer homotopy theory semester program but came to the program from different edges of the field. Traynor, a professor at Bryn Mawr College, specializes in symplectic and contact structures on manifolds, while Tanaka of Texas State University focuses on homotopy theory. But eventually their walks became a path to a joint research project.

#### From Soap Bubbles to Singularities: Exploring the Principle of Geometric Minimization

Imagine that you are blowing a soap bubble using a simple children’s wand. Why does the bubble that you create take the form of a sphere? Why not an ovoid? Or a cylindrical shape? Or something totally random? An answer is provided by the concept of “minimization.” In nature, certain phenomena tend toward minimization, meaning that they seem to “want” to take a path that uses minimal energy to return to an equilibrium state, or, in the case of the bubble, form a shape that has the least surface area possible within the system’s required constraints.

#### Climate Tipping Phenomena in Nonautonomous Paleoecosystems

IMSI hosted an Interdisciplinary Research Cluster entitled, “Climate Tipping Phenomena in Nonautonomous Paleoecosystems,” June 20-30, 2023, at the University of Chicago. The IRC focused on developing a framework for describing tipping points and regime shifts in a coupled global climate-biota system taking into account the temporal heterogeneity of the processes involved. The focus was the origin and timing of potential extinction triggers. There is evidence in the fossil record that the global ecosystems’ response to a trigger (i.e., increases in CO2) can be diverse, leading to a considerable biodiversity loss (i.e., a mass extinction) in some cases but having a relatively small effect in others.

#### Managing Hospital Emergency Rooms

The quantification of medical and health care has revolutionized human lives, with strong and long-lasting positive social and economic impact. This revolution stems from synergies among mathematics, statistics, data science, medicine, and machine learning (ML) and artificial intelligence (AI) which has prompted the creation of interdisciplinary areas across the various fields. Despite major progress in many scientific directions, there is a continuing need to develop existing areas and set the foundations for new ones. This need has been amplified by emergent events such as the recent COVID-19 pandemic.

#### The Promise of Computer-Assisted Mathematics

Many mathematicians predict that one of the most important mathematical stories in the 21st century will be the changing role of computers in assisting mathematicians in proving theorems, or even in proving theorems themselves. UCLA professor Terence Tao highlighted some of the history and potential future of machine-assisted mathematics at the 40th anniversary symposium for MSRI/SLMath held in April 2023.

#### Machine Assisted Proofs

(By Jordan Ellenberg) In February of 2023, IPAM hosted a workshop on Machine-Assisted Proof. This program had been in the works for more than a year, but ended up being held at a timely moment, when unexpectedly rapid development in large language models (LLMs) in late 2022 had brought focused national attention to the question of how machines could assist or even replace human ingenuity.

#### Slices of Polytopes

Given a 3-dimensional cube, the intersection with an affine hyperplane is always a polygon with 3,4,5, or 6 vertices. But how can one understand the slices of a general polytope? And which slice is "the best," e.g., is the slice of maximal volume?

#### 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.