Search Talks

Characterizing metastable states and saddle points in complex systems can shed light on a variety of equilibrium and out-of-equilibrium behaviors. In this talk, I will discuss the most versatile theoretical technique for this, the Kac–Rice method, and its application to a family of mean-field models, the spherical spin glasses. I will draw connections from this technique to random matrix theory and Faddeev–Popov gauge fixing. For the spherical models, writing down an effective action is straightforward but finding physical saddle points can be challenging. I will show how, for the lowest-energy states, a BRST supersymmetry can be used to simplify the action and map it to a more tractable equilibrium problem. Finally, I will discuss challenges to using these techniques in inference problems with non-Gaussian energy functions, and current work being done to address them.

Eukaryotic cells and bacteria can use the substrate to self-regulate their movement. Eukaryotic cells, such as MDCK epithelial cells, leave footprints on the substrate, which are then used to regulate their motility and exploratory behavior. On the other hand, Escherichia coli bacteria stop and adhere to the surface to break confinement when swimming close to the surface and explore their surroundings more efficiently. In this talk, I will explore these mechanisms through mathematical models highlighting how both cells and bacteria, at two different biological scales, use the environment to self-regulate the way they explore their surroundings.

From quenched physical systems to simple optimization algorithms, descent in an energy or cost landscape is thought to underpin many interesting phenomena. Likewise, the geometry of this landscape is thought to explain diverse outcomes of this descent: in those with few minima, the lowest are easily found; while in those with very many, getting stuck among a proliferation of high minima is overwhelmingly likely. In mean-field spin glasses and inference problems, we can hope to use geometry to explain when transitions between these behaviors occur and which kind of minima will typically be found. I will describe the past success of this approach in the simplest spin glass models and its recent failure in slightly less simple ones. I will share several new results on the statistics and arrangement of metastable states and saddle points which together indicate that traditional mean-field geometric tools may not be able to answer everything. Finally, I share some thoughts on what extensions might be needed to precisely predict where descent goes.