Göran Gustafsson Symposium in Mathematics, June 12-14, 2019

This is work with Jacques Rougemont (Geneva) and Tsvi Tlusty (Ulsan). The general context is an attempt to bring the language of mathematics closer to that of biology, where there exist lots of data but few conceptual methods.

Considering the evolution of functional proteins (proteins which do something useful), we view them as amorphous solids. It turns out that describing the protein as a (finite!) random network of connections, which are changed through evolution, a Green's function approach can explain many properties of proteins which biologists have observed. The talk needs no previous knowledge of biological concepts.

I will begin with a mathematical model of a group of excitatory and inhibitory neurons similar to those in local circuits in many parts of the cerebral cortex. An example of an emergent phenomenon in this complex dynamical system is an irregular rhythm in the gamma band, a rhythm detected all over the real brain. I will propose a mechanistic explanation and discuss implications. In the second part of the talk, I will present snapshots of ongoing work on a (realistic) computational model of the visual cortex. Dynamical responses to visual stimuli vary across the cortical surface. Viewing it as local circuits ``stitched together" one sees a rough analogy between steady-state neuronal responses and local thermal equilibria ideas in nonequilibrium statistical mechanics.

Ant colonies are one of nature’s pinnacles of group living and cooperative behavior. As such they provide a good model system from which one can try to generalize towards common principles of biological cooperation: the possibilities it provides and its limitations. This is, however, a difficult task. In this talk, I will present several examples which relate ant behaviors to theoretical concepts of different mathematical flavors.

The first talk will give the general context of trapping phenomena for dynamics in random environments, and introduce sub-diffusivity in the presence of disorder. We will start by the simpler universality class of slow dynamics, i.e. the Fractional Kinetics or equivalently the Bouchaud Trap model. This class contrains highly non-trivial models, as Metropolis dynamics for Spin Glasses, or biased random walks on super-critical percolation clusters. But this class is too simple to describe the ant in the labyrinth. We will thus introduce the more general class of Randomly Trapped Random Walks and their scaling limits. We will see how these new limit theorems can be applied to the case of random walks on critical random trees.

The second talk will present a general theorem giving the scaling limit for random walks on incipient critical structures. This scaling limit is a very complex object, the “BISE": the Brownian Motion on the Integrated Super-Brownian excursion. This talk will introduce the needed general limiting mathematical objects, and our general abstract convergence theorem. This theorem is valid under a set of 4 conditions which are believed to be true for all critical structures in high enough dimensions.

The third talk will show how the general approach does indeed include some interesting models, using and sharpening he classical but heavy tool of lace expansions. We will discuss the case of random walks on the trace of critical branching clusters, of lattice trees, and discuss the potential application of our approach to the usual critical percolation clusters in high dimension.

We will discuss several toy models for interface fluctuations and their large-scale behaviour, with an emphasis on trying to understand the "big picture". On the way, we find a very natural probabilistic object which can be interpreted as a renormalisation fixed point with infinitely many unstable directions.

I will consider the random operator obtained by perturbing the Laplacian by a white noise on a finite box. In dimension 1, this operator is related to models of random matrices. In higher dimension, the operator is ill-defined and requires renormalisation. I will present some results on the well-posedness of the operator, and on the behaviour of the spectrum when the size of the underlying box goes to infinity.

In this seminar I will describe how to derive rigorously the laws that rule the space-time evolution of the conserved quantities of a certain stochastic process. The goal is to describe the connection between the macroscopic equations and the microscopic system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly. Depending on the choice of the transition probability that particles obey, we will see that the macroscopic laws can be of different nature.  I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian.

The one dimensional KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations (e.g. the eponymous Kardar-Parisi-Zhang equation). It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data. The explanation is that on large scales everything should approach a special scaling invariant Markov process, the KPZ fixed point. It is obtained by solving one model in the class, TASEP, and passing to the limit. Both TASEP and the KPZ fixed point turn out to be completely integrable, and there are unexpected connections to dispersive PDE.   (Joint work with Konstantin Matetski and Daniel Remenik).