This talk presents a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and the market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the  order book evolution into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics. This talk is based on joint work with Rama Cont and Pierre Degond.

I’m excited to share with everyone some new, unpublished work that we are just in the process of wrapping up and could use everyone’s reactions. It is a reconciliation of evolutionary game theory and ecological dynamics that I have wrestled with since I moved from an evolution program into an ecology-heavy department. It always seemed like, depending on the problem I was thinking about, I had to change my perspective and approach it as either an evolutionary game theorist, or an ecologist; and yet I had this nagging feeling that, at its core, the problem was often one and the same, and therefore one theoretical framework should suffice. So when should one write down an n-type replicator equation and when should one write down an n-species Lotka-Volterra system; and what does it mean mathematically and biologically when one has made such a choice? In the process of reconciling, I also got a deeper appreciation of what is and is not a proper game, such as a Prisoner’s Dilemma. These findings can help shed light on previously puzzling empirical findings.

A growing plant is a fascinating system involving multiple fields. Biologically, it is a multi-cellular system controlled by bio-chemical networks. Physically, it is an example of an "active solid" whose element (cells) are active, performing mechanical work to drive the evolving geometry. Computationally, it is a distributed system, processing a multitude of local inputs into a coordinated developmental response. In this talk I will discuss how plants, a living information-processing organism, uses physical laws and biological mechanisms to alter its own shape, and negotiate its environment. Here I will focus on two examples reflecting the computational and mechanical aspects: (i) probing temporal integration in gravitropic responses reveals plants sum and subtract signals, (ii) the interplay between active growth-driven processes and passive mechanics.

We propose a unified theory of brain function called ‘Thermodynamics of Mind’ which provides a natural, parsimonious way to explain the underlying computational mechanisms. The theory uses tools from non-equilibrium thermodynamics to describe the hierarchical dynamics of brain states over time. Crucially, the theory combines correlative (model-free) measures with causal generative models to provide solid causal inference for the underlying brain mechanisms. The model-based framework is a powerful way to use regional neural dynamics within the hierarchical anatomical brain connectivity to understand the underlying mechanisms for shaping the temporal unfolding of whole-brain dynamics in brain states. As such this model-based framework fitted to empirical data can be exhaustively investigated to provide objectively strong causal evidence of the underlying brain mechanisms orchestrating brain states. 

Generating n-tuples of a group G, or in other words epimorphisms Fₙ→G are usually studied up to the natural right action of Aut(Fₙ) on Epi(Fₙ,G); here Fₙ is the free group of n generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any n ≥ k there is exactly one Nielsen class of generating n-tuples of Fₖ. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.

My talk will have two parts. First, I will tell you how a single cell produces light to survive; then, I will explain how a huddle of chloroplasts in cells form glasses to optimize plant life. Part I: Bioluminescence (light generation in living organisms) has mesmerized humans since thousands of years ago. I will first go over the recent progress in experimental and mathematical biophysics of single-cell bioluminescence (PRL 125 (2), 028102, 2020) and then will go beyond and present a lab-scale experiment and a continuum model of bioluminescent breaking waves. Part II: To remain efficient during photosynthesis, plants can re-arrange the internal structure of cells by the active motion of chloroplasts. I will show that the chloroplasts can behave like a densely packed light-sensitive active matter, whose non-gaussian athermal fluctuations can lead to various self-organization scenarios, including glassy dynamics under dim lights (PNAS 120 (3), 2216497120, 2023). To this end, I will also present a simple model that captures the dynamic of these biological glasses.