Event type: Seminar

In the first part of the talk, we discuss the multi-species competition in a spatial domain, particularly the result of A. Hastings and some recent progress on the conjecture by Dockery et al. concerning the evolution of slow dispersal, i.e. when other things are equal, the slowest diffuser can competitively exclude other competitors. We then discuss the effect of adding passive drift and how it changes the evolutionary dynamics so that fast/intermediate diffuser is selected. In the second part, we will discuss the effect of mutation, and the associated moving Dirac solutions in a nonlocal PDE model proposed by Perthame and Souganidis. This latter equation describes a population structured by space and a phenotypic trait, and can be understood as competition of infinitely many species with different rate of dispersal.

The observed gait of a swimmer arises from the interplay of internal force generation, the passive elastic properties of its body, and environmental features such as fluid viscosity, boundaries, and obstacles. One could question whether optimal swimming of a fish occurs when it is actuated at a frequency near a natural frequency determined by its material bending rigidity. We will share some insights that we have gained using computational models of a few systems that range from the Stokes regime to others where inertial forces are important.

Boolean networks are a common modeling tool for studying the phenotypic changes that cells undergo in response stimuli. Examples include cell differentiation during embryogenesis, metastasis of cancer cells, and apoptosis. In these models, genes and proteins are represented by nodes in a network, and each node is assigned a Boolean activity variable that evolves in discrete time-steps such that the attractors of the resulting dynamics correspond to phenotypes of interest. In this talk, I will focus on the techniques we use to analyze these discrete dynamical systems. These techniques rely on the construction and iterative reduction of an auxiliary network that encodes dynamical information as graph structure (an example is diagrammed in the accompanying figure). Our goals include finding all attractors, identifying key feedback loops that govern attractor selection, and driving the system to a desired attractor from an arbitrary initial state. I will briefly cover several recent applications of these methods to empirical and statistical models. I will also discuss ways in which these discrete models — and the techniques we use to analyze them — are related to their ODE counterparts.

I am a cancer researcher working on a framework for a new mathematical model of metastasis. The reason for making a new model is that, in mice, metastases can grow from cells that are still normal at the time of arriving to the future metastasis site. If the same happens in humans, it would be pretty important for cancer therapy. First, it would explain why anti-cancer therapies fail to prevent metastases in some patients: if the cells are yet non-malignant at the time of therapy, they would be spared by the treatment. Second, it may be possible to identify improved treatments based on the ability to kill these non-malignant cells. I hypothesize that dissemination of non-malignant epithelial cells occurs in parallel with tumor cells, and subsequent transformation at the ectopic sites is a source of some metastases. This scenario ties together two contradictory observations: that metastases are associated with large and rapidly growing primary tumors, while metastatic tumors themselves often take a long time to appear. During my recent sabbatical at the Institut des Hautes Etudes Scientifiques, together with Misha Gromov and his colleagues I started to interrogate publicly available data on mutation rates and/or profiles from cancer patients and healthy individuals to determine whether the earliest common ancestor predicted using phylogenetic methods in some primary-metastasis pairs has features of a non-malignant cell.

Predicting a previously unseen example from training examples is unsolvable without additional assumptions about the nature of the task at hand. A learner’s performance depends crucially on how its internal assumptions, or inductive biases, align with the task. I will present a theory that describes the inductive biases of neural networks using kernel methods and statistical mechanics. This theory elucidates an inductive bias to explain data with “simple functions, which are identified by solving a related kernel eigenfunction problem on the data distribution. This notion of simplicity allows us to characterize whether a network is compatible with a learning task, facilitating good generalization performance from a small number of training examples. I will present applications of this theory to artificial and biological neural systems, and real datasets.