Coleman and Vologodsky integration theories give canonical parallel transports for unipotent differential equations – Coleman on overconvergent spaces with good reduction and Vologodsky on algebraic varieties, both over p-adic fields. In Coleman’s theory the transport is via a path invariant under the action of Frobenius while in Vologodsky’s theory one adds a condition involving the monodromy operator. While both theories can be formulated in fairly similar terms, the precise relationship between them is a bit unclear.

In this talk, based on joint work in progress with Sarah Zerbes, I will describe some background on the two integration theories and I will describe the simplest non-trivial case – a holomorphic form on a curve with semi-stable reduction, where we can say what the relation should be. Time permitting I’ll discuss possible applications to syntomic regulators.

I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $ell$-adic sheaf on a smooth relative curve over a strictly henselian trait of characteristic $p ne ell$. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito’s refined Swan conductor with Kato’s Swan conductor with differential values, which is the key ingredient in Kato’s formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Delinge-Kato’s formula.

Thomas’ conjecture about the necessity of a positive circuit for the multistationarity has been proved both in the discrete and continuous setting. Nevertheless it has mostly been used in the discrete case since, as we will demonstrate, it is almost always satisfied by biochemical reaction networks. In order to work around this shortcoming, we will go back to the decomposition of the Jacobian matrix determinant and notice that, for dynamical systems corresponding to biochemical reactions, certain patterns allow the statement of a stronger necessary condition. We will illustrate how that new condition rules out the cases that trivially satisfied the classical condition of Thomas on a simple example of enzymatic reaction.

Morphogens are secreted molecules that influence the development of cells in embryonic tissue. Patterns of gene expression or signalling immediately downstream of many morphogens such as the bone morphogenetic protein (BMP) and decapentaplegic (Dpp) are highly reproducible and robust to perturbations. This is perhaps surprising in view of experimentally determined low concentration (approximately picomolar) range of Dpp, tight receptor binding and very slow kinetic rates, all of which should favor large fluctuations in signalling.

To shed light on this issue, we developed a stochastic model of Dpp signalling in Drosophila melanogaster and used the model to quantify the extent to which stochastic fluctuations would lead to errors in spatial patterning. This was done both with and without a surface-associated BMP-binding protein (SBP), which is an auxiliary protein that plays no direct role in signalling. By analyzing this model over a wide range of parameter values, we find that such SBPs are likely to play an important role in the reliability of morphogen patterning.

Research in quantitative evolutionary genomics and systems biology led to the discovery of several universal regularities connecting genomic and molecular phenomic variables. These universals include the log-normal distribution of the evolutionary rates of orthologous genes; the power law-like distributions of paralogous family size and node degree in various biological networks; the negative correlation between a gene’s sequence evolution rate and expression level; and differential scaling of functional classes of genes with genome size. The universals of genome evolution can be accounted for by simple mathematical models similar to those used in statistical physics, such as the birth-death-innovation model. These ‘laws’ of evolutionary genomics, analogously to the laws of statistical physics, appear to emerge from the Maximum Entropy principle which dictates that the probability distribution of any variable in a large ensemble of data or measurements tends to the distribution with the maximum entropy within the applicable constraints. In the case of genome evolution, these constraints are readily interpretable as effects of purifying selection against gene malfunction.