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.

The « very good » property for algebraic stacks was introduced by Beilinson and Drinfeld in their paper « The Quantization of Hitchin’s Integrable System and Hecke Eigensheaves ». They proved that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is « very good » as long as X has genus g > 1. We will introduce the « very good » property in the context of a group action on an algebraic variety, and prove it for a moduli space of parabolic bundles on P1 arising from quiver representations. As a special case, we will consider the « very good » property for the diagonal action of the group PGL(n) on a product of partial flag varieties and its relationship with the space of solutions to the additive Deligne-Simpson problem.

The principal approximation methods used to compute the inspiral of compact binary systems are the post-Newtonian (pN) expansion, in which an orbital angular velocity MΩ serves as the expansion parameter; and the self-force or extreme-mass-ratio-inspiral approach, in which the small parameter is the mass ratio m/M of the binary’s two components. We work in an overlapping regime where both approximations are valid and find numerical values of pN coefficients at orders beyond the reach of current analytical work. In this talk we present a novel analytic extraction of high-order pN parameters that govern quasi-circular binary systems using ultra-high accuracy numerical computations.

The problem reads: Is there a general principle(s) that underlies the processes of human learning the following?

1. Native languages.

2. Mathematics.

3. Bipedal locomotion.

We shall bring (speculative) evidence for the existence of such principle(s) and indicate a possible direction of (quasi)mathematical modeling of (conjectural) universal learning.

Starting from the known unfaithful spinorial representations of the compact subalgebra KE(10) of the hyperbolic Kac Moody algebra E(10), new fermionic `higher spin’ representations are constructed in a second quantized framework.