Georgia Institute of Technology College of Sciences

Morphogenesis is the biological process that causes cells, tissues, or organisms to develop their shape. The theory of morphogenesis, proposed by Alan Turning, is a chemical model where biological cells differentiate and form patterns through intercellular reaction-diffusion mechanisms. Various reaction-diffusion models can produce a chemical pattern that mimics natural patterns. However, while they provide a plausible prepattern, they do not describe a mechanism in which the pattern is expressed. An alternative model is a mechanical model of the skin, initially described by Murray, Oster, and Harris. This model used mechanical interactions between cells without a chemical prepattern to produce structures like those observed in a Turing model. In this talk, we derive a modified version of the Murray, Oster, and Harris model incorporating nonlinear deformation effects. Since it is observed in some experiments that chemicals present in developing skin can cause or disrupt pattern formation, the mechanical model is coupled with a single diffusing chemical. Furthermore, it is observed that the interaction between tissue deformations with a diffusing chemical can cause a previously undescribed instability. This instability could describe both the pattern’s chemical patterning and mechanical expression without the need for a reaction-diffusion system.

We show how the theory of Lorentzian polynomials extends to cones other than the positive orthant, and how this may be used to prove Hodge-Riemann relations of degree one for Chow rings. If time permits, we will show explicitly how the theory applies to volume polynomials of matroids and/or polytopes. Joint work with Petter Brändén.

We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least $4\pi$, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant, and is based on Gaussian correlation inequality. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.

We present a polynomial-time algorithm for robustly learning an unknown affine transformation of the standard hypercube from samples, an important and well-studied setting for independent component analysis (ICA). Total variation distance is the information-theoretically strongest possible notion of distance in our setting and our recovery guarantees in this distance are optimal up to the absolute constant factor multiplying the fraction of corruption. Our key innovation is a new approach to ICA (even to outlier-free ICA) that circumvents the difficulties in the classical method of moments and instead relies on a new geometric certificate of correctness of an affine transformation. Our algorithm is based on a new method that iteratively improves an estimate of the unknown affine transformation whenever the requirements of the certificate are not met.