Integrated Calendar

Color each vertex of an infinite graph blue with probability p and red with probability 1-p, independently among vertices. For which values of p is there an infinite connected component of blue vertices? The talk will focus on this classical percolation problem for the class of planar graphs. Recently, Itai Benjamini made several conjectures in this context, relating the percolation problem to the behavior of simple random walk on the graph. We will explain how partial answers to Benjamini's conjectures may be obtained using the theory of circle packings. Among the results is the fact that the critical percolation probability admits a universal lower bound for the class of recurrent plane triangulations. No previous knowledge on percolation or circle packings will be assumed.

The mixing time and spectral gap of a random walk on the symmetric group can sometimes be understood in terms of its low dimensional representations (e.g., Aldous' spectral gap conjecture). It turns out that under a mild degree condition involving the step of the group, the same holds for nilpotent groups w.r.t. their one dimensional representations: the spectral gap and the epsilon total variation mixing time of the walk on G are determined by those of the projection of the walk to the abelianization G/[G,G]. We'll discuss some applications concerning the cutoff phenomenon (= abrupt convergence to equilibrium) and the dependence (or lack of!) of the spectral gap and the mixing time on the choice of generators.  

As time permits we shall discuss a related result, confirming in the nilpotent setup a conjecture of Aldous and Diaconis concerning the occurrence of cutoff when a diverging number of generators are picked uniformly at random. Joint work with Zoe Huang.

A defining challenge in synthetic biology is the development of high-throughput and automated methodologies for precise design and engineering of cells. To address these challenges, we develop multiplex genome engineering technologies for versatile genome modification and evolution of bacterial and eukaryotic cells. We use these technologies to create genetic variants to reveal a causal understanding of complex phenotypes as well as engineer pathways and recode genomes. These Genomically Recoded Organisms (GROs) contain alternative genetic codes, in which codons have been eliminated from the genome of E. coli. GROs exhibit improved properties for incorporation of nonstandard amino acids that expand the chemical diversity of proteins or polymers, establish genetic isolation and multi-virus resistance, and enable the engineering of GROs to depend on synthetic amino acids for robust biocontainment strategies. We have also developed new computational-experimental technologies – computer aided design of synthetic genetic elements (CAD-SGE) – that permits the redesign, expression, and mobilization of biosynthetic pathways in diverse organisms for the discovery of new metabolites. This work increases the toolbox for genomic and cellular engineering with broad applications for new classes of enzymes, materials, and therapeutics.