Sebastian Roch (University of Wisconsin, Madison) (Two talks)
A Biased Talk Through Mathematical Phylogenomics, I & II
Abstract: The reconstruction of the Tree of Life is a classical problem in evolutionary biology that has benefited from numerous branches of mathematics, including probability, information theory, combinatorics, and geometry. Modern DNA sequencing technologies are producing a deluge of new genetic data -- transforming how we view the Tree of Life and how it is reconstructed. I will survey recent progress on some mathematical questions that arise in this context.
Rick Kenyon (Brown University) (Two talks)
Limit Shapes for Height Models, I & II
Abstract: We discuss the notion of "limit shape", compute a few examples including random partitions, random lozenge tilings, and the 5-vertex model. In 2+1 dimensions we describe a nice formula for limit shapes from gradient measures (of which the above are examples), which shows a certain linearity in the underlying limit shape theory.
Paul Bourgade (New York University)
Random Matrices and Logarithmically Correlated Fields Abstract: A connection between branching structures and characteristic polynomials of random matrices emerged in the past few years. We will illustrate this for two models of random matrices, corresponding to dimension 1 and 2 spectra: the Circular Unitary Ensemble and Ginibre random matrices. The discussed topics will include the second moment method, extrema, the Gaussian free field and Gaussian multiplicative chaos.
Firas Rassoul-Agha (University of Utah)
Shifted Weights and Restricted-Length Path in First-Passage Percolation
Abstract: We study standard first-passage percolation via related optimization problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. This is joint work with Arjun Krishnan and Timo Seppalainen.
Tianyi Zheng (University of California, San Diego)
Random Walks on Groups of Intermediate Volume GrowthAbstract: Groups of intermediate growth were first constructed by Grigorchuk in the early