The 40th Midwest Probability Colloquium

Conference Abstracts

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 FieldsA 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.Abstract:

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 Growth

1980s. Such groups have extraordinary algebraic and geometric properties, providing examples

and counterexamples to many questions in group theory. The main focus of the talk will be

on near optimal volume growth lower estimates of Grigorchuk groups coming from random walks

with nontrivial Poisson boundary on these groups. We will also discuss some results on joint

behavior of volume growth and random walk entropy/rate of escape.

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