# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Walking within growing domains: recurrence versus transience

When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary. This talk is based on joint works with Ruojun Huang, Vladas Sidoravicius and Tianyi Zheng.

##### TBA - Amir Ali Ahmadi

##### Trace reconstruction for the deletion channel

In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability $q$, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability?

##### TBA - Veit Elser

##### Methods of network comparison

The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path provides a notion of scale, which is vitally important in characterizing dominant modes of system behavior. Here, by combining topology with scale, we prove the existence of universal features which reveal the dominant scales of any network. We use these features to compare several canonical network types in the context of a social media discussion which evolves through the sharing of rumors, leaks and other news.

##### Mean estimation: median-of-means tournaments

One of the most basic problems in statistics is how to estimate the expected value of a distribution, based on a sample of independent random draws. When the goal is to minimize the length of a confidence interval, the usual empirical mean has a sub-optimal performance, especially for heavy-tailed distributions. In this talk we discuss some estimators that achieve a sub-Gaussian performance under general conditions. The multivariate scenario turns out to be more challenging. We present an estimator with near-optimal performance. We also discuss how these ideas extend to regression function estimation. The talk is based on joint work with Shahar Mendelson (Technion, Israel), Luc Devroye (Mcgill University, Canada), Matthieu Lerasle (CNRS, France) and Roberto Imbuzeiro Oliveira (IMPA, Brazil).

##### Symmetry methods for quantum variational principles and expectation value dynamics

Inspired by previous works by Kramer & Saraceno and Shi & Rabitz, this talk exploits symmetry methods for the variational formulation of different problems in physics and chemistry. First, I will use symmetry methods to provide new variational principles for the description of mixed quantum states, in various pictures including Schrödinger, Heisenberg, Dirac (interaction) and Wigner-Moyal. Then, after discussing Ehrenfest's mean-field model, I will modify its symmetry properties to provide a new variational principle for expectation value dynamics in general situations. Upon moving to the Hamiltonian approach, this construction provides a complete dynamical splitting between expectation values and quantum deviations.

##### The mathematics of charged liquid drops

In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. Mathematically, these problems all share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor drop fragmentation.

##### Two problems involving breakup of a liquid film

Understanding the breakup of a liquid film is complicated by the fact that there is no obvious instability driving breakup: surface tension favors a film of uniform thickness over a deformed one. Here, we identify two mechanisms driving a film toward (infinite time) pinch-off. In the first problem, we show how the rise of a bubble is arrested in a narrow tube, on account of the lubricating film pinching off. In the second problem, breakup of a free liquid film is driven by a strong temperature gradient across the pinch region.