# Mathematical Sciences Research Institute

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# MSRI-UP 2015: Geometric Combinatorics Motivated by the Social Sciences

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## Dr. Talea Mayo, Princeton University

### Hurricane Storm Surge Modeling for Risk Analysis

Hurricane storm surge risk analysis is a vital component of urban planning and design strategies. Analyses of risk typically rely on historical data, however, such data is often insufficient. Additionally, it is not representative of future risk, which will be greatly impacted as the climate changes. In this work, we analyze risk for several regions in the Northeastern United States by creating synthetic storm surge data representative of storm surges of various climates. Specifically, we use synthetic hurricanes developed from estimates of both the observed climate and the projected climate and hydrodynamic modeling to compute thousands of synthetic storm surges. With this method, we are able to calculate physically-based assessments of risk for several time periods over the next century. We find that the probability of current 100 year storm surges may increase by a factor of 10 by the end of the 21st century. As our method relies on storm surge modeling, we also explore mathematical methods of reducing uncertainties in hydrodynamic models. This work reveals the critical role of hydro dynamical modeling in storm surge mitigation.

## Dr. Gina-Maria Pomann, Duke University

### Statistical Image Analysis for the Study

A number of magnetic resonance (MR) imaging modalities can be used to measure the diffusion of water in the brain. An important question is which of these modalities are most useful for differentiating between MR images of patients with multiple sclerosis (MS) and those of healthy controls. We propose a hypothesis test that facilitates this differentiation while taking advantage of the functional nature of the data. The methods represent the data using a common orthogonal basis expansion and reduce the dimension of the testing problem in a way that enables the application of traditional nonparametric univariate testing procedures. This results in a procedure that is not only computationally inexpensive but also allows for testing of higher order moments in functional principal component factor loadings. Simulation studies are presented to demonstrate the strength and validity of our approach. We also provide a comparison to a competing method. The proposed methodology is then illustrated by applying it to a state-of-the art diffusion tensor imaging (DTI) study where the objective is to compare white matter tract profiles in healthy individuals and multiple sclerosis (MS) patients.

## Prof. Federico Ardila, San Francisco State University

### Moving robots efficiently using the combinatorics of CAT(0) cube complexes

For many discrete systems, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the space of possible positions of the system is a CAT(0) cube complex. When this is the case, we are able to describe this space completely in terms of a combinatorial object called a poset with inconsistent pairs (PIP). Using this PIP as a remote control, we find an algorithm to compute the optimal way of moving a robot from one position to another. We implement this general technique for several examples of interest.

This talk is based on joint work with Tia Baker, Hanner Bastidas, John Guo, Maxime Pouokam, and Rika Yatchak. It will assume no previous knowledge of CAT(0) cubical complexes.

## Dr. Bobby Wilson, University of Chicago

### Tangents to sigma-finite curves

In this talk, we will discuss the question of whether continuous, simple curves in Euclidean space with sigma-finite length have tangents at any points.  The results on $\sigma$-finite curves that we will discuss were initiated by the observation that the graph of a continuous function on [0,1] that satisfies a weak-Lipschitz property has sigma-finite one-dimensional Hausdorff measure. We will discuss our conclusion that every $\sigma$-finite curve has a tangent, in the pointwise sense, on a set of positive measure. This is joint work with M. Csornyei.

## Dr.  Khalilah Beal , University of California, Berkeley

### Viscosity Solution Methods and the Problem of Ruin

In this talk, we will investigate the problem of "collective ruin" in classical risk theory, which models the risk of an insurance business.  We model the probability that a reserve remains nonnegative with a scaled integrodifferential equation (IDE).  Introducing a gauge parameter allows consideration of solutions for large initial reserves and motivates our discussion of perturbation.  Using generalized solutions, as opposed to classical solutions, allows a broad class of collective risk problems to be studied.  We begin with a discussion of viscosity solutions.

The results of this talk extend work of Knessl, Mangel, Peters, and others who treat cases of the main problem with constant premiums and/or smooth distributions of claims.