# SIAM at the University of Utah

# The First SIAM Wasatch Student Chapters Conference

## Date: Saturday, April 21st, 2018

Location: LeRoy E. Cowles Building (LCB), University of Utah

Hosted by the SIAM student chapter at the University of Utah, the aim of this conference is to cultivate connections amongst early career applied math students and researchers and faculty members from Wasatch universities. The conference will consist of a series of informal presentations. Graduate students are especially encouraged to apply and present an introduction to their research interests to spur discussions between unacquainted attendees.

## Registration and Funding

- If you would like to attend and/or speak at the conference, please fill out the
registration form before March 14th. Funding for travel can be provided. If you have any questions, please contact Nathan Willis at
willis@math.utah.edu.

## Program

- Eigenvalues are fundamental in the study of linear partial differential operators and have important physical interpretations in a variety of settings. For example, they describe the frequency of wave propagation in electromagnetism
and fluid flow, vibrational frequencies of mechanical structures, and energy levels in quantum mechanics. Recently,
eigenvalues have also found interesting applications in data analysis. In this talk, I will discuss the dependence of eigenvalues on either

(i) a coefficient that appears in the partial differential operator, or

(ii) the shape of the domain on which the operator is defined.

In particular, we will consider problems where it is of interest to maximize or minimize an eigenvalue with respect to one of these two dependencies. I will assume some familiarity with partial differential equations and spectral theory, but this talk should be suitable for a broad mathematical audience.

- The study of animal movement is undergoing a revolution due to the boom in availability of individual telemetry data (GPS tracking) and the increase in resolution of remotely-sensed environmental data (landscape classifcation from satellite imagery). This data is a time series of correlated locations embedded in landscape patches of known type and varying effect on animal movement. A long history of mathematical research describes probabilistic consequences of animal movement using partial
differential equations. Solutions, with appropriate initial data, are probability density functions (PDFs) of future locations in a time series of individual telemetry data. Such a PDF can be used in a maximum likelihood estimation procedure to estimate animal movement parameters. The diffusion equation is commonly used but does not allow for variable landscape resistance to movement, spatial aggregation of populations in favorable habitats, or correlation in individual movement. All three are
significant for animals like mule deer. The ecological telegrapher's equation (ETE) naturally aggregates populations in preferred habitats and accommodates both correlated movement and variable landscape resistance. We use the ETE to derive a PDF describing individual movement. Using asymptotic techniques and homogenization over short scale variation we find a closed form solution to parameterize a landscape-level population model for mule deer in Southern Utah.

- Astrocytes are glial cells in the brain that each wrap aroundthousands of synapses. Neurotransmitters released by neurons can activate receptors on astrocytes, leading to the release of IP3, and initiating calcium transients in astrocytes. It is believed that these transients allow astrocytes to communicate with nearby neurons, but the mechanisms are still being investigated. One proposed pathway is through the sodium-calcium exchanger activated by the increase in astrocyte cytosolic
calcium. The exchanger activity affects the sodium-potassium pump, enabling astrocytes to regulate extracellular ion concentrations, and thus the neuronal excitability. We study the viability of such a communication pathway.

Using experimental data collected by our collaborators, we first study the calcium responses evoked by short puffs of ATP. We develop an open-cell, single compartment minimal ODE model that captures the experimentally observed diversity of calcium transients. By varying the strength of calcium channels, we manipulate the underlying bifurcation structure of the ODE system, thus examining the specific roles of individual calcium fluxes and making testable predictions. Building off of this understanding, we then introduce sodium and potassium fluxes. Altering the strengths of these new fluxes, we explore the impacts calcium transients in astrocytes can have on extracellular ionic concentrations and the firing patterns of neighboring neurons in healthy and pathological states.

- Predicting potential relationships between nodes in a network is commonly known as the link prediction problem. Many approaches to solving this problem have been proposed, each having varying levels of accuracy depending on the network. We introduce a new method for link prediction called Effective Transition. This method is based on the idea of the transition matrix and uses isoradial reductions to compute scores between nodes. We seek to apply this method to large social, technological, and natural networks and determine in which cases its accuracy is most competitive.

- An important component of population ecology is understanding the impact on animal movement. Modeling animal movement helps us understand how this impact manifests itself in animal behavior and habitat selection as well as human impact on wildlife. Correlated movement and heterogeneous landscapes add complexity to these models and are often neglected. How do landscape features condition population movement and habitat choice? I answer this question by showing that motility does
play a significant role in population dynamics. The ecological telegrapherâ€™s equation (ETE) incorporates both variable landscape and correlated movement. The solution to the ETE predicts the PDF of future locations. I use this PDF in a maximum
likelihood process to parameterize the ETE with simulated data. In this work, I develop code to generate test data by simulating trajectories with a correlated random walk on a simulated terrain. I determine the accuracy of the MLE procedure by
parameterizing the ETE with the simulated data and comparing with the assigned values in the simulated landscape. Applying the same MLE procedure to actual telemetry data from mule deer in Southern Utah will yield the impact of different
landscape types on mule deer.

- Traditionally, evolution, or changes in gene frequencies, is perceived as occurring very slowly compared to ecological rates of change (e.g., changes in population abundances). Recent empirical studies suggest that ecology and evolution may change at similar rates. However, there is debate about how fast evolution can be relative to ecology. We present a method based on the theory of fast-slow dynamical systems to estimate the relative rates of ecological and evolutionary change from abundance and phenotypic time series data. When applied to a suite of empirical data sets, we find in many cases that ecology and evolution have comparable time scales. These results show that the traditional assumption of slow evolution does not always hold in empirical systems. In addition, this reinforces the idea that a new theory addressing the concurrent
interactions between ecological and evolutionary processes (i.e., eco-evolutionary dynamics) is needed.

- We study a model of cargo transport by non-processive molecular motors. In our model, the motors change states by random discrete events (corresponding to stepping and binding/unbinding), while the cargo position follows a stochastic differential equation (SDE) that depends on the discrete states. The resulting system for the cargo position is consequently an SDE that randomly switches according to a Markov jump process. To study this system we

(1) cast it in a renewal theory framework and generalize the renewal reward theorem and

(2) decompose the continuous and discrete sources of stochasticity and exploit a resulting pair of disparate timescales.

With these mathematical tools, we obtain explicit formulas for experimentally measurable quantities, such as cargo velocity and run length. Analyzing these formulas then yields some predictions regarding so-called non-processive clustering. We will also discuss the possibility of generalizing these ideas to studying higher order moments of switching SDEs. This is joint work with Sean Lawley and Jim Keener.

- The sloshing problem of an incompressible, inviscid, irrotational fluid is considered in a half-space covered by an ice sheet with a fishing hole, including effects due to surface tension on the free surface. The problem is restricted to the case of constant contact angle and time-harmonic solutions of the linearised problem are sought. This reduces the sloshing problem to a mixed Steklov-Neumann eigenvalue problem, where the eigenvalue (sloshing frequency) appears in the free surface boundary conditions. We reformulate the problem in terms of an integral eigenvalue problem (Fredholm integro-differential equation of the second kind) and show that the spectrum is discrete. The integral eigenvalue problem is solved numerically using Boundary Element Method (BEM) and we recover existing numerical results due to Henrici et.al (1970) in the limit of zero surface tension. As opposed to the zero surface tension case, where the ``high spots", the maximal elevation of the free surface of the sloshing fluid, is attained at the interior of the free surface, we observe numerically that the location of the high spot moves to the boundary of the free surface beyond a critical surface tension threshold. This is joint work with Nathan Willis, Christel Hohenegger and Braxton Osting.

## Organizer and Support

- This conference is organized by the University of Utah SIAM Student Chapter. Support for this event is provided by SIAM.