## Mathematics Department

Undergraduate Research Symposium

Spring 2017

## Monday, May 1. 9:15-11:30 am in LCB 323

**9:15-9:30 Sarah Melancon**

Mentor: Adam Boocher

Intro to Research: Commutative Algebra in the Polynomial Ring

**9:30 -9:45 Dylan Soller**

Mentor: Adam Boocher

Intro to Research: commutative algebra

**9:45- 10:00 Max Carlson**

Mentor: Christel Hohenegger, Braxton Osting

Improving the Numerical Method for Approximating Solutions to the Free Surface Sloshing Model

**10:00-10:15 Weston Barton**

Mentor: Tom Alberts

A Direct Bijective Proof of the Hook Length Formula

**10:15-10:30 Michael (Barrett) Williams**

Mentor: Elena Cherkaev

Ice Diffusivity

**10:30 -10:45 Willem Collier**

Mentor: Arjun Krishnan

The RSK Algorithm and the Marcenko-Pastur Law

**10:45-11:00 Peter Harpending**

Mentor: Elena Cherkaev

The Geometry of Fractional Calculus

**11:00- 11:15 Rebecca Hardenbrook**

Mentor: Ken Golden

Bounds on the thermal conductivity of sea ice in the presence of fluid convection

## Abstracts

**Weston Barton**

Mentor: Tom Alberts

A Direct Bijective Proof of the Hook Length Formula

Mentor: Tom Alberts

A Direct Bijective Proof of the Hook Length Formula

The hook length formula, discovered in 1954 by J. S. Frame, G. de B. Robinson, and R. M. Thrall, gives the number of standard Young tableaux of a given shape. The original proof was rather complicated, as have many of the proofs developed since. In 1997 Novelli, Pak, and Stoyanovskii in developed a direct bijection between the set of Standard Young Tableaux and the set of ordered pairs consisting of one non-standard Young tableaux and a set of new objects—Hook functions. This bijection may provide insight into counting subclasses of standard Young tableau, such as those with a particular Schützenberger Path.

**Max Carlson**

Mentor: Christel Hohenegger and Braxton Osting

Improving the Numerical Method for Approximating Solutions to the Free Surface Sloshing Model

Mentor: Christel Hohenegger and Braxton Osting

Improving the Numerical Method for Approximating Solutions to the Free Surface Sloshing Model

By further improving the numerical free surface sloshing model to include non-axisymmetric containers and fully 3D computations, the limitations of the current method have become clear. This semester, I explore parallel programming algorithms to improve the performance and accuracy of the numerical model I've developed previously. In addition, with the type of containers that can be analyzed being expanded, I will discuss parametric container shapes and how they can be used to further analyze the model.

**Willem Collier**

Mentor: Arjun Krishnan

The RSK Algorithm and the Marcenko-Pastur Law

The Robinson-Schensted-Knuth (RSK) algorithm is a bijection between matrices with non- negative entries and pairs of Young Tableaux. The lengths of the rows of the Young diagram obtained through the RSK algorithm serve as eigenvalues of a certain growth process on the plane called last-passage percolation. When the matrix consists of geometrically distributed random variables, it has been shown that the empirical law of the lengths of the rows of the Young diagram converges to the Marchenko-Pastur distribution. We explore this convergence numerically, and see if it generalizes to random variables that are not geometrically distributed.

Mentor: Arjun Krishnan

The RSK Algorithm and the Marcenko-Pastur Law

**Rebecca Hardenbrook**

Mentor: Ken Golden

Bounds on the thermal conductivity of sea ice in the presence of fluid convection

Mentor: Ken Golden

Bounds on the thermal conductivity of sea ice in the presence of fluid convection

Sea ice forms the thin boundary layer between the ocean and atmosphere in the polar regions of Earth. As such, ocean-atmosphere heat exchange is largely controlled by the thermal conductivity of sea ice. However, frozen seawater is a complex composite material consisting of an ice host containing brine and air inclusions. Moreover, the volume fraction, geometry and connectivity of the brine inclusions changes significantly with temperature, and if the temperature of the sea ice exceeds a critical temperature, fluid can flow through the ice which can enhance thermal transport. Calculating the thermal conductivity of sea ice is thus a very challenging problem requiring sophisticated mathematics and computation, and little is known from either an experimental or theoretical perspective about this key parameter. The underlying equation describing the physics of this system is known as the advection-diffusion equation. Here we exploit Stieltjes integral representations for the effective or homogenized thermal conductivity of sea ice in the presence of a fluid flow field to obtain rigorous bounds on this key geophysical parameter. In particular, we assume a periodic convective flow field and analytically calculate the moments of the spectral measure for the system, which is at the heart of the integral representation. The moments are then used to obtain the first bounds on the thermal conductivity of sea ice which incorporate fluid velocity effects.

**Peter Harpending**

Mentor: Elena Cherkaev

The Geometry of Fractional Calculus

Mentor: Elena Cherkaev

The Geometry of Fractional Calculus

Fractional calculus is the study of differentiation and integration to non-integer orders. We explore geometrical interpretations of frac- tional calculus operations, and compare them to their integer-order counderparts. This analysis leads to a rather beautiful result regard- ing paths in the spectrum of the fractional derivative operator.

**Sarah Melancon**

Mentor: Adam Boocher

Commutative Algebra in the Polynomial Ring

Mentor: Adam Boocher

Commutative Algebra in the Polynomial Ring

Given a surface described by parametric equations, we may want to determine a set of polynomials which vanish on the surface. We will learn how to find these polynomials using ideas from commutative algebra such as rings, ideals, and varieties.

**Barrett Williams**

Mentor: Elena Cherkaev

Ice Diffusivity

Mentor: Elena Cherkaev

Ice Diffusivity

Marginal ice zones (MIZ) are areas of ice that are in the transition from an ice shelf to the open ocean. MIZ can be comprised of large and small ice floes, which can be found in both the Arctic and Antarctic Oceans. We will be starting with the diffusion equation and will build a model that will represent the ice concentration data that we have, using our model we will be able to find diffusivity. This allows us to test the accuracy of our method by comparing the results from the simulated ice concentration data and the actual data collected. The results of this general problem will show us how the ice diffuses on a spatial basis, allowing us to see how the MIZ move spatially based on the current concentrations.