WXML

Autumn 2017

Graduate and undergraduate WXML applications for Autumn 2017 are now open! The application deadline is September 17, 2017.

WXML teams will be able to use Mary Gates Hall 295 for meetings, on TuTh from 5-7pm.

WXML general meetings will be from 5-6pm on Thursdays: 9/28 (opening meeting), 11/2 (mid-quarter meeting), and 11/30 (final meeting), all in MGH 295. The WXML Autumn 2017 Open House will be on 12/7, from 5-7pm, location TBA.

New Projects:

Simulation of Randomized Brownian Bridges

Project Mentor: Tim Leung (AMath)

The Brownian bridge is an interesting stochastic process with many applications. This project concerns the simulation of Brownian bridges with random, rather than constant, end points. Specifically, for any randomized Brownian bridge (rBb) with a given distribution of the random end point, we seek to simulate its sample paths and examine its path behaviors. We will first develop simulation algorithms for rBbs in one dimension, and then turn to rBbs in higher dimensions in potentially incorporate correlations among bridges. Moreover, an application to algorithmic trading will be discussed. This is an advanced project, where some knowledge of stochastic processes and R or MATLAB will be ideal.

Simulating randomly mixing fluids

Project Mentor: Soumik Pal (Math)

Mixing viscous fluids in 2 or 3 dimensions is big business (literally). It is used in diverse industries such as polymer processing, pharmaceutical industry, and food processing. Most such mixing procedures are of deterministic designs which have patchy success. On the other hand, random mixing of fluids is occurring in nature all around us, whether it is mixing pollutant in the air or garbage in the ocean or when one mixes cake batter. Very little is known on how long such random mixing procedures take to obtain a sufficient degree of homogeneity. We will study some common random mixing procedures by simulations, particularly focussing on the effect of the boundary of the container, optimal mixing procedures, and the mysterious ‘cut-off’ phenomenon where there is a sudden transition from almost no-mixing to almost complete mixing in very little time. This is an intermediate project, where some knowledge of PDE would be helpful, but not required, and some knowledge of programming (Mathematica/Matlab/Python) will be very useful.

Counting k-tuples in discrete sets

Project Mentor: Jayadev Athreya (Math)

Graduate Mentor: Sam Fairchild

We will count pairs, triples, and k-tuples of integer vectors in the plane, 3-space, and higher dimensions, using both number theory and programming. We’ll also explore other interesting sets arising from dynamical systems and billiards. This project will involve number theory, geometry, and probability, and will be a great introduction to those subjects. This is an elementary project, where some knowledge of basic programming (Python) will be useful.

Rotation Random Walks

Project Mentor: Jayadev Athreya and Chris Hoffman (Math)

Graduate Mentor: Anthony Sanchez

We’ll study a simple random walk on the circle given by rotating clockwise or counterclockwise by a fixed angle, based on flipping a fair coin. We’ll try and understand the fine distribution of this walk, using both numerical experiments and theory. This is an intermediate project, where some knowledge of probability theory and some basic programming (Python) will be useful.

Uniformity of solutions of Diophantine Equations

Project Mentor: Amos Turchet

Graduate Mentor: Travis Scholl

Team Members: Blanca Viña Patiño, Daria Micovic, Rohan Hiatt

Given an equation in two variables with integer coefficients, how many solutions can you find where both values of the variables are rational numbers? This apparently easy question turned out to be very deep and very interesting from the point of view of Number Theory and Algebraic Geometry. In this project we will investigate the behavior of such solutions set for special equations representing so called, higher genus curves. We will gather some statistic on the size of the solutions set trying to address an important open problem on the uniformity of these sizes. This is an intermediate project, completion of Math 308 and some familiarity with Sage/CoCalc/Python will be useful.

Continuing Projects:

Number Theory and Noise

Project Mentor: Matthew Conroy (Math)

Graduate Mentor: Gabriel Dorfsman-Hopkins and Nikolas Eptaminitakis

Team Members: Hannah Van Wyk, Emily Flanagan and Penny Espinoza

 

Conditional path sampling of metastable states

Project Mentor: Panos Stinis (AMath)

Graduate Mentor: Jacob Price (AMath)

Team Members: Jesse Rivera and Landon Shorack