Williams College, Mathematics and Statistics

Program ID: SMALLREU-SMALL2021CONNECTIONSTHROUGHDIOPHANTINEEQ [#1017]
Program Title: Williams College SMALL 2021 REU CONNECTIONS THROUGH DIOPHANTINE EQUATIONS
Program Type: Undergraduate program
Program Location: Williamstown, Massachusetts 01267, United States [map]
Subject Area: Mathematics
Application Deadline: 2021/02/03 11:59PMhelp popup (posted 2020/10/20, listed until 2021/04/20)
Program Description:    

The SMALL Undergraduate Research Project is a nine-week residential summer program in which undergraduates investigate open research problems in mathematics. One of the largest programs of its kind in the country, SMALL is supported in part by a National Science Foundation grant for Research Experiences for Undergraduates and by the Science Center of Williams College. Around 500 students have participated in the project since its inception in 1988. In the next few weeks we will have a listing of what professors are participating and what their groups will be; when you apply you will need to rank your preferences.

Students work in small groups directed by individual faculty members. Many participants have published papers and presented talks at research conferences based on work done in SMALL. Some have gone on to complete PhD’s in Mathematics. During off hours, students enjoy the many attractions of summer in the Berkshires: hiking, biking, plays, concerts, etc. Weekly lunches, teas, and casual sporting events bring SMALL students together with faculty and other students spending the summer doing research at Williams College.

THE SMALL PROGRAM WILL HAVE SEVERAL GROUPS THIS SUMMER: CONNECTIONS THROUGH DIOPHANTINE EQUATIONS (THIS GROUP), NUMBER THEORY/PROBABILITY, CHIP FIRING GAMES, AND OTHERS TO BE DETERMINED; HOWEVER, PLEASE APPLY JUST TO YOUR TOP CHOICE. DO NOT APPLY TO THE MAIN PROGRAM, DO NOT APPLY TO MULTIPLE GROUPS; IF YOU DO EITHER YOUR APPLICATION WILL NOT BE READ. WE ARE HAVING YOU APPLY TO YOUR TOP CHOICE TO FACILITATE ADMINISTRATIVE TASKS. IF YOU HAVE ANY QUESTIONS EMAIL THE DIRECTOR AT smalldirector@williams.edu. WHEN YOU FILL OUT THE ADDITIONAL FORM LISTED BELOW, YOU CAN PUT YOUR OTHER CHOICES. DEADLINE IS WEDNESDAY FEBRUARY 3rd, 5pm US EASTERN.

Title: Connections through Diophantine Equations

Advisor: Eva Goedhart (https://www.evagoedhartphd.com/)

Project Description: Number Theory, Algebra, Combinatorics, and Analysis all get used to solve Diophantine equations, polynomial equations with integer coefficients. A focus of study for hundreds of years, the study of Diophantine equations remains a vibrant area of research. It has yielded a multitude of beautiful results and has wide ranging applications in other areas of mathematics, in cryptography, and in the natural sciences.

Through research in Diophantine equations, we see many connections between mathematical fields. This project strives to make those connections as well as establish new collaborations between people. This includes the people in the groups as well as other Diophantine equations researchers from around the world.

Projects in Diophantine equations range from finding all solutions to Lebesgue-Naggell type Diophantine equations, using the existence of primitive divisors of Lucas and Lehmer numbers, to solving exponential Diophantine equations, using bounds on inequalities and continued fractions. See the references for a few examples.

Given local and global uncertainties of these times, if this project runs it will be held virtually.

References:

[1] E. Goedhart and H. G. Grundman, “On the Diophantine equation NX^2 + 2^L 3^M = Y^N ”, J. Number Theory 141 (2014), 214-224. (https://arxiv.org/abs/1304.6413)

[2] E. Goedhart and H. G. Grundman, “Diophantine approximation and the equation (a^2 cx^k − 1)(b^2 cy^k − 1) = (abcz^k − 1)^2”, J. Number Theory, 154 (2015), 74-81. (https://arxiv.org/abs/1411.1984)

[3] E. Goedhart and H. G. Grundman, “On the Diophantine equation X^(2N) +2^(2α) 5^(2β) Y^(2γ) = Z^5”, Period. Math. Hung. 75 (2017), no. 2, 196–200. (https://arxiv.org/abs/1409.2463)

Applications are due Wednesday, February 3rd by 5pm Eastern. Please remember to fill out and upload the additional requested information, which is available as a word file at https://math.williams.edu/files/2019/10/SMALLApplicationDocGeneral.doc and as a pdf at https://math.williams.edu/files/2019/10/SMALLApplicationDocGeneral.pdf


Application Materials Required:
Submit the following items online at this website to complete your application:
And anything else requested in the program description.

Further Info:
http://math.williams.edu/small/
413-597-3293
 
Department of Math/Stats
Williams College
33 Stetson Court
Williamstown, MA 01267

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