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 BETWEEN COMBINATORICS, GEOMETRY, AND DATA SCIENCE: THEORY AND COMPUTATION, KNOT THEORY (THIS GROUP), NUMBER THEORY/PROBABILITY, AND THE STEPPING STONE PUZZLE AND RELATED PROBLEMS; 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. DESCRIPTION: In 1978, Bill Thurston showed that knots fall into three categories: torus knots, satellite knots and hyperbolic knots. Hyperbolic knots have a hyperbolic volume associated with them that can be used to distinguish between them. In recent work, SMALL students and I have been exploring ways to determine volumes (see https://arxiv.org/pdf/2111.06319.pdf) and have extended the idea of hyperbolicity to virtual knots (which are an extension of knots analogous of the extension of the real numbers to the complex numbers) (see https://arxiv.org/pdf/1904.06385.pdf and https://arxiv.org/pdf/2110.09859.pdf) and also to the torus and satellite knots (see https://arxiv.org/pdf/1912.09435.pdf). We will further explore these extensions and attempt to extend even further to include various other situations such as knotoids (open knots) and multi-crossing knots. Applications are due Wednesday, February 2nd 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 |