# Research: Number Theory

## Summer School 2020: Ergodic Theory with Applications to Continued Fractions

2020 2019 2018 2017 2016 2015 2014 2013 2012

From May 18 to May 22 2020 the University of North Carolina Greensboro will host the UNCG Summer School in Computational Number Theory and Algebra: **Ergodic Theory with Applications to Continued Fractions**. The summer school will take place in a live online format. The hours of the school will be during the daytimes of the Eastern Daylight Time Zone (EDT, i.e. UTC -4h, see schedule below).

Ergodic theory is the study of the long term behavior of points and sets under iterations of a measure preserving transformation. What better way to learn the subject than see it in action through its applications to number theory ?

This program is targeted primarily at early stage graduate students in mathematics with an interest in dynamical systems and number theory.

Questions? Email us at uncg2020summerschool@gmail.com

### Speakers

- Daniel Glasscock, UMass Lowell
- Claire Merriman, Ohio State University
- Donald Robertson, University of Manchester
- Clifford Smyth, UNCG

### Lecture Notes

### Current Schedule

The hours of the school will be during the daytimes of the Eastern Daylight Time Zone (EDT, i.e. UTC -4h).

Monday, 5/18

09:00 Introductions

09:30 Lecture I

10:00 Problem Session I

11:30 Discussion Session I

12:00 End of session

13:00 Lecture II

13:30 Problem Session II

15:00 Discussion Session II

15:30 End of day

Tuesday, 5/19

09:00 Lecture III

09:30 Problem Session III

11:00 Discussion Session III

11:30 End of session

13:00 Lecture IV

13:30 Problem Session IV

15:00 Discussion Session IV

15:30 End of day

Wednesday, 5/20

09:00 Lecture V

09:30 Problem Session V

11:00 Discussion Session V

11:30 End of session

13:00 Lecture VI

13:30 Problem Session VI

15:00 Discussion Session VI

15:30 End of day

16:00 Social session

17:00 End social session

Thursday, 5/21

09:00 Lecture VII

09:30 Problem Session VII

11:00 Discussion Session VII

11:30 End of session

13:00 Lecture VIII

13:30 Problem Session VIII

15:00 Discussion Session VIII

15:30 Talk: Sideways Dynamics, by Karl Petersen, UNC Chapel Hill, retired

Friday, 5/22

09:00 Lecture IX

09:30 Problem Session IX

11:00 Discussion Session IX

11:30 End of session

13:00 Lecture X

13:30 Problem Session X

15:00 Discussion Session X

15:30 Closing remarks

16:00 End of summer school

Non-participants are welcome to listen and participate in lectures, but not in the breakout/problem sessions. The breakout sessions are deliberately kept small in order to be manageable.

### Topical Outline

After the first 7 topics have been covered the participants will split up into three groups

in each of which one of the last three topics will be tackled.

- Basic definitions, examples, and heuristics of ergodic theory
- Irrational rotations of the torus (as the first example connection between ergodic theory and number theory)
- Base-2 digit statistics (as the second example connection between ergodic theory and number theory)
- Basics of ergodic theory
- Basics of finite continued fractions
- Infinite continued fractions as formal objects and as real numbers
- Basics of the Gauss map
- Digit statistics (applications of the ergodic theorem)
- Basics of diophantine approximation
- Continued fraction convergents in Diophantine approximation

### Organizers

Talia Fernos, Sebastian Pauli, Filip Saidak, Clifford Smyth, Brett Tangedal, Dan Yasaki

### Acknowledgements

The summer school in computational number theory is supported by UNCG and the NSF.