Research: Number Theory

Summer School 2018: Algorithms for Extensions of Large Degree

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UNCG Summer School in Computational Number Theory 2018

From May 28 to June 1, 2018, the University of North Carolina at
Greensboro will host the UNCG Summer School in Computational Number
Theory: Algorithms for Extensions of Large Degree.

Algorithms for extensions of global and local fields are the backbone of computational number theory.
Improvements in computing power have made it feasible to conduct computations in larger and larger degree.
As the complexity of algorithms for field extensions depends on the degree of the extensions,
this has increased the interest in asymptotically fast algorithms.
Among others we will consider algorithms for integral bases computation and ideal arithmetic.

Speakers

Application

We expect to be able to provide support for approximately ten graduate students (travel, dorm accomodation, food).
Interested students should complete the online
application
and have a letter of reference sent via email to batanged@uncg.edu by
March 22, 2018. We will respond with funding decisions by April 13, 2018.

Schedule

All talks will take place in room 213 in the Petty Building (campus
maps and directions
). The problem sessions will take place in Petty 213 and the computer lab in Petty 211.

Time Sunday 5/27 Monday 5/28 Tuesday 5/29 Wednesday 5/30 Thursday 5/31 Friday 6/1
9:00 Arrival Welcome Coffee
9:30 Participants: Introductions
slides (pdf)
Peter Bruin: Curves and function fields
Notes: Curves, Function Fields, Picard Groups I, II, and III (pdf)
Jordi Guàrdia: MacLane valuations vs Okutsu frames Claus Fieker: Some hope Peter Bruin: Algorithms for curves and Picard groups
Cocalc Worksheet
10:45 Coffee
11:15 Claus Fieker: The curse of the large degree

Classgroup Computations in Large Degree (pdf)
Claus Fieker: The power of asymptotics Peter Bruin: Differentials, \(L(X,D)\), Riemann-Roch Jordi Guàrdia: Montes algorithm and applications Jordi Guàrdia: Montes algorithm examples
12:30 Lunch
14:00 Jordi Guàrdia: The origins: Kummer, Dedekind, Ore Problem Session Excursion Problem Session
15:30 Problem Session I (Claus (pdf),
Jordi (pdf))
17:00 Student Reports Student Reports
Old Town Draught House Hillbilly Hideaway Grashoppers


group photo 2018

Materials

For notes and problems please see the schedule. The following are software package, background reading, and references ordered by speaker.

  • Peter Bruin:
  • Claus Fieker:
    • Hecke, a software package for algebraic number theory maintained by Claus Fieker and Tommy Hofmann written
      in the Julia programming language
      and based on the
      computer algebra package
      Nemo.
  • Jordi Guàrdia:
    • +Ideals, a package for ideal arithmetic in number fields for Magma

Participants

  1. Vishal Arul (MIT)
  2. Chad Awtrey (Elon University)
  3. Alex Best (Boston University)
  4. Peter Bruin (Universiteit Leiden)
  5. Benjamin Carrillo (Arizona State University)
  6. Lance Everhart (UNCG)
  7. Endrit Fejzullahu (University of Florida)
  8. Claus Fieker (Technische Universität Kaiserslautern)
  9. Nathan Fontes (UNCG)
  10. Richard Gottesman (University of California, Santa Cruz)
  11. Jordi Guàrdia (Universitat Politècnica de Catalunya)
  12. Cole Love (UNCG)
  13. Jonathan Milstead (UNCG)
  14. Michael Musty (Dartmouth College)
  15. Sebastian Pauli (UNCG)
  16. James Rudzinski (UNCG)
  17. Sandi Rudzinski (UNCG)
  18. Filip Saidak (UNCG)
  19. Bernd Sing (University of the West Indies)
  20. Carlo Sircana (Technische Universität Kaiserslautern)
  21. Brett Tangedal (UNCG)
  22. Shuai Wei (Clemson University)
  23. Dan Yasaki (UNCG)
  24. William Youmans (University of South Florida)
  25. Anthoula Zervou (University of Paderborn)
  26. Dena Zhu (Duke University)

Organizers

Acknowledgements

The summer school in computational number theory is supported by UNCG
and the NSF (DMS-1602025).