for this workshop
Nonstandard methods in combinatorial number theory
American Institute of Mathematics, San Jose, California
Mauro Di Nasso, Isaac Goldbring, and Martino Lupini
This workshop, sponsored by AIM and the NSF, hopes to further develop the use of nonstandard methods in combinatorial number theory and Ramsey theory. For example, recently nonstandard methods have proven useful in problems about configurations of sumsets in sets of positive density as well as partition regularity of equations. Our aim is to continue to explore these and other directions leading to a greater understanding of the role that nonstandard methods could play in this area of combinatorics. Examples of specific questions include:
- Are there specific ways in which nonstandard models can provide further insight into the investigation of the dichotomy between structure and randomness prevalent in many of the arguments in this area, e.g. in the proof of Furstenberg's multiple recurrence theorem?
- Which sumsets configurations can be found in sets of positive density?
- Could such methods be another building block in trying to extend powerful theorems such as Szemeredi's theorem further into the realm of sets with zero density?
- How can iterated nonstandard extensions continue to be used to prove partition regularity of different classes of equations?
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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