#### Event Title

Banach-Mazur Games and Determinacy

#### Location

Diamond 323

#### Start Date

1-5-2014 10:00 AM

#### End Date

1-5-2014 11:00 AM

#### Project Type

Presentation- Restricted to Campus Access

#### Description

A Banach-Mazur Interval game is a contest between two players that pick successive compact subintervals of the real line, together defining a sequence of nested compact intervals. Beforehand, player one is given a winning subset of the real line, and player two is given the complement. The object of the game for player one (player two) is to steer the sequence of intervals towards (or away from) the winning set. The game is decided by whether or not the infinite intersection of the sequence of intervals has an element of the winning set or not. Two questions that immediately spring to mind are the following: (i) For which winning sets can player one or player two always win? (ii) Are there winning sets where neither player one or two can always win? I will answer these questions, and in the end, generalize the notion of a Banach-Mazur game to spaces different from the real numbers. This leads to more interesting questions of determinacy which I hope to illustrate for the audience.

#### Faculty Sponsor

Leo Livshits

#### Sponsoring Department

Colby College. Mathematics and Statistics Dept.

#### CLAS Field of Study

Natural Sciences

#### Event Website

http://www.colby.edu/clas

#### ID

325

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Banach-Mazur Games and Determinacy

Diamond 323

A Banach-Mazur Interval game is a contest between two players that pick successive compact subintervals of the real line, together defining a sequence of nested compact intervals. Beforehand, player one is given a winning subset of the real line, and player two is given the complement. The object of the game for player one (player two) is to steer the sequence of intervals towards (or away from) the winning set. The game is decided by whether or not the infinite intersection of the sequence of intervals has an element of the winning set or not. Two questions that immediately spring to mind are the following: (i) For which winning sets can player one or player two always win? (ii) Are there winning sets where neither player one or two can always win? I will answer these questions, and in the end, generalize the notion of a Banach-Mazur game to spaces different from the real numbers. This leads to more interesting questions of determinacy which I hope to illustrate for the audience.

https://digitalcommons.colby.edu/clas/2014/program/375