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Continuous Representations: What goes right and what goes wrong? Supplementary Slides Job Rock Department of Mathematics Brandeis University Joint with Kiyoshi Igusa and Gordana Todorov Representation Theory and Related Topics Seminar 3 May

  1. Continuous Representations: What goes right and what goes wrong? Supplementary Slides Job Rock Department of Mathematics Brandeis University Joint with Kiyoshi Igusa and Gordana Todorov Representation Theory and Related Topics Seminar 3 May 2019 Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

  2. Topics 1 Continuous Clusters 2 Continuous Mutation Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

  3. Topics 1 Continuous Clusters 2 Continuous Mutation Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

  4. Examples Example Let a < b ∈ R and T = { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { [ x , a ) , { x } : −∞ < x < a } ∪ { ( a , x ] , { x } : a < x < b } ∪ { ( b , x ] , { x } : b < x } . T is a continuous cluster with three continuous fountains . Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 1 / 9

  5. Examples Example Let a < b ∈ R and T = { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { [ x , a ) , { x } : −∞ < x < a } ∪ { ( a , x ] , { x } : a < x < b } ∪ { ( b , x ] , { x } : b < x } . T is a continuous cluster with three continuous fountains . We can visualize T as a set of noncrossing partitions of R : Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 1 / 9

  6. Structure The types of compatible sets that make up a continuous cluster are: • Discrete • Nests • Continuous fountains (a type of nest) • Antifountains (a type of nest) • Simples Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 2 / 9

  7. Structure The types of compatible sets that make up a continuous cluster are: • Discrete • Nests • Continuous fountains (a type of nest) • Antifountains (a type of nest) • Simples Example Discrete Antifountain Nest Continuous Fountain Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 2 / 9

  8. Structure Example Here is another picture of a continuous cluster. Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 3 / 9

  9. Structure Example Here is another picture of a continuous cluster. Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 3 / 9

  10. Topics 1 Continuous Clusters 2 Continuous Mutation Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 3 / 9

  11. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  12. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  13. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  14. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  15. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  16. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  17. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  18. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  19. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  20. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  21. Examples Example (Continuous Flip) Let a , b ∈ R such that a < 0 < b . Let T be { ( −∞ , + ∞ ) , ( −∞ , a ) , ( a , b ) , ( −∞ , b ) , ( b , + ∞ ) } ∪ { ( −∞ , x ] , { x } , ( a , y ] , { y } , ( b , z ] , { z }} for all x < a < y < b < z . We’re going to “flip” this blue continuous fountain at a to a purple continuous fountain at b . µ ( a , y ] �→ [ y , b ) and f ( a , y ] = b − y b − a = g [ y , b ) Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 4 / 9

  22. Examples Example (Lamination) Here is a picture: µ t = h ( x ) ( −∞ x ) { x } Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 5 / 9

  23. Examples Example (Transfinite Mutation) · · · · · · -3 -2 -1 0 1 2 3 Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

  24. Examples Example (Transfinite Mutation) · · · · · · -3 -2 -1 0 1 2 3 Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

  25. Examples Example (Transfinite Mutation) · · · · · · -3 -2 -1 0 1 2 3 Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

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