Continuous Representations: What goes right and what goes wrong? - - PowerPoint PPT Presentation

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continuous representations what goes right and what goes

Continuous Representations: What goes right and what goes wrong? - - PowerPoint PPT Presentation

Feb 27, 2023 35 likes •327 views

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

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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 2019

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

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Topics

1 Continuous Clusters 2 Continuous Mutation

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

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Topics

1 Continuous Clusters 2 Continuous Mutation

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 0 / 9

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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

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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

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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

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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 Continuous Fountain Antifountain Nest

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 2 / 9

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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

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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

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Topics

1 Continuous Clusters 2 Continuous Mutation

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 3 / 9

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 6 / 9

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 7 / 9

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Examples

Example (Transfinite Mutation)

3
2
1

1 2 3 · · · · · ·

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 8 / 9

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The End

Thank you!

Job Rock (w/ Igusa and Todorov) Continuous Representations at Northeastern 03.05.2019 9 / 9